(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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"Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] Chapter 4(1) :[font = section; inactive; Cclosed; preserveAspect; startGroup] Problem 3 :[font = text; inactive; preserveAspect] For each of the functions given the partial sums of the Fourier series representation (which are derived in the solution manual, but of course can also be obtained from Mathematica) are plotted with the function for an increasing number of terms in the partial sum. :[font = subsection; inactive; preserveAspect; startGroup] Problem 3(a) :[font = input; preserveAspect; startGroup] f[x]=x(Pi-x); b[n_]=(2/Pi)Integrate[f[x]Sin[n x],{x,0,Pi}] :[font = output; output; inactive; preserveAspect; endGroup] (2*(2/n^3 - (2*Cos[n*Pi])/n^3 - (Pi*Sin[n*Pi])/n^2))/ Pi ;[o] 2 2 Cos[n Pi] Pi Sin[n Pi] 2 (-- - ----------- - ------------) 3 3 2 n n n ----------------------------------- Pi :[font = text; inactive; preserveAspect] The values of the first few coefficients are :[font = input; preserveAspect; startGroup] b[{1,2,3,4,5,6,7,8,9,10}] :[font = output; output; inactive; preserveAspect; endGroup] {8/Pi, 0, 8/(27*Pi), 0, 8/(125*Pi), 0, 8/(343*Pi), 0, 8/(729*Pi), 0} ;[o] 8 8 8 8 8 {--, 0, -----, 0, ------, 0, ------, 0, ------, 0} Pi 27 Pi 125 Pi 343 Pi 729 Pi :[font = text; inactive; preserveAspect] which shows that the b[n] vanish for even values of n. :[font = text; inactive; preserveAspect] The terms of the Fourier series are then generated as a list and then Apply is used to sum the terms of the list. :[font = input; preserveAspect] Clear[FCoefficients,PartialSum] FCoefficients[x_,n_]:=Table[b[k]Sin[k x],{k,1,n}]; PartialSum[x_,n_]:=Apply[Plus,FCoefficients[x,n]] :[font = text; inactive; preserveAspect] The function x(Pi-x) is now plotted with the partial sums. :[font = input; preserveAspect; startGroup] Plot[{x(Pi-x),PartialSum[x,1]},{x,0,Pi}, PlotStyle->{{Thickness[0.005]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=x(Pi-x), n=1", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.231144 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=x\(Pi-x\), n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.5)] .01131 .13029 1 0 Msboxa [(1)] .01131 .24586 1 0 Msboxa [(1.5)] .01131 .36143 1 0 Msboxa [(2)] .01131 .477 1 0 Msboxa [(2.5)] .01131 .59258 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 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Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Plot[{x(Pi-x),PartialSum[x,3]},{x,0,Pi}, PlotStyle->{{Thickness[0.005]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=x(Pi-x), n=3", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.238552 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=x\(Pi-x\), n=3)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.5)] .01131 .13399 1 0 Msboxa [(1)] .01131 .25327 1 0 Msboxa [(1.5)] .01131 .37254 1 0 Msboxa [(2)] .01131 .49182 1 0 Msboxa [(2.5)] .01131 .6111 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.5)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(1)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(1.5)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(2)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(2.5)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(3)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(f[x]=x\(Pi-x\), n=3)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .02381 .13399 m .03006 .13399 L s P [(0.5)] .01131 .13399 1 0 Mshowa p .002 w .02381 .25327 m .03006 .25327 L s P [(1)] .01131 .25327 1 0 Mshowa p .002 w .02381 .37254 m .03006 .37254 L s P [(1.5)] .01131 .37254 1 0 Mshowa p .002 w .02381 .49182 m .03006 .49182 L s P [(2)] .01131 .49182 1 0 Mshowa p .002 w .02381 .6111 m .03006 .6111 L s P [(2.5)] .01131 .6111 1 0 Mshowa p .001 w .02381 .03857 m .02756 .03857 L s P p .001 w .02381 .06243 m .02756 .06243 L s P p .001 w .02381 .08628 m .02756 .08628 L s P p .001 w .02381 .11014 m .02756 .11014 L s P p .001 w .02381 .15785 m .02756 .15785 L s P p .001 w .02381 .1817 m .02756 .1817 L s P p .001 w .02381 .20556 m .02756 .20556 L s P p .001 w .02381 .22941 m .02756 .22941 L s P p .001 w .02381 .27712 m .02756 .27712 L s P p .001 w .02381 .30098 m .02756 .30098 L s P p .001 w .02381 .32483 m .02756 .32483 L 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.59833 L .52976 .59773 L .53968 .5962 L .5496 .59423 L .55952 .5918 L .57937 .58558 L .59921 .57745 L .61905 .56733 L .65873 .5408 L .69841 .50526 L .7381 .46017 L Mistroke .77778 .4053 L .81746 .34095 L .85714 .26797 L .89683 .18785 L .93651 .10262 L .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = text; inactive; preserveAspect; endGroup] We see that even the first term in the series reproduces the function quite well over the entire range (0,Pi), and that the partial sum is almost indistinguishable from the original function. This is due to the smoothness of the odd periodic extension of the function. :[font = subsection; inactive; preserveAspect; startGroup] Problem 3(b) :[font = input; preserveAspect] Clear[f,b,FCoefficients,PartialSum] f[x]=x; b[n_]=(2/Pi)Integrate[f[x]Sin[n x],{x,0,Pi}]; FCoefficients[x_,n_]:=Table[b[k]Sin[k x],{k,1,n}]; PartialSum[x_,n_]:=Apply[Plus,FCoefficients[x,n]] :[font = text; inactive; preserveAspect] The thicker line is used below for the function and the lighter line for the partial sums. :[font = input; preserveAspect; startGroup] Plot[{x,PartialSum[x,1]},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=x, n=1", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.187358 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=x, n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.5)] .01131 .10839 1 0 Msboxa [(1)] .01131 .20207 1 0 Msboxa [(1.5)] .01131 .29575 1 0 Msboxa [(2)] .01131 .38943 1 0 Msboxa [(2.5)] .01131 .48311 1 0 Msboxa [(3)] .01131 .57679 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.5)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(1)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(1.5)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(2)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(2.5)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(3)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(f[x]=x, n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .02381 .10839 m .03006 .10839 L s P [(0.5)] .01131 .10839 1 0 Mshowa p .002 w .02381 .20207 m .03006 .20207 L s P [(1)] .01131 .20207 1 0 Mshowa p .002 w .02381 .29575 m .03006 .29575 L s P [(1.5)] .01131 .29575 1 0 Mshowa p .002 w .02381 .38943 m .03006 .38943 L s P [(2)] .01131 .38943 1 0 Mshowa p .002 w .02381 .48311 m .03006 .48311 L s P [(2.5)] .01131 .48311 1 0 Mshowa p .002 w .02381 .57679 m .03006 .57679 L s P [(3)] .01131 .57679 1 0 Mshowa p .001 w .02381 .03345 m .02756 .03345 L s P p .001 w .02381 .05219 m .02756 .05219 L s P p .001 w .02381 .07092 m .02756 .07092 L s P p .001 w .02381 .08966 m .02756 .08966 L s P p .001 w .02381 .12713 m .02756 .12713 L s P p .001 w .02381 .14587 m .02756 .14587 L s P p .001 w .02381 .1646 m .02756 .1646 L s P p .001 w .02381 .18334 m .02756 .18334 L s P p .001 w .02381 .22081 m .02756 .22081 L s P p .001 w .02381 .23955 m .02756 .23955 L s P p .001 w .02381 .25828 m .02756 .25828 L s P p .001 w .02381 .27702 m .02756 .27702 L s P p .001 w .02381 .31449 m .02756 .31449 L s P p .001 w .02381 .33322 m .02756 .33322 L s P p .001 w .02381 .35196 m .02756 .35196 L s P p .001 w .02381 .3707 m .02756 .3707 L s P p .001 w .02381 .40817 m .02756 .40817 L s P p .001 w .02381 .4269 m .02756 .4269 L s P p .001 w .02381 .44564 m .02756 .44564 L s P p .001 w .02381 .46438 m .02756 .46438 L s P p .001 w .02381 .50185 m .02756 .50185 L s P p .001 w .02381 .52058 m .02756 .52058 L s P p .001 w .02381 .53932 m .02756 .53932 L s P p .001 w .02381 .55805 m .02756 .55805 L s P p .001 w .02381 .59553 m .02756 .59553 L s P p .001 w .02381 .61426 m .02756 .61426 L s P [(f)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P p p p .0075 w .02381 .01472 m .06349 .03924 L .10317 .06377 L .14286 .08829 L .18254 .11282 L .22222 .13734 L .2619 .16187 L .30159 .18639 L .34127 .21092 L .38095 .23544 L .42063 .25997 L .46032 .28449 L .5 .30902 L .53968 .33354 L .57937 .35807 L .61905 .38259 L .65873 .40712 L .69841 .43164 L .7381 .45617 L .77778 .48069 L .81746 .50522 L .85714 .52974 L .89683 .55427 L .93651 .57879 L .97619 .60332 L s P P p p .005 w .02381 .01472 m .06349 .06363 L .10317 .1117 L .14286 .15811 L .18254 .20207 L .22222 .24283 L .2619 .27968 L .30159 .312 L .34127 .33923 L .38095 .36091 L .40079 .36955 L .42063 .37666 L .44048 .38223 L .4504 .38443 L .46032 .38623 L .47024 .38763 L .4752 .38818 L .48016 .38863 L .48512 .38898 L .4876 .38912 L .49008 .38923 L .49256 .38932 L .4938 .38935 L .49504 .38938 L .49628 .3894 L .49752 .38942 L .49876 .38943 L .5 .38943 L .50124 .38943 L .50248 .38942 L .50372 .3894 L .50496 .38938 L .5062 .38935 L .50744 .38932 L .50992 .38923 L .5124 .38912 L .51488 .38898 L .51984 .38863 L .5248 .38818 L .52976 .38763 L .53968 .38623 L .5496 .38443 L .55952 .38223 L .57937 .37666 L .61905 .36091 L .65873 .33923 L .69841 .312 L .7381 .27968 L .77778 .24283 L .81746 .20207 L Mistroke .85714 .15811 L .89683 .1117 L .93651 .06363 L .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Plot[{x,PartialSum[x,5]},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=x, n=5", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.185931 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=x, n=5)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.5)] .01131 .10768 1 0 Msboxa [(1)] .01131 .20065 1 0 Msboxa [(1.5)] .01131 .29361 1 0 Msboxa [(2)] .01131 .38658 1 0 Msboxa [(2.5)] .01131 .47954 1 0 Msboxa [(3)] .01131 .57251 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.5)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(1)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(1.5)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(2)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(2.5)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(3)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(f[x]=x, n=5)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .02381 .10768 m .03006 .10768 L s P [(0.5)] .01131 .10768 1 0 Mshowa p .002 w .02381 .20065 m .03006 .20065 L s P [(1)] .01131 .20065 1 0 Mshowa p .002 w .02381 .29361 m .03006 .29361 L s P [(1.5)] .01131 .29361 1 0 Mshowa p .002 w .02381 .38658 m .03006 .38658 L s P [(2)] .01131 .38658 1 0 Mshowa p .002 w .02381 .47954 m .03006 .47954 L s P [(2.5)] .01131 .47954 1 0 Mshowa p .002 w .02381 .57251 m .03006 .57251 L s P [(3)] .01131 .57251 1 0 Mshowa p .001 w .02381 .03331 m .02756 .03331 L s P p .001 w .02381 .0519 m .02756 .0519 L s P p .001 w .02381 .07049 m .02756 .07049 L s P p .001 w .02381 .08909 m .02756 .08909 L s P p .001 w .02381 .12627 m .02756 .12627 L s P p .001 w .02381 .14487 m .02756 .14487 L s P p .001 w .02381 .16346 m .02756 .16346 L s P p .001 w .02381 .18205 m .02756 .18205 L s P p .001 w .02381 .21924 m .02756 .21924 L s P p .001 w .02381 .23783 m .02756 .23783 L s P p .001 w .02381 .25643 m .02756 .25643 L s P p .001 w .02381 .27502 m .02756 .27502 L s P p .001 w .02381 .31221 m .02756 .31221 L s P p .001 w .02381 .3308 m .02756 .3308 L s P p .001 w .02381 .34939 m .02756 .34939 L s P p .001 w .02381 .36798 m .02756 .36798 L s P p .001 w .02381 .40517 m .02756 .40517 L s P p .001 w .02381 .42376 m .02756 .42376 L s P p .001 w .02381 .44236 m .02756 .44236 L s P p .001 w .02381 .46095 m .02756 .46095 L s P p .001 w .02381 .49814 m .02756 .49814 L s P p .001 w .02381 .51673 m .02756 .51673 L s P p .001 w .02381 .53532 m .02756 .53532 L s P p .001 w .02381 .55392 m .02756 .55392 L s P p .001 w .02381 .5911 m .02756 .5911 L s P p .001 w .02381 .6097 m .02756 .6097 L s P [(f)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P p p p .0075 w .02381 .01472 m .06349 .03905 L .10317 .06339 L .14286 .08773 L .18254 .11207 L .22222 .13641 L .2619 .16075 L .30159 .18508 L .34127 .20942 L .38095 .23376 L .42063 .2581 L .46032 .28244 L .5 .30678 L .53968 .33111 L .57937 .35545 L .61905 .37979 L .65873 .40413 L .69841 .42847 L .7381 .45281 L .77778 .47714 L .81746 .50148 L .85714 .52582 L .89683 .55016 L .93651 .5745 L .97619 .59884 L s P P p p .005 w .02381 .01472 m .06349 .06136 L .08333 .08108 L .10317 .09697 L .12302 .10854 L .13294 .11269 L .1379 .11437 L .14286 .11581 L .15278 .118 L .15774 .11878 L .1627 .11937 L .16518 .1196 L .16766 .11979 L .17014 .11994 L .17262 .12006 L .17386 .12011 L .1751 .12015 L .17634 .12018 L .17758 .12021 L .17882 .12023 L .18006 .12024 L .1813 .12025 L .18254 .12026 L .18378 .12025 L .18502 .12025 L .18626 .12023 L .1875 .12022 L .19246 .12013 L .19742 .12001 L .20238 .11988 L .20486 .11982 L .20734 .11977 L .20982 .11973 L .21106 .11972 L .2123 .11971 L .21354 .1197 L .21478 .1197 L .21602 .11971 L .21726 .11972 L .2185 .11973 L .21974 .11976 L .22098 .11979 L .22222 .11982 L .2247 .11992 L .22718 .12005 L .22966 .12021 L .23214 .12042 L .2371 .12095 L .24206 .12167 L .24702 .1226 L Mistroke .25198 .12375 L .2619 .12679 L .27183 .13091 L .28175 .13618 L .30159 .15032 L .32143 .1691 L .34127 .19184 L .38095 .24387 L .40079 .26972 L .42063 .29305 L .44048 .31227 L .4504 .31996 L .46032 .32624 L .46528 .32884 L .47024 .33106 L .4752 .33293 L .48016 .33443 L .48264 .33504 L .48512 .33557 L .4876 .33602 L .49008 .33637 L .49256 .33665 L .4938 .33676 L .49504 .33684 L .49628 .33691 L .49752 .33696 L .49876 .33699 L .5 .337 L .50124 .33699 L .50248 .33696 L .50372 .33691 L .50496 .33685 L .5062 .33677 L .50744 .33667 L .50992 .33643 L .51488 .33575 L .51984 .33484 L .52976 .33246 L .53968 .32952 L .55952 .3231 L .56944 .32022 L .5744 .31899 L .57937 .31796 L .58433 .31716 L .58681 .31686 L .58805 .31673 L .58929 .31663 L .59053 .31654 L .59177 .31647 L .59301 .31643 L Mistroke .59425 .3164 L .59549 .31639 L .59673 .31641 L .59797 .31644 L .59921 .3165 L .60045 .31659 L .60169 .31669 L .60417 .31698 L .60665 .31736 L .60913 .31785 L .61409 .31914 L .61905 .32088 L .62897 .3258 L .63889 .33272 L .64881 .34174 L .65873 .35286 L .67857 .38119 L .69841 .41655 L .7381 .49865 L .75794 .53847 L .76786 .55629 L .77778 .57198 L .7877 .58502 L .79266 .59039 L .79762 .59491 L .80258 .59851 L .80754 .60115 L .81002 .60209 L .81126 .60246 L .8125 .60277 L .81374 .60301 L .81498 .60318 L .81622 .60328 L .81746 .60332 L .8187 .60328 L .81994 .60318 L .82118 .603 L .82242 .60275 L .8249 .60204 L .82738 .60103 L .82986 .59973 L .83234 .59812 L .8373 .59398 L .84226 .58859 L .84722 .58192 L .85714 .56469 L .86706 .54221 L .87698 .51451 L .89683 .44392 L .93651 .25057 L Mistroke .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Plot[{x,PartialSum[x,10]},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=x, n=10", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.172413 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=x, n=10)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.5)] .01131 .10092 1 0 Msboxa [(1)] .01131 .18713 1 0 Msboxa [(1.5)] .01131 .27333 1 0 Msboxa [(2)] .01131 .35954 1 0 Msboxa [(2.5)] .01131 .44575 1 0 Msboxa [(3)] .01131 .53195 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.5)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(1)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(1.5)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(2)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(2.5)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(3)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(f[x]=x, n=10)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .02381 .10092 m .03006 .10092 L s P [(0.5)] .01131 .10092 1 0 Mshowa p .002 w .02381 .18713 m .03006 .18713 L s P [(1)] .01131 .18713 1 0 Mshowa p .002 w .02381 .27333 m .03006 .27333 L s P [(1.5)] .01131 .27333 1 0 Mshowa p .002 w .02381 .35954 m .03006 .35954 L s P [(2)] .01131 .35954 1 0 Mshowa p .002 w .02381 .44575 m .03006 .44575 L s P [(2.5)] .01131 .44575 1 0 Mshowa p .002 w .02381 .53195 m .03006 .53195 L s P [(3)] .01131 .53195 1 0 Mshowa p .001 w .02381 .03196 m .02756 .03196 L s P p .001 w .02381 .0492 m .02756 .0492 L s P p .001 w .02381 .06644 m .02756 .06644 L s P p .001 w .02381 .08368 m .02756 .08368 L s P p .001 w .02381 .11816 m .02756 .11816 L s P p .001 w .02381 .1354 m .02756 .1354 L s P p .001 w .02381 .15265 m .02756 .15265 L s P p .001 w .02381 .16989 m .02756 .16989 L s P p .001 w .02381 .20437 m .02756 .20437 L s P p .001 w .02381 .22161 m .02756 .22161 L s P p .001 w .02381 .23885 m .02756 .23885 L s P p .001 w .02381 .25609 m .02756 .25609 L s P p .001 w .02381 .29058 m .02756 .29058 L s P p .001 w .02381 .30782 m .02756 .30782 L s P p .001 w .02381 .32506 m .02756 .32506 L s P p .001 w .02381 .3423 m .02756 .3423 L s P p .001 w .02381 .37678 m .02756 .37678 L s P p .001 w .02381 .39402 m .02756 .39402 L s P p .001 w .02381 .41126 m .02756 .41126 L s P p .001 w .02381 .42851 m .02756 .42851 L s P p .001 w .02381 .46299 m .02756 .46299 L s P p .001 w .02381 .48023 m .02756 .48023 L s P p .001 w .02381 .49747 m .02756 .49747 L s P p .001 w .02381 .51471 m .02756 .51471 L s P p .001 w .02381 .54919 m .02756 .54919 L s P p .001 w .02381 .56644 m .02756 .56644 L s P p .001 w .02381 .58368 m .02756 .58368 L s P p .001 w .02381 .60092 m .02756 .60092 L s P [(f)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P p p p .0075 w .02381 .01472 m .06349 .03728 L .10317 .05985 L .14286 .08242 L .18254 .10499 L .22222 .12756 L .2619 .15013 L .30159 .1727 L .34127 .19527 L .38095 .21783 L .42063 .2404 L .46032 .26297 L .5 .28554 L .53968 .30811 L .57937 .33068 L .61905 .35325 L .65873 .37582 L .69841 .39838 L .7381 .42095 L .77778 .44352 L .81746 .46609 L .85714 .48866 L .89683 .51123 L .93651 .5338 L .97619 .55637 L s P P p p .005 w .02381 .01472 m .02505 .01472 L .02629 .01472 L .02753 .01472 L .02877 .01473 L .03001 .01474 L .03125 .01476 L .03249 .01479 L .03373 .01483 L .03621 .01493 L .03745 .015 L .03869 .01508 L .04117 .0153 L .04365 .01558 L .04861 .01638 L .05109 .01692 L .05357 .01755 L .05853 .01914 L .06349 .02117 L .07341 .02668 L .08333 .03409 L .10317 .05362 L .12302 .07608 L .13294 .08681 L .14286 .0964 L .15278 .10436 L .15774 .10764 L .1627 .11043 L .16766 .11272 L .17262 .11453 L .1751 .11526 L .17758 .11588 L .18006 .1164 L .18254 .11683 L .18502 .11716 L .1875 .11741 L .18874 .11751 L .18998 .11759 L .19122 .11766 L .19246 .11771 L .1937 .11774 L .19494 .11777 L .19618 .11778 L .19742 .11778 L .19866 .11777 L .1999 .11776 L .20114 .11773 L .20238 .11771 L .20734 .11758 L .20982 .11752 L Mistroke .21106 .1175 L .2123 .11748 L .21354 .11748 L .21478 .11747 L .21602 .11748 L .21726 .1175 L .2185 .11754 L .21974 .11758 L .22098 .11764 L .22222 .11772 L .2247 .11793 L .22718 .11822 L .22966 .11859 L .23214 .11906 L .2371 .12031 L .24206 .12201 L .24702 .12421 L .25198 .12692 L .2619 .13389 L .28175 .15328 L .30159 .17667 L .32143 .19862 L .33135 .20741 L .33631 .21106 L .34127 .21416 L .34623 .2167 L .35119 .21868 L .35367 .21947 L .35615 .22013 L .35863 .22067 L .36111 .22108 L .36235 .22124 L .36359 .22138 L .36483 .22149 L .36607 .22158 L .36731 .22164 L .36855 .22168 L .36979 .22169 L .37103 .22169 L .37227 .22167 L .37351 .22162 L .37599 .22149 L .37723 .2214 L .37847 .2213 L .38095 .22106 L .39087 .21992 L .39583 .2194 L .39831 .21919 L .39955 .21911 L .40079 .21904 L Mistroke .40203 .21899 L .40327 .21896 L .40451 .21894 L .40575 .21895 L .40699 .21898 L .40823 .21903 L .40947 .2191 L .41071 .21921 L .41319 .21949 L .41567 .2199 L .41815 .22043 L .42063 .22109 L .4256 .22283 L .43056 .22517 L .44048 .2317 L .4504 .24063 L .46032 .25165 L .48016 .27742 L .5 .30262 L .50992 .31287 L .51488 .3171 L .51984 .32068 L .5248 .32355 L .52728 .32472 L .52976 .3257 L .53224 .32652 L .53472 .32715 L .53596 .32741 L .5372 .32762 L .53844 .32779 L .53968 .32792 L .54092 .32801 L .54216 .32807 L .5434 .32808 L .54464 .32806 L .54588 .328 L .54712 .32791 L .54836 .32779 L .5496 .32763 L .55952 .32545 L .56448 .3239 L .56944 .32222 L .5744 .32052 L .57937 .31895 L .58433 .31765 L .58681 .31713 L .58805 .31691 L .58929 .31672 L .59053 .31657 L .59177 .31645 L Mistroke .59301 .31636 L .59425 .31631 L .59549 .3163 L .59673 .31632 L .59797 .31639 L .59921 .31651 L .60045 .31666 L .60169 .31687 L .60417 .31741 L .60665 .31815 L .60913 .3191 L .61409 .32161 L .61905 .32499 L .62897 .33432 L .63889 .34683 L .65873 .3784 L .67857 .41111 L .68849 .42465 L .69345 .43019 L .69841 .43477 L .70089 .43667 L .70337 .4383 L .70585 .43965 L .70833 .44072 L .70957 .44115 L .71081 .4415 L .71205 .44179 L .71329 .442 L .71453 .44215 L .71577 .44222 L .71701 .44222 L .71825 .44216 L .71949 .44202 L .72073 .44182 L .72197 .44155 L .72321 .44122 L .72817 .43926 L .73065 .43793 L .73313 .43639 L .7381 .43274 L .75794 .41393 L .76786 .40502 L .77282 .40151 L .7753 .40009 L .77778 .39892 L .77902 .39844 L .78026 .39804 L .7815 .39771 L .78274 .39747 L .78398 .3973 L Mistroke .78522 .39722 L .78646 .39723 L .7877 .39733 L .78894 .39752 L .79018 .39781 L .79266 .39867 L .7939 .39925 L .79514 .39994 L .79762 .40162 L .80258 .40624 L .80754 .41259 L .81746 .43039 L .82738 .45435 L .8373 .48304 L .85714 .54515 L .86706 .57249 L .87202 .58375 L .87698 .59283 L .87946 .59642 L .88194 .59931 L .88318 .60048 L .88442 .60146 L .88566 .60223 L .8869 .60281 L .88814 .60317 L .88938 .60332 L .89062 .60325 L .89187 .60295 L .89311 .60243 L .89435 .60167 L .89559 .60067 L .89683 .59943 L .89931 .59621 L .90179 .59197 L .90675 .58035 L .91171 .56442 L .91667 .54407 L .92659 .49013 L .93651 .41917 L .95635 .23429 L .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Plot[{x,PartialSum[x,50]},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=x, n=50", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.161614 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=x, n=50)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.5)] .01131 .09552 1 0 Msboxa [(1)] .01131 .17633 1 0 Msboxa [(1.5)] .01131 .25714 1 0 Msboxa [(2)] .01131 .33794 1 0 Msboxa [(2.5)] .01131 .41875 1 0 Msboxa [(3)] .01131 .49956 1 0 Msboxa [(3.5)] .01131 .58037 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.5)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(1)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(1.5)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(2)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(2.5)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(3)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(f[x]=x, n=50)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .02381 .09552 m .03006 .09552 L s P [(0.5)] .01131 .09552 1 0 Mshowa p .002 w .02381 .17633 m .03006 .17633 L s P [(1)] .01131 .17633 1 0 Mshowa p .002 w .02381 .25714 m .03006 .25714 L s P [(1.5)] .01131 .25714 1 0 Mshowa p .002 w .02381 .33794 m .03006 .33794 L s P [(2)] .01131 .33794 1 0 Mshowa p .002 w .02381 .41875 m .03006 .41875 L s P [(2.5)] .01131 .41875 1 0 Mshowa p .002 w .02381 .49956 m .03006 .49956 L s P [(3)] .01131 .49956 1 0 Mshowa p .002 w .02381 .58037 m .03006 .58037 L s P [(3.5)] .01131 .58037 1 0 Mshowa p .001 w .02381 .03088 m .02756 .03088 L s P p .001 w .02381 .04704 m .02756 .04704 L s P p .001 w .02381 .0632 m .02756 .0632 L s P p .001 w .02381 .07936 m .02756 .07936 L s P p .001 w .02381 .11168 m .02756 .11168 L s P p .001 w .02381 .12785 m .02756 .12785 L s P p .001 w .02381 .14401 m .02756 .14401 L s P p .001 w .02381 .16017 m .02756 .16017 L s P p .001 w .02381 .19249 m .02756 .19249 L s P p .001 w .02381 .20865 m .02756 .20865 L s P p .001 w .02381 .22481 m .02756 .22481 L s P p .001 w .02381 .24098 m .02756 .24098 L s P p .001 w .02381 .2733 m .02756 .2733 L s P p .001 w .02381 .28946 m .02756 .28946 L s P p .001 w .02381 .30562 m .02756 .30562 L s P p .001 w .02381 .32178 m .02756 .32178 L s P p .001 w .02381 .35411 m .02756 .35411 L s P p .001 w .02381 .37027 m .02756 .37027 L s P p .001 w .02381 .38643 m .02756 .38643 L s P p .001 w .02381 .40259 m .02756 .40259 L s P p .001 w .02381 .43491 m .02756 .43491 L s P p .001 w .02381 .45107 m .02756 .45107 L s P p .001 w .02381 .46724 m .02756 .46724 L s P p .001 w .02381 .4834 m .02756 .4834 L s P p .001 w .02381 .51572 m .02756 .51572 L s P p .001 w .02381 .53188 m .02756 .53188 L s P p .001 w .02381 .54804 m .02756 .54804 L s P p .001 w .02381 .5642 m .02756 .5642 L s P p .001 w .02381 .59653 m .02756 .59653 L s P p .001 w .02381 .61269 m .02756 .61269 L s P [(f)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P p p p .0075 w .02381 .01472 m .06349 .03587 L .10317 .05703 L .14286 .07818 L .18254 .09934 L .22222 .12049 L .2619 .14165 L .30159 .1628 L .34127 .18396 L .38095 .20511 L .42063 .22627 L .46032 .24742 L .5 .26858 L .53968 .28973 L .57937 .31089 L .61905 .33204 L .65873 .3532 L .69841 .37435 L .7381 .39551 L .77778 .41667 L .81746 .43782 L .85714 .45898 L .89683 .48013 L .93651 .50129 L .97619 .52244 L s P P p p .005 w .02381 .01472 m .06349 .03484 L .10317 .05506 L .14286 .07546 L .18254 .09613 L .22222 .11712 L .2619 .13845 L .30159 .16013 L .34127 .18213 L .38095 .20439 L .42063 .22682 L .46032 .24934 L .5 .27181 L .53968 .29411 L .57937 .31611 L .61905 .33767 L .65873 .35868 L .69841 .37901 L .7381 .39852 L .77778 .41703 L .81746 .43419 L .8187 .43246 L .81994 .43095 L .82118 .42976 L .82242 .42898 L .82366 .42866 L .8249 .42886 L .82614 .42962 L .82738 .43092 L .83234 .44097 L .8373 .45473 L .83978 .46076 L .84102 .46317 L .84226 .46506 L .8435 .46637 L .84474 .46705 L .84598 .46711 L .84722 .46655 L .84846 .46541 L .8497 .46376 L .85218 .4593 L .85466 .45409 L .85714 .44924 L .85838 .44728 L .85962 .44581 L .86086 .44491 L .8621 .44467 L .86334 .44515 L .86458 .44636 L .86582 .44831 L Mistroke .86706 .45094 L .87202 .46659 L .8745 .47561 L .87698 .48373 L .87822 .48706 L .87946 .48973 L .8807 .49164 L .88194 .4927 L .88318 .49288 L .88442 .49218 L .88566 .49063 L .8869 .48831 L .88814 .48532 L .88938 .4818 L .89187 .47385 L .89435 .46599 L .89559 .46259 L .89683 .4598 L .89807 .45779 L .89931 .45671 L .90055 .45665 L .90179 .45768 L .90303 .45981 L .90427 .46304 L .90551 .46727 L .90675 .4724 L .91171 .49809 L .91419 .51071 L .91543 .51605 L .91667 .52044 L .91791 .52367 L .91915 .52558 L .92039 .52606 L .92163 .52505 L .92287 .52255 L .92411 .51864 L .92659 .50715 L .92907 .49232 L .93155 .47666 L .93403 .46311 L .93527 .45804 L .93651 .45455 L .93775 .45294 L .93899 .45343 L .94023 .45615 L .94147 .46119 L .94395 .47799 L .94519 .48941 L .94643 .50246 L .94891 .5317 L Mistroke .95139 .56147 L .95387 .58657 L .95511 .59562 L .95635 .60143 L .95759 .60332 L .95883 .60071 L .96007 .59308 L .96131 .58 L .96379 .53639 L .96503 .5056 L .96627 .4689 L .97123 .26923 L .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = text; inactive; preserveAspect; endGroup] The partial sums are seen to converge to f(x)=x everywhere except in a neighborhood of x=Pi, where the odd periodic extension of the function has a discontinuity. This is the Gibbs phenomenon discussed in Problem 8. :[font = subsection; inactive; preserveAspect; startGroup] Problem 3(c) :[font = input; preserveAspect] Clear[f,b,FCoefficients,PartialSum] f[x]=1; b[n_]=(2/Pi)Integrate[f[x]Sin[n x],{x,0,Pi}]; FCoefficients[x_,n_]:=Table[b[k]Sin[k x],{k,1,n}]; PartialSum[x_,n_]:=Apply[Plus,FCoefficients[x,n]] :[font = input; preserveAspect; startGroup] Plot[{1,PartialSum[x,1]},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=1, n=1" AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.462288 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(f[x]=1, n=1 AxesLabel -> {x, f})] .5 .61803 0 -2 Msboxa [(0.2)] .01131 .10717 1 0 Msboxa [(0.4)] .01131 .19963 1 0 Msboxa [(0.6)] .01131 .29209 1 0 Msboxa [(0.8)] .01131 .38455 1 0 Msboxa [(1)] .01131 .477 1 0 Msboxa [(1.2)] .01131 .56946 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.5)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(1)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(1.5)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(2)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(2.5)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(3)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 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.95635 .60275 L .95759 .60332 L .95883 .59946 L .96007 .59069 L .96131 .57663 L .96379 .5316 L .96503 .50042 L .96627 .46354 L .97123 .26536 L .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = text; inactive; preserveAspect; endGroup; endGroup] As in Problem 3(c), the partial sums converge to f(x)=1everywhere except in neighborhoods of the points x=0 and x=1, where the odd periodic extension of the function has discontinuities. :[font = section; inactive; Cclosed; preserveAspect; startGroup] Problem 5 :[font = text; inactive; preserveAspect] The partial sums for Fourier series for Cos[k x] for noninteger k are, from Problem 4, given by :[font = input; preserveAspect] Clear[a,FCoefficients,PartialSum] f[k,x]=Cos[k x]; a[k_,n_]=(1/Pi)Integrate[f[x]Cos[n x],{x,-Pi,Pi}]; FCoefficients[k_,x_,N_]:=Table[a[k,n]Cos[n x],{n,1,N}]; PartialSum[k_,x_,N_]:=a[k,0]/2+ Apply[Plus,FCoefficients[k,x,N]] :[font = text; inactive; preserveAspect] We choose k=1/2 and plot Cos[k x] together with several partial sums. We will use the heavier line to indicate the function Cos[k x]. Notice that the comparison is made over the interval (-Pi,Pi). :[font = input; Cclosed; preserveAspect; startGroup] Plot[{Cos[x/2],PartialSum[Pi/4,x,1]},{x,-Pi,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"Cos[x/2], n=1", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.151576 0.217616 0.361262 [ [(-3)] .04527 .21762 0 2 Msboxa [(-2)] .19685 .21762 0 2 Msboxa [(-1)] .34842 .21762 0 2 Msboxa [(1)] .65158 .21762 0 2 Msboxa [(2)] .80315 .21762 0 2 Msboxa [(3)] .95473 .21762 0 2 Msboxa [(x)] 1.025 .21762 -1 0 Msboxa [(Cos[x/2], n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(-0.5)] .4875 .03698 1 0 Msboxa [(-0.25)] .4875 .1273 1 0 Msboxa [(0.25)] .4875 .30793 1 0 Msboxa [(0.5)] .4875 .39825 1 0 Msboxa [(0.75)] .4875 .48856 1 0 Msboxa [(1)] .4875 .57888 1 0 Msboxa [(f)] .5 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .04527 .21762 m .04527 .22387 L s P [(-3)] .04527 .21762 0 2 Mshowa p .002 w .19685 .21762 m .19685 .22387 L s P [(-2)] .19685 .21762 0 2 Mshowa p .002 w .34842 .21762 m .34842 .22387 L s P [(-1)] .34842 .21762 0 2 Mshowa p .002 w .65158 .21762 m .65158 .22387 L s P [(1)] .65158 .21762 0 2 Mshowa p .002 w .80315 .21762 m .80315 .22387 L s P [(2)] .80315 .21762 0 2 Mshowa p .002 w .95473 .21762 m .95473 .22387 L s P [(3)] .95473 .21762 0 2 Mshowa p .001 w .07559 .21762 m .07559 .22137 L s P p .001 w .1059 .21762 m .1059 .22137 L s P p .001 w .13622 .21762 m .13622 .22137 L s P p .001 w .16653 .21762 m .16653 .22137 L s P p .001 w .22716 .21762 m .22716 .22137 L s P p .001 w .25748 .21762 m .25748 .22137 L s P p .001 w .28779 .21762 m .28779 .22137 L s P p .001 w .31811 .21762 m .31811 .22137 L s P p .001 w .37874 .21762 m .37874 .22137 L s P p .001 w .40905 .21762 m .40905 .22137 L s P p .001 w .43937 .21762 m .43937 .22137 L s P p .001 w .46968 .21762 m .46968 .22137 L s P p .001 w .53032 .21762 m .53032 .22137 L s P p .001 w .56063 .21762 m .56063 .22137 L s P p .001 w .59095 .21762 m .59095 .22137 L s P p .001 w .62126 .21762 m .62126 .22137 L s P p .001 w .68189 .21762 m .68189 .22137 L s P p .001 w .71221 .21762 m .71221 .22137 L s P p .001 w .74252 .21762 m .74252 .22137 L s P p .001 w .77284 .21762 m .77284 .22137 L s P p .001 w .83347 .21762 m .83347 .22137 L s P p .001 w .86378 .21762 m .86378 .22137 L s P p .001 w .8941 .21762 m .8941 .22137 L s P p .001 w .92441 .21762 m .92441 .22137 L s P p .001 w .01496 .21762 m .01496 .22137 L s P p .001 w .98504 .21762 m .98504 .22137 L s P [(x)] 1.025 .21762 -1 0 Mshowa p .002 w 0 .21762 m 1 .21762 L s P [(Cos[x/2], n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .5 .03698 m .50625 .03698 L s P [(-0.5)] .4875 .03698 1 0 Mshowa p .002 w .5 .1273 m .50625 .1273 L s P [(-0.25)] .4875 .1273 1 0 Mshowa p .002 w .5 .30793 m .50625 .30793 L s P [(0.25)] .4875 .30793 1 0 Mshowa p .002 w .5 .39825 m .50625 .39825 L s P [(0.5)] .4875 .39825 1 0 Mshowa p .002 w .5 .48856 m .50625 .48856 L s P [(0.75)] .4875 .48856 1 0 Mshowa p .002 w .5 .57888 m .50625 .57888 L s P [(1)] .4875 .57888 1 0 Mshowa p .001 w .5 .05505 m .50375 .05505 L s P p .001 w .5 .07311 m .50375 .07311 L s P p .001 w .5 .09117 m .50375 .09117 L s P p .001 w .5 .10924 m .50375 .10924 L s P p .001 w .5 .14536 m .50375 .14536 L s P p .001 w .5 .16343 m .50375 .16343 L s P p .001 w .5 .18149 m .50375 .18149 L s P p .001 w .5 .19955 m .50375 .19955 L s P p .001 w .5 .23568 m .50375 .23568 L s P p .001 w .5 .25374 m .50375 .25374 L s P p .001 w .5 .27181 m .50375 .27181 L s P p .001 w .5 .28987 m .50375 .28987 L s P p .001 w .5 .32599 m .50375 .32599 L s P p .001 w .5 .34406 m .50375 .34406 L s P p .001 w .5 .36212 m .50375 .36212 L s P p .001 w .5 .38018 m .50375 .38018 L s P p .001 w .5 .41631 m .50375 .41631 L s P p .001 w .5 .43437 m .50375 .43437 L s P p .001 w .5 .45244 m .50375 .45244 L s P p .001 w .5 .4705 m .50375 .4705 L s P p .001 w .5 .50663 m .50375 .50663 L s P p .001 w .5 .52469 m .50375 .52469 L s P p .001 w .5 .54275 m .50375 .54275 L s P p .001 w .5 .56081 m .50375 .56081 L s P p .001 w .5 .01892 m .50375 .01892 L s P p .001 w .5 .00086 m .50375 .00086 L s P p .001 w .5 .59694 m .50375 .59694 L s P p .001 w .5 .615 m .50375 .615 L s P [(f)] .5 .61803 0 -4 Mshowa p .002 w .5 0 m .5 .61803 L s P P p p p .0075 w .02381 .21762 m .06349 .26477 L .10317 .31112 L .14286 .35586 L .18254 .39825 L .22222 .43754 L .2619 .47307 L .30159 .50422 L .34127 .53048 L .38095 .55138 L .42063 .56657 L .44048 .57194 L .4504 .57405 L .46032 .57579 L .47024 .57714 L .4752 .57767 L .48016 .5781 L .48512 .57844 L .4876 .57858 L .49008 .57868 L .49256 .57877 L .4938 .5788 L .49504 .57883 L 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.60269 L .49256 .60296 L .4938 .60307 L .49504 .60316 L .49628 .60323 L .49752 .60328 L .49876 .60331 L .5 .60332 L .50124 .60331 L .50248 .60328 L .50372 .60323 L .50496 .60316 L .5062 .60307 L .50744 .60296 L .50992 .60269 L .5124 .60233 L .51488 .6019 L .51984 .6008 L .5248 .59939 L Mistroke .52976 .59766 L .53968 .59329 L .55952 .58092 L .57937 .56389 L .61905 .51712 L .65873 .45617 L .69841 .38519 L .7381 .30902 L .77778 .23285 L .81746 .16187 L .85714 .10091 L .87698 .07553 L .89683 .05414 L .91667 .03712 L .92659 .03033 L .93651 .02474 L .94643 .02037 L .95139 .01865 L .95635 .01723 L .96131 .01613 L .96379 .0157 L .96627 .01535 L .96875 .01507 L .96999 .01496 L .97123 .01487 L .97247 .0148 L .97371 .01475 L .97495 .01472 L .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; Cclosed; preserveAspect; startGroup] Plot[{Cos[x/2],PartialSum[1/2,x,2]},{x,-Pi,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"Cos[x/2], n=2", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.151576 0.0147151 0.588604 [ [(-3)] .04527 .01472 0 2 Msboxa [(-2)] .19685 .01472 0 2 Msboxa [(-1)] .34842 .01472 0 2 Msboxa [(1)] .65158 .01472 0 2 Msboxa [(2)] .80315 .01472 0 2 Msboxa [(3)] .95473 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(Cos[x/2], n=2)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.2)] .4875 .13244 1 0 Msboxa [(0.4)] .4875 .25016 1 0 Msboxa [(0.6)] .4875 .36788 1 0 Msboxa [(0.8)] .4875 .4856 1 0 Msboxa [(1)] .4875 .60332 1 0 Msboxa [(f)] .5 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .04527 .01472 m .04527 .02097 L s P [(-3)] .04527 .01472 0 2 Mshowa p .002 w .19685 .01472 m .19685 .02097 L s P [(-2)] .19685 .01472 0 2 Mshowa p .002 w .34842 .01472 m .34842 .02097 L s P [(-1)] .34842 .01472 0 2 Mshowa p .002 w .65158 .01472 m .65158 .02097 L s P [(1)] .65158 .01472 0 2 Mshowa p .002 w .80315 .01472 m .80315 .02097 L s P [(2)] .80315 .01472 0 2 Mshowa p .002 w .95473 .01472 m .95473 .02097 L s P [(3)] .95473 .01472 0 2 Mshowa p .001 w .07559 .01472 m .07559 .01847 L s P p .001 w .1059 .01472 m .1059 .01847 L s P p .001 w .13622 .01472 m .13622 .01847 L s P p .001 w .16653 .01472 m .16653 .01847 L s P p .001 w .22716 .01472 m .22716 .01847 L s P p .001 w .25748 .01472 m .25748 .01847 L s P p .001 w .28779 .01472 m .28779 .01847 L s P p .001 w .31811 .01472 m .31811 .01847 L s P p .001 w .37874 .01472 m .37874 .01847 L s P p .001 w .40905 .01472 m .40905 .01847 L s P p .001 w .43937 .01472 m .43937 .01847 L s P p .001 w .46968 .01472 m .46968 .01847 L s P p .001 w .53032 .01472 m .53032 .01847 L s P p .001 w .56063 .01472 m .56063 .01847 L s P p .001 w .59095 .01472 m .59095 .01847 L s P p .001 w .62126 .01472 m .62126 .01847 L s P p .001 w .68189 .01472 m .68189 .01847 L s P p .001 w .71221 .01472 m .71221 .01847 L s P p .001 w .74252 .01472 m .74252 .01847 L s P p .001 w .77284 .01472 m .77284 .01847 L s P p .001 w .83347 .01472 m .83347 .01847 L s P p .001 w .86378 .01472 m .86378 .01847 L s P p .001 w .8941 .01472 m .8941 .01847 L s P p .001 w .92441 .01472 m .92441 .01847 L s P p .001 w .01496 .01472 m .01496 .01847 L s P p .001 w .98504 .01472 m .98504 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(Cos[x/2], n=2)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .5 .13244 m .50625 .13244 L s P [(0.2)] .4875 .13244 1 0 Mshowa p .002 w .5 .25016 m .50625 .25016 L s P [(0.4)] .4875 .25016 1 0 Mshowa p .002 w .5 .36788 m .50625 .36788 L s P [(0.6)] .4875 .36788 1 0 Mshowa p .002 w .5 .4856 m .50625 .4856 L s P [(0.8)] .4875 .4856 1 0 Mshowa p .002 w .5 .60332 m .50625 .60332 L s P [(1)] .4875 .60332 1 0 Mshowa p .001 w .5 .03826 m .50375 .03826 L s P p .001 w .5 .0618 m .50375 .0618 L s P p .001 w .5 .08535 m .50375 .08535 L s P p .001 w .5 .10889 m .50375 .10889 L s P p .001 w .5 .15598 m .50375 .15598 L s P p .001 w .5 .17952 m .50375 .17952 L s P p .001 w .5 .20307 m .50375 .20307 L s P p .001 w .5 .22661 m .50375 .22661 L s P p .001 w .5 .2737 m .50375 .2737 L s P p .001 w .5 .29724 m .50375 .29724 L s P p .001 w .5 .32079 m .50375 .32079 L s P p .001 w .5 .34433 m .50375 .34433 L s P p .001 w .5 .39142 m .50375 .39142 L s P p .001 w .5 .41497 m .50375 .41497 L s P p .001 w .5 .43851 m .50375 .43851 L s P p .001 w .5 .46205 m .50375 .46205 L s P p .001 w .5 .50914 m .50375 .50914 L s P p .001 w .5 .53269 m .50375 .53269 L s P p .001 w .5 .55623 m .50375 .55623 L s P p .001 w .5 .57977 m .50375 .57977 L s P [(f)] .5 .61803 0 -4 Mshowa p .002 w .5 0 m .5 .61803 L s P P p p p .0075 w .02381 .01472 m .06349 .09154 L .10317 .16706 L .14286 .23996 L .18254 .30902 L .22222 .37303 L .2619 .43092 L .30159 .48169 L .34127 .52446 L .38095 .55851 L .42063 .58326 L .44048 .59201 L .4504 .59546 L .46032 .59828 L .47024 .60048 L .4752 .60135 L .48016 .60206 L .48512 .60261 L .4876 .60283 L .49008 .603 L .49256 .60314 L .4938 .6032 L .49504 .60324 L .49628 .60327 L .49752 .6033 L .49876 .60331 L .5 .60332 L .50124 .60331 L .50248 .6033 L .50372 .60327 L .50496 .60324 L .5062 .6032 L .50744 .60314 L .50992 .603 L .5124 .60283 L .51488 .60261 L .51984 .60206 L .5248 .60135 L .52976 .60048 L .53968 .59828 L .5496 .59546 L .55952 .59201 L .57937 .58326 L .61905 .55851 L .65873 .52446 L .69841 .48169 L .7381 .43092 L .77778 .37303 L .81746 .30902 L .85714 .23996 L Mistroke .89683 .16706 L .93651 .09154 L .97619 .01472 L Mfstroke P P p p .005 w .02381 .08966 m .02505 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.43939 L .77778 .36804 L .81746 .28951 L .85714 .21279 L .87698 .17831 L .89683 .14811 L .91667 .12331 L .92659 .11324 L .93651 .10486 L .94643 .09826 L .95139 .09565 L .95635 .0935 L .96131 .09182 L .96379 .09116 L .96627 .09062 L .96875 .0902 L .96999 .09003 L .97123 .0899 L .97247 .08979 L .97371 .08972 L .97495 .08967 L .97619 .08966 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; Cclosed; preserveAspect; startGroup] Plot[{Cos[x/2],PartialSum[1/2,x,3]},{x,-Pi,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"Cos[x/2], n=3", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.151576 0.0147151 0.581321 [ [(-3)] .04527 .01472 0 2 Msboxa [(-2)] .19685 .01472 0 2 Msboxa [(-1)] .34842 .01472 0 2 Msboxa [(1)] .65158 .01472 0 2 Msboxa [(2)] .80315 .01472 0 2 Msboxa [(3)] .95473 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(Cos[x/2], n=3)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.2)] .4875 .13098 1 0 Msboxa [(0.4)] .4875 .24724 1 0 Msboxa [(0.6)] .4875 .36351 1 0 Msboxa [(0.8)] .4875 .47977 1 0 Msboxa [(1)] .4875 .59604 1 0 Msboxa [(f)] .5 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .04527 .01472 m .04527 .02097 L s P [(-3)] .04527 .01472 0 2 Mshowa p .002 w .19685 .01472 m .19685 .02097 L s P [(-2)] .19685 .01472 0 2 Mshowa p .002 w .34842 .01472 m .34842 .02097 L s P [(-1)] .34842 .01472 0 2 Mshowa p .002 w .65158 .01472 m .65158 .02097 L s P [(1)] .65158 .01472 0 2 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.01472 m .74252 .01847 L s P p .001 w .77284 .01472 m .77284 .01847 L s P p .001 w .83347 .01472 m .83347 .01847 L s P p .001 w .86378 .01472 m .86378 .01847 L s P p .001 w .8941 .01472 m .8941 .01847 L s P p .001 w .92441 .01472 m .92441 .01847 L s P p .001 w .01496 .01472 m .01496 .01847 L s P p .001 w .98504 .01472 m .98504 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(Cos[x/2], n=5)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .5 .13183 m .50625 .13183 L s P [(0.2)] .4875 .13183 1 0 Mshowa p .002 w .5 .24894 m .50625 .24894 L s P [(0.4)] .4875 .24894 1 0 Mshowa p .002 w .5 .36606 m .50625 .36606 L s P [(0.6)] .4875 .36606 1 0 Mshowa p .002 w .5 .48317 m .50625 .48317 L s P [(0.8)] .4875 .48317 1 0 Mshowa p .002 w .5 .60029 m .50625 .60029 L s P [(1)] .4875 .60029 1 0 Mshowa p .001 w .5 .03814 m .50375 .03814 L s P p .001 w .5 .06156 m .50375 .06156 L s P p .001 w .5 .08498 m .50375 .08498 L s P p .001 w .5 .10841 m .50375 .10841 L s P p .001 w .5 .15525 m .50375 .15525 L s P p .001 w .5 .17867 m .50375 .17867 L s P p .001 w .5 .2021 m .50375 .2021 L s P p .001 w .5 .22552 m .50375 .22552 L s P p .001 w .5 .27237 m .50375 .27237 L s P p .001 w .5 .29579 m .50375 .29579 L s P p .001 w .5 .31921 m .50375 .31921 L s P p .001 w .5 .34263 m .50375 .34263 L s P p .001 w .5 .38948 m .50375 .38948 L s P p .001 w .5 .4129 m .50375 .4129 L s P p .001 w .5 .43633 m .50375 .43633 L s P p .001 w .5 .45975 m .50375 .45975 L s P p .001 w .5 .50659 m .50375 .50659 L s P p .001 w .5 .53002 m .50375 .53002 L s P p .001 w .5 .55344 m .50375 .55344 L s P p .001 w .5 .57686 m .50375 .57686 L s P [(f)] .5 .61803 0 -4 Mshowa p .002 w .5 0 m .5 .61803 L s P P p p p .0075 w .02381 .01472 m .06349 .09115 L .10317 .16627 L .14286 .2388 L .18254 .3075 L .22222 .37119 L .2619 .42878 L .30159 .47928 L .34127 .52183 L .38095 .55571 L .42063 .58033 L .44048 .58903 L .4504 .59246 L .46032 .59528 L .47024 .59747 L .4752 .59833 L .48016 .59903 L .48512 .59958 L .4876 .5998 L .49008 .59997 L .49256 .60011 L .4938 .60016 L .49504 .60021 L .49628 .60024 L .49752 .60027 L .49876 .60028 L .5 .60029 L .50124 .60028 L .50248 .60027 L .50372 .60024 L .50496 .60021 L .5062 .60016 L .50744 .60011 L .50992 .59997 L .5124 .5998 L .51488 .59958 L .51984 .59903 L .5248 .59833 L .52976 .59747 L .53968 .59528 L .5496 .59246 L .55952 .58903 L .57937 .58033 L .61905 .55571 L .65873 .52183 L .69841 .47928 L .7381 .42878 L .77778 .37119 L .81746 .3075 L .85714 .2388 L Mistroke .89683 .16627 L .93651 .09115 L .97619 .01472 L Mfstroke P P p p .005 w .02381 .0486 m .02505 .04864 L .02629 .04874 L .02753 .04891 L .02877 .04915 L .03125 .04983 L .03373 .05077 L .03869 .05346 L .04365 .05719 L .05357 .06757 L .06349 .08147 L .10317 .15986 L .14286 .24399 L .18254 .31155 L .22222 .3681 L .2619 .42537 L .30159 .48116 L .32143 .50503 L .34127 .525 L .38095 .55468 L .42063 .57727 L .44048 .58723 L .4504 .59168 L .46032 .59561 L .47024 .59886 L .4752 .60018 L .48016 .60129 L .48512 .60217 L .4876 .60252 L .49008 .60281 L .49256 .60303 L .4938 .60312 L .49504 .60319 L .49628 .60325 L .49752 .60329 L .49876 .60331 L .5 .60332 L .50124 .60331 L .50248 .60329 L .50372 .60325 L .50496 .60319 L .5062 .60312 L .50744 .60303 L .50992 .60281 L .51488 .60217 L .51984 .60129 L .52976 .59886 L .53968 .59561 L .57937 .57727 L .61905 .55468 L Mistroke .63889 .54124 L .65873 .525 L .69841 .48116 L .7381 .42537 L .77778 .3681 L .81746 .31155 L .85714 .24399 L .89683 .15986 L .91667 .11748 L .92659 .09833 L .93651 .08147 L .94643 .06757 L .95139 .06191 L .95635 .05719 L .96131 .05346 L .96379 .05199 L .96627 .05077 L .96875 .04983 L .96999 .04945 L .97123 .04915 L .97247 .04891 L .97371 .04874 L .97495 .04864 L .97619 .0486 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = text; inactive; preserveAspect] The partial sums converge rapidly everywhere to Cos[k x] except near x=-Pi and x=Pi. Because we constructed an even periodic extension of this function, for thisd choice of k, the function is continuous everywhere but has a discontinuity in the first derivative at x=Pi and x=-Pi. :[font = text; inactive; preserveAspect] The partial products of Sin[x] is constructed as :[font = input; preserveAspect] PartialProduct[x_,N_]:=x Product[ 1-(x^2/(n^2 Pi^2)),{n,1,N}] :[font = text; inactive; preserveAspect] The first few partial products are given by :[font = input; Cclosed; preserveAspect; startGroup] PartialProduct[x,1] :[font = output; output; inactive; preserveAspect; endGroup] x*(1 - x^2/Pi^2) ;[o] 2 x x (1 - ---) 2 Pi :[font = input; Cclosed; preserveAspect; startGroup] Expand[%] :[font = output; output; inactive; preserveAspect; endGroup] x - x^3/Pi^2 ;[o] 3 x x - --- 2 Pi :[font = input; Cclosed; preserveAspect; startGroup] PartialProduct[x,2] :[font = output; output; inactive; preserveAspect; endGroup] x*(1 - x^2/Pi^2)*(1 - x^2/(4*Pi^2)) ;[o] 2 2 x x x (1 - ---) (1 - -----) 2 2 Pi 4 Pi :[font = input; Cclosed; preserveAspect; startGroup] Expand[%] :[font = output; output; inactive; preserveAspect; endGroup] x - (5*x^3)/(4*Pi^2) + x^5/(4*Pi^4) ;[o] 3 5 5 x x x - ----- + ----- 2 4 4 Pi 4 Pi :[font = input; Cclosed; preserveAspect; startGroup] PartialProduct[x,3] :[font = output; output; inactive; preserveAspect; endGroup] x*(1 - x^2/Pi^2)*(1 - x^2/(4*Pi^2))*(1 - x^2/(9*Pi^2)) ;[o] 2 2 2 x x x x (1 - ---) (1 - -----) (1 - -----) 2 2 2 Pi 4 Pi 9 Pi :[font = input; Cclosed; preserveAspect; startGroup] Expand[%] :[font = output; output; inactive; preserveAspect; endGroup] x - (49*x^3)/(36*Pi^2) + (7*x^5)/(18*Pi^4) - x^7/(36*Pi^6) ;[o] 3 5 7 49 x 7 x x x - ------ + ------ - ------ 2 4 6 36 Pi 18 Pi 36 Pi :[font = text; inactive; preserveAspect] To compare the development of this product as the number of terms increase, we compare with the series representation of Sin[x]. :[font = input; Cclosed; preserveAspect; startGroup] Series[Sin[x],{x,0,10}] :[font = output; output; inactive; preserveAspect; endGroup] SeriesData[x, 0, {1, 0, -1/6, 0, 1/120, 0, -1/5040, 0, 1/362880}, 1, 11, 1] ;[o] 3 5 7 9 x x x x 11 x - -- + --- - ---- + ------ + O[x] 6 120 5040 362880 :[font = input; preserveAspect] or, :[font = input; Cclosed; preserveAspect; startGroup] N[%] :[font = output; output; inactive; preserveAspect; endGroup] SeriesData[x, 0, {1., 0, -0.1666666666666666, 0, 0.00833333333333333, 0, -0.0001984126984126984, 0, 2.755731922398589*10^-6}, 1, 11, 1] ;[o] 3 5 7 1. x - 0.16667 x + 0.0083333 x - 0.00019841 x + -6 9 11 2.7557 10 x + O[x] :[font = text; inactive; preserveAspect] The corresponding expansions of the partial products are :[font = input; Cclosed; preserveAspect; startGroup] N[Expand[PartialProduct[x,1]]] :[font = output; output; inactive; preserveAspect; endGroup] x - 0.1013211836423378*x^3 ;[o] 3 x - 0.101321 x :[font = input; Cclosed; preserveAspect; startGroup] N[Expand[PartialProduct[x,3]]] :[font = output; output; inactive; preserveAspect; endGroup] x - 0.1379093888465153*x^3 + 0.003992326432377242*x^5 - 0.00002889337425821812*x^7 ;[o] 3 5 7 x - 0.137909 x + 0.00399233 x - 0.0000288934 x :[font = input; Cclosed; preserveAspect; startGroup] N[Expand[PartialProduct[x,5]]] :[font = output; output; inactive; preserveAspect; endGroup] x - 0.1482948101698549*x^3 + 0.005450238495629288*x^5 - 0.00007389480466539287*x^7 + 4.025327458987606*10^-7*x^9 - 7.415471685320512*10^-10*x^11 ;[o] 3 5 7 x - 0.148295 x + 0.00545024 x - 0.0000738948 x + -7 9 -10 11 4.02533 10 x - 7.41547 10 x :[font = input; Cclosed; preserveAspect; startGroup] N[Expand[PartialProduct[x,10]]] :[font = output; output; inactive; preserveAspect; endGroup] x - 0.1570243008924943*x^3 + 0.006774231352855197*x^5 - 0.0001258886723238589*x^7 + 1.215300464538591*10^-6*x^9 - 6.699722152217645*10^-9*x^11 + 2.209197671316394*10^-11*x^13 - 4.407143507365955*10^-14*x^15 + 5.18286131127126*10^-17*x^17 - 3.290311044561922*10^-20*x^19 + 8.65917427497326*10^-24*x^21 ;[o] 3 5 7 x - 0.157024 x + 0.00677423 x - 0.000125889 x + -6 9 -9 11 1.2153 10 x - 6.69972 10 x + -11 13 -14 15 2.2092 10 x - 4.40714 10 x + -17 17 -20 19 5.18286 10 x - 3.29031 10 x + -24 21 8.65917 10 x :[font = text; inactive; preserveAspect] To see the pattern of convergence of the partial products, we plot them with the function Sin[x]: :[font = input; Cclosed; preserveAspect; startGroup] Plot[{Sin[x],PartialProduct[x,1]},{x,-Pi,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"Sin[x], n=1", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.151576 0.309017 0.243386 [ [(-3)] .04527 .30902 0 2 Msboxa [(-2)] .19685 .30902 0 2 Msboxa [(-1)] .34842 .30902 0 2 Msboxa [(1)] .65158 .30902 0 2 Msboxa [(2)] .80315 .30902 0 2 Msboxa [(3)] .95473 .30902 0 2 Msboxa [(x)] 1.025 .30902 -1 0 Msboxa [(Sin[x], n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(-1)] .4875 .06563 1 0 Msboxa [(-0.5)] .4875 .18732 1 0 Msboxa [(0.5)] .4875 .43071 1 0 Msboxa [(1)] .4875 .5524 1 0 Msboxa [(f)] .5 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .04527 .30902 m .04527 .31527 L s P [(-3)] .04527 .30902 0 2 Mshowa p .002 w .19685 .30902 m .19685 .31527 L s P [(-2)] .19685 .30902 0 2 Mshowa p .002 w .34842 .30902 m .34842 .31527 L s P [(-1)] .34842 .30902 0 2 Mshowa p .002 w .65158 .30902 m .65158 .31527 L s P [(1)] .65158 .30902 0 2 Mshowa p .002 w .80315 .30902 m .80315 .31527 L s P [(2)] .80315 .30902 0 2 Mshowa p .002 w .95473 .30902 m .95473 .31527 L s P [(3)] .95473 .30902 0 2 Mshowa p .001 w .07559 .30902 m .07559 .31277 L s P p .001 w .1059 .30902 m .1059 .31277 L s P p .001 w .13622 .30902 m .13622 .31277 L s P p .001 w .16653 .30902 m .16653 .31277 L s P p .001 w .22716 .30902 m .22716 .31277 L s P p .001 w .25748 .30902 m .25748 .31277 L s P p .001 w .28779 .30902 m .28779 .31277 L s P p .001 w .31811 .30902 m .31811 .31277 L s P p .001 w .37874 .30902 m .37874 .31277 L s P p .001 w .40905 .30902 m .40905 .31277 L s P p .001 w .43937 .30902 m .43937 .31277 L s P p .001 w .46968 .30902 m .46968 .31277 L s P p .001 w .53032 .30902 m .53032 .31277 L s P p .001 w .56063 .30902 m .56063 .31277 L s P p .001 w .59095 .30902 m .59095 .31277 L s P p .001 w .62126 .30902 m .62126 .31277 L s P p .001 w .68189 .30902 m .68189 .31277 L s P p .001 w .71221 .30902 m .71221 .31277 L s P p .001 w .74252 .30902 m .74252 .31277 L s P p .001 w .77284 .30902 m .77284 .31277 L s P p .001 w .83347 .30902 m .83347 .31277 L s P p .001 w .86378 .30902 m .86378 .31277 L s P p .001 w .8941 .30902 m .8941 .31277 L s P p .001 w .92441 .30902 m .92441 .31277 L s P p .001 w .01496 .30902 m .01496 .31277 L s P p .001 w .98504 .30902 m .98504 .31277 L s P [(x)] 1.025 .30902 -1 0 Mshowa p .002 w 0 .30902 m 1 .30902 L s P [(Sin[x], n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .5 .06563 m .50625 .06563 L s P [(-1)] .4875 .06563 1 0 Mshowa p .002 w .5 .18732 m .50625 .18732 L s P [(-0.5)] .4875 .18732 1 0 Mshowa p .002 w .5 .43071 m .50625 .43071 L s P [(0.5)] .4875 .43071 1 0 Mshowa p .002 w .5 .5524 m .50625 .5524 L s P [(1)] .4875 .5524 1 0 Mshowa p .001 w .5 .08997 m .50375 .08997 L s P p .001 w .5 .11431 m .50375 .11431 L s P p .001 w .5 .13865 m .50375 .13865 L s P p .001 w .5 .16299 m .50375 .16299 L s P p .001 w .5 .21166 m .50375 .21166 L s P p .001 w .5 .236 m .50375 .236 L s P p .001 w .5 .26034 m .50375 .26034 L s P p .001 w .5 .28468 m .50375 .28468 L s P p .001 w .5 .33336 m .50375 .33336 L s P p .001 w .5 .35769 m .50375 .35769 L s P p .001 w .5 .38203 m .50375 .38203 L s P p .001 w .5 .40637 m .50375 .40637 L s P p .001 w .5 .45505 m .50375 .45505 L s P p .001 w .5 .47939 m .50375 .47939 L s P p .001 w .5 .50373 m .50375 .50373 L s P p .001 w .5 .52806 m .50375 .52806 L s P p .001 w .5 .04129 m .50375 .04129 L s P p .001 w .5 .01695 m .50375 .01695 L s P p .001 w .5 .57674 m .50375 .57674 L s P p .001 w .5 .60108 m .50375 .60108 L s P [(f)] .5 .61803 0 -4 Mshowa p .002 w .5 0 m .5 .61803 L s P P p p p .0075 w .02381 .30902 m .06349 .24602 L .10317 .18732 L .14286 .13692 L .1627 .11593 L .18254 .09824 L .20238 .08416 L .2123 .07855 L .22222 .07392 L .23214 .07031 L .2371 .06888 L .24206 .06771 L .24702 .0668 L .2495 .06644 L .25198 .06615 L .25446 .06592 L .2557 .06583 L .25694 .06576 L .25818 .0657 L .25942 .06566 L .26066 .06564 L .2619 .06563 L .26314 .06564 L .26438 .06566 L .26563 .0657 L .26687 .06576 L .26935 .06592 L .27183 .06615 L .27431 .06644 L .27679 .0668 L .28175 .06771 L .28671 .06888 L .29167 .07031 L .30159 .07392 L .32143 .08416 L .34127 .09824 L .38095 .13692 L .42063 .18732 L .46032 .24602 L .5 .30902 L .53968 .37201 L .57937 .43071 L .61905 .48112 L .63889 .50211 L .65873 .5198 L .67857 .53388 L .68849 .53949 L .69841 .54411 L .70833 .54773 L .71329 .54915 L Mistroke .71825 .55032 L .72321 .55123 L .72569 .55159 L .72817 .55188 L .73065 .55211 L .73189 .5522 L .73313 .55227 L .73438 .55233 L .73562 .55237 L .73686 .5524 L .7381 .5524 L .73934 .5524 L .74058 .55237 L .74182 .55233 L .74306 .55227 L .74554 .55211 L .74802 .55188 L .7505 .55159 L .75298 .55123 L .75794 .55032 L .7629 .54915 L .76786 .54773 L .77778 .54411 L .79762 .53388 L .81746 .5198 L .85714 .48112 L .89683 .43071 L .93651 .37201 L .97619 .30902 L Mfstroke P P p p .005 w .02381 .30902 m .06349 .19707 L .08333 .15221 L .10317 .11432 L .12302 .08307 L .14286 .05813 L .15278 .04791 L .1627 .03915 L .17262 .0318 L .18254 .02582 L .19246 .02117 L .19742 .01933 L .20238 .0178 L .20734 .01659 L .2123 .01568 L .21478 .01534 L .21726 .01507 L .2185 .01497 L .21974 .01488 L .22098 .01481 L .22222 .01476 L .22346 .01473 L .2247 .01472 L .22594 .01472 L .22718 .01474 L .22842 .01478 L .22966 .01484 L .23214 .015 L .23462 .01524 L .2371 .01555 L .24206 .01637 L .24702 .01745 L .25198 .01881 L .2619 .02228 L .27183 .02676 L .28175 .03218 L .30159 .04574 L .34127 .08246 L .38095 .12981 L .42063 .18512 L .46032 .24574 L .5 .30902 L .53968 .37229 L .57937 .43291 L .61905 .48823 L .65873 .53557 L .67857 .55543 L .69841 .5723 L .71825 .58585 L .72817 .59128 L Mistroke .7381 .59575 L .74306 .59762 L .74802 .59923 L .75298 .60058 L .75794 .60167 L .76042 .60211 L .7629 .60249 L .76538 .60279 L .76786 .60303 L .7691 .60312 L .77034 .6032 L .77158 .60325 L .77282 .60329 L .77406 .60332 L .7753 .60332 L .77654 .6033 L .77778 .60327 L .77902 .60322 L .78026 .60315 L .78274 .60296 L .78522 .60269 L .7877 .60235 L .79266 .60144 L .79762 .60023 L .80754 .59686 L .81746 .59221 L .82738 .58623 L .8373 .57888 L .85714 .55991 L .87698 .53496 L .89683 .50371 L .93651 .42097 L .97619 .30902 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; Cclosed; preserveAspect; startGroup] Plot[{Sin[x],PartialProduct[x,3]},{x,-Pi,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"Sin[x], n=3", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.151576 0.309017 0.272879 [ [(-3)] .04527 .30902 0 2 Msboxa [(-2)] .19685 .30902 0 2 Msboxa [(-1)] .34842 .30902 0 2 Msboxa [(1)] .65158 .30902 0 2 Msboxa [(2)] .80315 .30902 0 2 Msboxa [(3)] .95473 .30902 0 2 Msboxa [(x)] 1.025 .30902 -1 0 Msboxa [(Sin[x], n=3)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(-1)] .4875 .03614 1 0 Msboxa [(-0.5)] .4875 .17258 1 0 Msboxa [(0.5)] .4875 .44546 1 0 Msboxa [(1)] .4875 .5819 1 0 Msboxa [(f)] .5 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .04527 .30902 m .04527 .31527 L s P [(-3)] .04527 .30902 0 2 Mshowa p .002 w .19685 .30902 m .19685 .31527 L s P [(-2)] .19685 .30902 0 2 Mshowa p .002 w .34842 .30902 m .34842 .31527 L s P [(-1)] .34842 .30902 0 2 Mshowa p .002 w .65158 .30902 m .65158 .31527 L s P [(1)] .65158 .30902 0 2 Mshowa p .002 w .80315 .30902 m .80315 .31527 L s P [(2)] .80315 .30902 0 2 Mshowa p .002 w .95473 .30902 m .95473 .31527 L s P [(3)] .95473 .30902 0 2 Mshowa p .001 w .07559 .30902 m .07559 .31277 L s P p .001 w .1059 .30902 m .1059 .31277 L s P p .001 w .13622 .30902 m .13622 .31277 L s P p .001 w .16653 .30902 m .16653 .31277 L s P p .001 w .22716 .30902 m .22716 .31277 L s P p .001 w .25748 .30902 m .25748 .31277 L s P p .001 w .28779 .30902 m .28779 .31277 L s P p .001 w .31811 .30902 m .31811 .31277 L s P p .001 w .37874 .30902 m .37874 .31277 L s P p .001 w .40905 .30902 m .40905 .31277 L s P p .001 w .43937 .30902 m .43937 .31277 L s P p .001 w .46968 .30902 m .46968 .31277 L s P p .001 w .53032 .30902 m .53032 .31277 L s P p .001 w .56063 .30902 m .56063 .31277 L s P p .001 w .59095 .30902 m .59095 .31277 L s P p .001 w .62126 .30902 m .62126 .31277 L s P p .001 w .68189 .30902 m .68189 .31277 L s P p .001 w .71221 .30902 m .71221 .31277 L s P p .001 w .74252 .30902 m .74252 .31277 L s P p .001 w .77284 .30902 m .77284 .31277 L s P p .001 w .83347 .30902 m .83347 .31277 L s P p .001 w .86378 .30902 m .86378 .31277 L s P p .001 w .8941 .30902 m .8941 .31277 L s P p .001 w .92441 .30902 m .92441 .31277 L s P p .001 w .01496 .30902 m .01496 .31277 L s P p .001 w .98504 .30902 m .98504 .31277 L s P [(x)] 1.025 .30902 -1 0 Mshowa p .002 w 0 .30902 m 1 .30902 L s P [(Sin[x], n=3)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .5 .03614 m .50625 .03614 L s P [(-1)] .4875 .03614 1 0 Mshowa p .002 w .5 .17258 m .50625 .17258 L s P [(-0.5)] .4875 .17258 1 0 Mshowa p .002 w .5 .44546 m .50625 .44546 L s P [(0.5)] .4875 .44546 1 0 Mshowa p .002 w .5 .5819 m .50625 .5819 L s P [(1)] .4875 .5819 1 0 Mshowa p .001 w .5 .06343 m .50375 .06343 L s P p .001 w .5 .09071 m .50375 .09071 L s P p .001 w .5 .118 m .50375 .118 L s P p .001 w .5 .14529 m .50375 .14529 L s P p .001 w .5 .19987 m .50375 .19987 L s P p .001 w .5 .22715 m .50375 .22715 L s P p .001 w .5 .25444 m .50375 .25444 L s P p .001 w .5 .28173 m .50375 .28173 L s P p .001 w .5 .3363 m .50375 .3363 L s P p .001 w .5 .36359 m .50375 .36359 L s P p .001 w .5 .39088 m .50375 .39088 L s P p .001 w .5 .41817 m .50375 .41817 L s P p .001 w .5 .47274 m .50375 .47274 L s P p .001 w .5 .50003 m .50375 .50003 L s P p .001 w .5 .52732 m .50375 .52732 L s P p .001 w .5 .55461 m .50375 .55461 L s P p .001 w .5 .00885 m .50375 .00885 L s P p .001 w .5 .60918 m .50375 .60918 L s P [(f)] .5 .61803 0 -4 Mshowa p .002 w .5 0 m .5 .61803 L s P P p p p .0075 w .02381 .30902 m .06349 .23839 L .10317 .17258 L .14286 .11606 L .1627 .09253 L .18254 .0727 L .20238 .05691 L .2123 .05062 L .22222 .04544 L .23214 .04138 L .2371 .03978 L .24206 .03847 L .24702 .03745 L .2495 .03705 L .25198 .03672 L .25446 .03647 L .2557 .03637 L .25694 .03628 L .25818 .03622 L .25942 .03617 L .26066 .03615 L .2619 .03614 L .26314 .03615 L .26438 .03617 L .26563 .03622 L .26687 .03628 L .26935 .03647 L .27183 .03672 L .27431 .03705 L .27679 .03745 L .28175 .03847 L .28671 .03978 L .29167 .04138 L .30159 .04544 L .32143 .05691 L .34127 .0727 L .38095 .11606 L .42063 .17258 L .46032 .23839 L .5 .30902 L .53968 .37964 L .57937 .44546 L .61905 .50197 L .63889 .52551 L .65873 .54534 L .67857 .56112 L .68849 .56741 L .69841 .5726 L .70833 .57665 L .71329 .57825 L Mistroke .71825 .57956 L .72321 .58058 L .72569 .58098 L .72817 .58131 L .73065 .58157 L .73189 .58167 L .73313 .58175 L .73438 .58181 L .73562 .58186 L .73686 .58189 L .7381 .5819 L .73934 .58189 L .74058 .58186 L .74182 .58181 L .74306 .58175 L .74554 .58157 L .74802 .58131 L .7505 .58098 L .75298 .58058 L .75794 .57956 L .7629 .57825 L .76786 .57665 L .77778 .5726 L .79762 .56112 L .81746 .54534 L .85714 .50197 L .89683 .44546 L .93651 .37964 L .97619 .30902 L Mfstroke P P p p .005 w .02381 .30902 m .06349 .21913 L .10317 .14255 L .12302 .11026 L .14286 .08239 L .1627 .05916 L .18254 .04072 L .20238 .02719 L .2123 .02227 L .21726 .02027 L .22222 .01858 L .22718 .0172 L .23214 .01612 L .23462 .0157 L .2371 .01535 L .23958 .01508 L .24082 .01497 L .24206 .01488 L .2433 .01481 L .24454 .01476 L .24578 .01473 L .24702 .01472 L .24826 .01472 L .2495 .01474 L .25074 .01479 L .25198 .01485 L .25446 .01503 L .25694 .01528 L .25942 .0156 L .2619 .016 L .26687 .01702 L .27183 .01833 L .28175 .0218 L .29167 .0264 L .30159 .03209 L .32143 .04662 L .34127 .06511 L .38095 .11261 L .42063 .1715 L .46032 .23825 L .5 .30902 L .53968 .37978 L .57937 .44654 L .61905 .50543 L .63889 .5308 L .65873 .55292 L .67857 .57141 L .69841 .58594 L .70833 .59163 L .71825 .59623 L Mistroke .72817 .59971 L .73313 .60101 L .7381 .60203 L .74058 .60243 L .74306 .60276 L .74554 .60301 L .74678 .60311 L .74802 .60319 L .74926 .60325 L .7505 .60329 L .75174 .60331 L .75298 .60332 L .75422 .60331 L .75546 .60327 L .7567 .60322 L .75794 .60315 L .76042 .60295 L .7629 .60268 L .76786 .60191 L .77282 .60083 L .77778 .59945 L .7877 .59577 L .79762 .59085 L .81746 .57731 L .8373 .55888 L .85714 .53565 L .89683 .47549 L .93651 .39891 L .97619 .30902 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; Cclosed; preserveAspect; startGroup] Plot[{Sin[x],PartialProduct[x,5]},{x,-Pi,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"Sin[x], n=5", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.151576 0.309017 0.280754 [ [(-3)] .04527 .30902 0 2 Msboxa [(-2)] .19685 .30902 0 2 Msboxa [(-1)] .34842 .30902 0 2 Msboxa [(1)] .65158 .30902 0 2 Msboxa [(2)] .80315 .30902 0 2 Msboxa [(3)] .95473 .30902 0 2 Msboxa [(x)] 1.025 .30902 -1 0 Msboxa [(Sin[x], n=5)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(-1)] .4875 .02826 1 0 Msboxa [(-0.5)] .4875 .16864 1 0 Msboxa [(0.5)] .4875 .44939 1 0 Msboxa [(1)] .4875 .58977 1 0 Msboxa [(f)] .5 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .04527 .30902 m .04527 .31527 L s P [(-3)] .04527 .30902 0 2 Mshowa p .002 w .19685 .30902 m .19685 .31527 L s P [(-2)] .19685 .30902 0 2 Mshowa p .002 w .34842 .30902 m .34842 .31527 L s P [(-1)] .34842 .30902 0 2 Mshowa p .002 w .65158 .30902 m .65158 .31527 L s P [(1)] .65158 .30902 0 2 Mshowa p .002 w .80315 .30902 m .80315 .31527 L s P [(2)] .80315 .30902 0 2 Mshowa p .002 w .95473 .30902 m .95473 .31527 L s P [(3)] .95473 .30902 0 2 Mshowa p .001 w .07559 .30902 m .07559 .31277 L s P p .001 w .1059 .30902 m .1059 .31277 L s P p .001 w .13622 .30902 m .13622 .31277 L s P p .001 w .16653 .30902 m .16653 .31277 L s P p .001 w .22716 .30902 m .22716 .31277 L s P p .001 w .25748 .30902 m .25748 .31277 L s P p .001 w .28779 .30902 m .28779 .31277 L s P p .001 w .31811 .30902 m .31811 .31277 L s P p .001 w .37874 .30902 m .37874 .31277 L s P p .001 w .40905 .30902 m .40905 .31277 L s P p .001 w .43937 .30902 m .43937 .31277 L s P p .001 w .46968 .30902 m .46968 .31277 L s P p .001 w .53032 .30902 m .53032 .31277 L s P p .001 w .56063 .30902 m .56063 .31277 L s P p .001 w .59095 .30902 m .59095 .31277 L s P p .001 w .62126 .30902 m .62126 .31277 L s P p .001 w .68189 .30902 m .68189 .31277 L s P p .001 w .71221 .30902 m .71221 .31277 L s P p .001 w .74252 .30902 m .74252 .31277 L s P p .001 w .77284 .30902 m .77284 .31277 L s P p .001 w .83347 .30902 m .83347 .31277 L s P p .001 w .86378 .30902 m .86378 .31277 L s P p .001 w .8941 .30902 m .8941 .31277 L s P p .001 w .92441 .30902 m .92441 .31277 L s P p .001 w .01496 .30902 m .01496 .31277 L s P p .001 w .98504 .30902 m .98504 .31277 L s P [(x)] 1.025 .30902 -1 0 Mshowa p .002 w 0 .30902 m 1 .30902 L s P [(Sin[x], n=5)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .5 .02826 m .50625 .02826 L s P [(-1)] .4875 .02826 1 0 Mshowa p .002 w .5 .16864 m .50625 .16864 L s P [(-0.5)] .4875 .16864 1 0 Mshowa p .002 w .5 .44939 m .50625 .44939 L s P [(0.5)] .4875 .44939 1 0 Mshowa p .002 w .5 .58977 m .50625 .58977 L s P [(1)] .4875 .58977 1 0 Mshowa p .001 w .5 .05634 m .50375 .05634 L s P p .001 w .5 .08441 m .50375 .08441 L s P p .001 w .5 .11249 m .50375 .11249 L s P p .001 w .5 .14056 m .50375 .14056 L s P p .001 w .5 .19672 m .50375 .19672 L s P p .001 w .5 .22479 m .50375 .22479 L s P p .001 w .5 .25287 m .50375 .25287 L s P p .001 w .5 .28094 m .50375 .28094 L s P p .001 w .5 .33709 m .50375 .33709 L s P p .001 w .5 .36517 m .50375 .36517 L s P p .001 w .5 .39324 m .50375 .39324 L s P p .001 w .5 .42132 m .50375 .42132 L s P p .001 w .5 .47747 m .50375 .47747 L s P p .001 w .5 .50554 m .50375 .50554 L s P p .001 w .5 .53362 m .50375 .53362 L s P p .001 w .5 .5617 m .50375 .5617 L s P p .001 w .5 .00019 m .50375 .00019 L s P p .001 w .5 .61785 m .50375 .61785 L s P [(f)] .5 .61803 0 -4 Mshowa p .002 w .5 0 m .5 .61803 L s P P p p p .0075 w .02381 .30902 m .06349 .23635 L .10317 .16864 L .14286 .11049 L .1627 .08628 L .18254 .06588 L .20238 .04963 L 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.7629 .58602 L .76786 .58438 L .77778 .5802 L .79762 .5684 L .81746 .55216 L .85714 .50754 L .89683 .44939 L .93651 .38168 L .97619 .30902 L Mfstroke P P p p .005 w .02381 .30902 m .06349 .22433 L .10317 .14973 L .14286 .08911 L .1627 .065 L .18254 .04542 L .20238 .03055 L .2123 .02493 L .22222 .02054 L .22718 .0188 L .23214 .01737 L .2371 .01625 L .23958 .0158 L .24206 .01543 L .24454 .01514 L .24578 .01502 L .24702 .01492 L .24826 .01484 L .2495 .01478 L .25074 .01474 L .25198 .01472 L .25322 .01472 L .25446 .01473 L .2557 .01477 L .25694 .01482 L .25942 .01498 L .26066 .0151 L .2619 .01523 L .26438 .01554 L .26687 .01593 L .27183 .01694 L .27679 .01824 L .28175 .01985 L .29167 .02392 L .30159 .02915 L .32143 .04293 L .34127 .06093 L .38095 .10823 L .42063 .16793 L .46032 .23626 L .5 .30902 L .53968 .38177 L .57937 .4501 L .61905 .5098 L .63889 .53522 L .65873 .55711 L .67857 .5751 L .68849 .58254 L .69841 .58889 L .70833 .59411 L Mistroke .71825 .59819 L .72321 .59979 L .72817 .60109 L .73065 .60164 L .73313 .6021 L .73562 .60249 L .7381 .60281 L .74058 .60305 L .74182 .60314 L .74306 .60321 L .7443 .60327 L .74554 .6033 L .74678 .60332 L .74802 .60332 L .74926 .60329 L .7505 .60325 L .75174 .60319 L .75298 .60311 L .75546 .6029 L .75794 .6026 L .7629 .60179 L .76786 .60067 L .77778 .5975 L .7877 .5931 L .79762 .58748 L .81746 .57261 L .8373 .55303 L .85714 .52893 L .89683 .46831 L .93651 .3937 L .97619 .30902 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; Cclosed; preserveAspect; startGroup] Plot[{Sin[x],PartialProduct[x,10]},{x,-Pi,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"Sin[x], n=10", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.151576 0.309017 0.287246 [ [(-3)] .04527 .30902 0 2 Msboxa [(-2)] .19685 .30902 0 2 Msboxa [(-1)] .34842 .30902 0 2 Msboxa [(1)] .65158 .30902 0 2 Msboxa [(2)] .80315 .30902 0 2 Msboxa [(3)] .95473 .30902 0 2 Msboxa [(x)] 1.025 .30902 -1 0 Msboxa [(Sin[x], n=10)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(-1)] .4875 .02177 1 0 Msboxa [(-0.5)] .4875 .16539 1 0 Msboxa [(0.5)] .4875 .45264 1 0 Msboxa [(1)] .4875 .59626 1 0 Msboxa [(f)] .5 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .04527 .30902 m .04527 .31527 L s P [(-3)] .04527 .30902 0 2 Mshowa p .002 w .19685 .30902 m .19685 .31527 L s P [(-2)] .19685 .30902 0 2 Mshowa p .002 w .34842 .30902 m .34842 .31527 L s P [(-1)] .34842 .30902 0 2 Mshowa p .002 w .65158 .30902 m .65158 .31527 L s P [(1)] .65158 .30902 0 2 Mshowa p .002 w .80315 .30902 m .80315 .31527 L s P [(2)] .80315 .30902 0 2 Mshowa p .002 w .95473 .30902 m .95473 .31527 L s P [(3)] .95473 .30902 0 2 Mshowa p .001 w .07559 .30902 m .07559 .31277 L s P p .001 w .1059 .30902 m .1059 .31277 L s P p .001 w .13622 .30902 m .13622 .31277 L s P p .001 w .16653 .30902 m .16653 .31277 L s P p .001 w .22716 .30902 m .22716 .31277 L s P p .001 w .25748 .30902 m .25748 .31277 L s P p .001 w .28779 .30902 m .28779 .31277 L s P p .001 w .31811 .30902 m .31811 .31277 L s P p .001 w .37874 .30902 m .37874 .31277 L s P p .001 w .40905 .30902 m .40905 .31277 L s P p .001 w .43937 .30902 m .43937 .31277 L s P p .001 w .46968 .30902 m .46968 .31277 L s P p .001 w .53032 .30902 m .53032 .31277 L s P p .001 w .56063 .30902 m .56063 .31277 L s P p .001 w .59095 .30902 m .59095 .31277 L s P p .001 w .62126 .30902 m .62126 .31277 L s P p .001 w .68189 .30902 m .68189 .31277 L s P p .001 w .71221 .30902 m .71221 .31277 L s P p .001 w .74252 .30902 m .74252 .31277 L s P p .001 w .77284 .30902 m .77284 .31277 L s P p .001 w .83347 .30902 m .83347 .31277 L s P p .001 w .86378 .30902 m .86378 .31277 L s P p .001 w .8941 .30902 m .8941 .31277 L s P p .001 w .92441 .30902 m .92441 .31277 L s P p .001 w .01496 .30902 m .01496 .31277 L s P p .001 w .98504 .30902 m .98504 .31277 L s P [(x)] 1.025 .30902 -1 0 Mshowa p .002 w 0 .30902 m 1 .30902 L s P [(Sin[x], n=10)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .5 .02177 m .50625 .02177 L s P [(-1)] .4875 .02177 1 0 Mshowa p .002 w .5 .16539 m .50625 .16539 L s P [(-0.5)] .4875 .16539 1 0 Mshowa p .002 w .5 .45264 m .50625 .45264 L s P [(0.5)] .4875 .45264 1 0 Mshowa p .002 w .5 .59626 m .50625 .59626 L s P [(1)] .4875 .59626 1 0 Mshowa p .001 w .5 .0505 m .50375 .0505 L s P p .001 w .5 .07922 m .50375 .07922 L s P p .001 w .5 .10794 m .50375 .10794 L s P p .001 w .5 .13667 m .50375 .13667 L s P p .001 w .5 .19412 m .50375 .19412 L s P p .001 w .5 .22284 m .50375 .22284 L s P p .001 w .5 .25157 m .50375 .25157 L s P p .001 w .5 .28029 m .50375 .28029 L s P p .001 w .5 .33774 m .50375 .33774 L s P p .001 w .5 .36647 m .50375 .36647 L s P p .001 w .5 .39519 m .50375 .39519 L s P p .001 w .5 .42392 m .50375 .42392 L s P p .001 w .5 .48136 m .50375 .48136 L s P p .001 w .5 .51009 m .50375 .51009 L s P p .001 w .5 .53881 m .50375 .53881 L s P p .001 w .5 .56754 m .50375 .56754 L s P [(f)] .5 .61803 0 -4 Mshowa p .002 w .5 0 m .5 .61803 L s P P p p p .0075 w .02381 .30902 m .06349 .23467 L .10317 .16539 L .14286 .1059 L .1627 .08113 L .18254 .06025 L .20238 .04364 L .2123 .03702 L .22222 .03156 L .23214 .02729 L .2371 .02561 L .24206 .02423 L .24702 .02315 L .2495 .02273 L .25198 .02239 L .25446 .02212 L .2557 .02201 L .25694 .02192 L .25818 .02186 L .25942 .02181 L .26066 .02178 L .2619 .02177 L .26314 .02178 L .26438 .02181 L .26563 .02186 L .26687 .02192 L .26935 .02212 L .27183 .02239 L .27431 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.81746 .56853 L .85714 .52331 L .89683 .46246 L .93651 .38956 L .97619 .30902 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = text; inactive; preserveAspect] Apart from the points where x=n Pi, for integer n, where both Sin[x] and the partial products vanish, the convergence is fastest near the origin and moves outward. This is confirmed by the comparisons between the partial products and the series expansion of Sin[x] near x=0, where it was seen that the coefficients of the various powers of x in the partial products approach those of the series representation in ascending order of powers of x. We can identify the regions where the convergence is slowest by plotting the difference of Sin[x] and the partial products. :[font = input; Cclosed; preserveAspect; startGroup] Plot[{Sin[x]-PartialProduct[x,10], Sin[x]-PartialProduct[x,20], Sin[x]-PartialProduct[x,30], Sin[x]-PartialProduct[x,40], Sin[x]-PartialProduct[x,50]},{x,-Pi,Pi}, PlotRange->All,AxesLabel->{x,"Difference "}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.151576 0.309017 7.53535 [ [(-3)] .04527 .30902 0 2 Msboxa [(-2)] .19685 .30902 0 2 Msboxa [(-1)] .34842 .30902 0 2 Msboxa [(1)] .65158 .30902 0 2 Msboxa [(2)] .80315 .30902 0 2 Msboxa [(3)] .95473 .30902 0 2 Msboxa [(x)] 1.025 .30902 -1 0 Msboxa [(-0.04)] .4875 .0076 1 0 Msboxa [(-0.02)] .4875 .15831 1 0 Msboxa [(0.02)] 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.4876 .30903 L .48884 .30902 L .49008 .30902 L .49132 .30902 L .49256 .30902 L .4938 .30902 L .49504 .30902 L .5 .30902 L .50124 .30902 L .50248 .30902 L .50372 .30902 L .50496 .30902 L .5062 .30902 L .50744 .30902 L .50868 .30901 L .50992 .30901 L .5124 .30901 L .51364 .30901 L .51488 .309 L .51736 .30899 L .51984 .30898 L .52232 .30897 L .5248 .30895 L .52976 .3089 L .53472 .30884 L .53968 .30875 L .54464 .30864 L .5496 .3085 L .55952 .30812 L .56944 .30761 L .57937 .30694 L .59921 .30507 L .61905 .30242 L .63889 .29894 L .65873 .29464 L .69841 .28395 L .7381 .27162 L .77778 .25981 L .79762 .25496 L .80754 .25297 L .81746 .25133 L .82738 .2501 L .83234 .24965 L .83482 .24947 L Mistroke .8373 .24933 L .83978 .24921 L .84102 .24916 L .84226 .24912 L .8435 .24909 L .84474 .24907 L .84598 .24906 L .84722 .24905 L .84846 .24906 L .8497 .24907 L .85094 .24909 L .85218 .24912 L .85342 .24916 L .85466 .24921 L .85714 .24934 L .85962 .2495 L .8621 .2497 L .86706 .25021 L .87202 .25089 L .87698 .25173 L .8869 .25393 L .89683 .25685 L .91667 .26496 L .93651 .27629 L .95635 .29097 L .97619 .30902 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = text; inactive; preserveAspect] Thus, although the convergence is seen to be uniform, the region near x=2.25 is seen to be the slowest to converge. :[font = text; inactive; preserveAspect] To determine the number of terms in the partial product required to attain convergence at x=Pi/2 to within 1%, we have :[font = input; preserveAspect] PerCent[n_]:=N[Abs[ (Sin[Pi/2]-PartialProduct[Pi/2,n])/Sin[Pi/2]]] :[font = text; inactive; preserveAspect] We then simply compute this quantity until it is less then 0.01. :[font = input; preserveAspect] h[n_]:=If[Abs[PerCent[n]]<0.01,Print[n]] :[font = input; Cclosed; preserveAspect; startGroup] Do[h[n],{n,1,30}] :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 25 :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 26 :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 27 :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 28 :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 29 :[font = print; output; inactive; preserveAspect; endGroup] No Input Form Generated ;[o] 30 :[font = text; inactive; preserveAspect; endGroup] Thus, beyond 25 terms the partial products converge to Sin[Pi/2} to with 1%. :[font = section; inactive; Cclosed; preserveAspect; startGroup] Problem 7 :[font = text; inactive; preserveAspect] For each of the functions in Problem 3, we can use Mathematica to compute the require quantities for input into Parsevals theorem. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Problem 7(a) :[font = input; Cclosed; preserveAspect; startGroup] Clear[f,b,FCoefficients,PartialSum] f[x]=x(Pi-x); b[n_]=(2/Pi)Simplify[Integrate[f[x]Sin[n x],{x,0,Pi}]/.{ Sin[n Pi]->0,Cos[n Pi]->(-1)^n}] (2/Pi)Integrate[(f[x])^2,{x,0,Pi}] :[font = output; output; inactive; preserveAspect] (4*(1 - (-1)^n))/(n^3*Pi) ;[o] n 4 (1 - (-1) ) ------------- 3 n Pi :[font = output; output; inactive; preserveAspect; endGroup] Pi^4/15 ;[o] 4 Pi --- 15 :[font = input; preserveAspect] Clear[FCoefficients,PartialSum] FCoefficients[N_]:=Table[(1/(2n+1)^6),{n,0,N}]; PartialSum[N_]:=Apply[Plus,FCoefficients[N]] :[font = input; Cclosed; preserveAspect; startGroup] MatrixForm[Table[{n,N[Pi^6/960],N[PartialSum[n]]}, {n,1,10}]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 1.001447076640942, 1.001371742112483}, {2, 1.001447076640942, 1.001435742112483}, {3, 1.001447076640942, 1.001444241972235}, {4, 1.001447076640942, 1.001446123648658}, {5, 1.001447076640942, 1.001446688122588}, {6, 1.001447076640942, 1.001446895298799}, {7, 1.001447076640942, 1.001446983090295}, {8, 1.001447076640942, 1.001447024519487}, {9, 1.001447076640942, 1.001447045775333}, {10, 1.001447076640942, 1.001447057434949}}] ;[o] 1 1.00145 1.00137 2 1.00145 1.00144 3 1.00145 1.00144 4 1.00145 1.00145 5 1.00145 1.00145 6 1.00145 1.00145 7 1.00145 1.00145 8 1.00145 1.00145 9 1.00145 1.00145 10 1.00145 1.00145 :[font = text; inactive; preserveAspect] The convergence is seen to be very rapid, with the partial sums converging tofive decimal places after only four terms. To have a visual display of the convergence, we form a Table of values and then use ListPlot. :[font = input; Cclosed; preserveAspect; startGroup] PS=Table[PartialSum[n],{n,1,50}]; ListPlot[PS,PlotRange->{0,Pi^6/960},AxesLabel->{n,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0190476 -8.99888e-18 0.617141 [ [(0)] .02381 0 0 2 Msboxa [(10)] .21429 0 0 2 Msboxa [(20)] .40476 0 0 2 Msboxa [(30)] .59524 0 0 2 Msboxa [(40)] .78571 0 0 2 Msboxa [(50)] .97619 0 0 2 Msboxa [(n)] 1.025 0 -1 0 Msboxa [(0.2)] .01131 .12343 1 0 Msboxa [(0.4)] .01131 .24686 1 0 Msboxa [(0.6)] .01131 .37028 1 0 Msboxa [(0.8)] .01131 .49371 1 0 Msboxa [(1)] .01131 .61714 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 0 m .02381 .00625 L s P [(0)] .02381 0 0 2 Mshowa p .002 w .21429 0 m .21429 .00625 L s P [(10)] .21429 0 0 2 Mshowa p .002 w .40476 0 m .40476 .00625 L s P [(20)] .40476 0 0 2 Mshowa p .002 w .59524 0 m .59524 .00625 L s P [(30)] .59524 0 0 2 Mshowa p .002 w .78571 0 m .78571 .00625 L s P [(40)] .78571 0 0 2 Mshowa p .002 w .97619 0 m .97619 .00625 L s P [(50)] .97619 0 0 2 Mshowa p .001 w .0619 0 m .0619 .00375 L s P p .001 w .1 0 m .1 .00375 L s P p .001 w .1381 0 m .1381 .00375 L s P p .001 w .17619 0 m .17619 .00375 L s P p .001 w .25238 0 m .25238 .00375 L s P p .001 w .29048 0 m .29048 .00375 L s P p .001 w .32857 0 m .32857 .00375 L s P p .001 w .36667 0 m .36667 .00375 L s P p .001 w .44286 0 m .44286 .00375 L s P p .001 w .48095 0 m .48095 .00375 L s P p .001 w .51905 0 m .51905 .00375 L s P p .001 w .55714 0 m .55714 .00375 L s P p .001 w .63333 0 m .63333 .00375 L s P p .001 w .67143 0 m .67143 .00375 L s P p .001 w .70952 0 m .70952 .00375 L s P p .001 w .74762 0 m .74762 .00375 L s P p .001 w .82381 0 m .82381 .00375 L s P p .001 w .8619 0 m .8619 .00375 L s P p .001 w .9 0 m .9 .00375 L s P p .001 w .9381 0 m .9381 .00375 L s P [(n)] 1.025 0 -1 0 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w .02381 .12343 m .03006 .12343 L s P [(0.2)] .01131 .12343 1 0 Mshowa p .002 w .02381 .24686 m .03006 .24686 L s P [(0.4)] .01131 .24686 1 0 Mshowa p .002 w .02381 .37028 m .03006 .37028 L s P [(0.6)] .01131 .37028 1 0 Mshowa p .002 w .02381 .49371 m .03006 .49371 L s P [(0.8)] .01131 .49371 1 0 Mshowa p .002 w .02381 .61714 m .03006 .61714 L s P [(1)] .01131 .61714 1 0 Mshowa p .001 w .02381 .02469 m .02756 .02469 L s P p .001 w .02381 .04937 m .02756 .04937 L s P p .001 w .02381 .07406 m .02756 .07406 L s P p .001 w .02381 .09874 m .02756 .09874 L s P p .001 w .02381 .14811 m .02756 .14811 L s P p .001 w .02381 .1728 m .02756 .1728 L s P p .001 w .02381 .19749 m .02756 .19749 L s P p .001 w .02381 .22217 m .02756 .22217 L s P p .001 w .02381 .27154 m .02756 .27154 L s P p .001 w .02381 .29623 m .02756 .29623 L s P p .001 w .02381 .32091 m .02756 .32091 L s P p .001 w .02381 .3456 m .02756 .3456 L s P p .001 w .02381 .39497 m .02756 .39497 L s P p .001 w .02381 .41966 m .02756 .41966 L s P p .001 w .02381 .44434 m .02756 .44434 L s P p .001 w .02381 .46903 m .02756 .46903 L s P p .001 w .02381 .5184 m .02756 .5184 L s P p .001 w .02381 .54308 m .02756 .54308 L s P p .001 w .02381 .56777 m .02756 .56777 L s P p .001 w .02381 .59246 m .02756 .59246 L s P [(f)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p .008 w .04286 .61799 Mdot .0619 .61803 Mdot .08095 .61803 Mdot .1 .61803 Mdot .11905 .61803 Mdot .1381 .61803 Mdot .15714 .61803 Mdot .17619 .61803 Mdot .19524 .61803 Mdot .21429 .61803 Mdot .23333 .61803 Mdot .25238 .61803 Mdot .27143 .61803 Mdot .29048 .61803 Mdot .30952 .61803 Mdot .32857 .61803 Mdot .34762 .61803 Mdot .36667 .61803 Mdot .38571 .61803 Mdot .40476 .61803 Mdot .42381 .61803 Mdot .44286 .61803 Mdot .4619 .61803 Mdot .48095 .61803 Mdot .5 .61803 Mdot .51905 .61803 Mdot .5381 .61803 Mdot .55714 .61803 Mdot .57619 .61803 Mdot .59524 .61803 Mdot .61429 .61803 Mdot .63333 .61803 Mdot .65238 .61803 Mdot .67143 .61803 Mdot .69048 .61803 Mdot .70952 .61803 Mdot .72857 .61803 Mdot .74762 .61803 Mdot .76667 .61803 Mdot .78571 .61803 Mdot .80476 .61803 Mdot .82381 .61803 Mdot .84286 .61803 Mdot .8619 .61803 Mdot .88095 .61803 Mdot .9 .61803 Mdot .91905 .61803 Mdot .9381 .61803 Mdot .95714 .61803 Mdot .97619 .61803 Mdot P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Problem 7(b) :[font = input; Cclosed; preserveAspect; startGroup] Clear[f,b,FCoefficients,PartialSum] f[x]=x; b[n_]=(2/Pi)Simplify[Integrate[f[x]Sin[n x],{x,0,Pi}]/.{ Sin[n Pi]->0,Cos[n Pi]->(-1)^n}] (2/Pi)Integrate[(f[x])^2,{x,0,Pi}] :[font = output; output; inactive; preserveAspect] (-2*(-1)^n)/n ;[o] n -2 (-1) -------- n :[font = output; output; inactive; preserveAspect; endGroup] (2*Pi^2)/3 ;[o] 2 2 Pi ----- 3 :[font = input; preserveAspect] Clear[FCoefficients,PartialSum] FCoefficients[N_]:=Table[1/n^2,{n,1,N}]; PartialSum[N_]:=Apply[Plus,FCoefficients[N]] :[font = input; Cclosed; preserveAspect; startGroup] MatrixForm[Table[{n,N[Pi^2/6],N[PartialSum[n]]}, {n,1,100,10}]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 1.644934066848226, 1.}, {11, 1.644934066848226, 1.558032193976458}, {21, 1.644934066848226, 1.598430817609168}, {31, 1.644934066848226, 1.613190700327924}, {41, 1.644934066848226, 1.620838847004556}, {51, 1.644934066848226, 1.625517201134024}, {61, 1.644934066848226, 1.628674262469401}, {71, 1.644934066848226, 1.630948280828643}, {81, 1.644934066848226, 1.63266428212784}, {91, 1.644934066848226, 1.634005213876653}}] ;[o] 1 1.64493 1. 11 1.64493 1.55803 21 1.64493 1.59843 31 1.64493 1.61319 41 1.64493 1.62084 51 1.64493 1.62552 61 1.64493 1.62867 71 1.64493 1.63095 81 1.64493 1.63266 91 1.64493 1.63401 :[font = input; Cclosed; preserveAspect; startGroup] {1000,N[Pi^2/6],N[PartialSum[1000]]} :[font = output; output; inactive; preserveAspect; endGroup] {1000, 1.644934066848226, 1.6439345666815598} ;[o] {1000, 1.64493, 1.6439345666815598} :[font = input; Cclosed; preserveAspect; startGroup] {5000,N[Pi^2/6],N[PartialSum[5000]]} :[font = output; output; inactive; preserveAspect; endGroup] {5000, 1.644934066848226, 1.6447340868468931} ;[o] {5000, 1.64493, 1.6447340868468931} :[font = text; inactive; preserveAspect] The convergence is much slower than in Problem 7(a), as even after 1000 terms, the partial sums have converged only to two decimal places. After 5000 terms, the series has converged only to three decimal places. :[font = input; Cclosed; preserveAspect; startGroup] PS=Table[PartialSum[n],{n,1,50}]; ListPlot[PS,PlotRange->{0,Pi^2/6},AxesLabel->{n,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0190476 -1.16281e-17 0.37572 [ [(0)] .02381 0 0 2 Msboxa [(10)] .21429 0 0 2 Msboxa [(20)] .40476 0 0 2 Msboxa [(30)] .59524 0 0 2 Msboxa [(40)] .78571 0 0 2 Msboxa [(50)] .97619 0 0 2 Msboxa [(n)] 1.025 0 -1 0 Msboxa [(0.25)] .01131 .09393 1 0 Msboxa [(0.5)] .01131 .18786 1 0 Msboxa [(0.75)] .01131 .28179 1 0 Msboxa [(1)] .01131 .37572 1 0 Msboxa [(1.25)] .01131 .46965 1 0 Msboxa [(1.5)] .01131 .56358 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 0 m .02381 .00625 L s P [(0)] .02381 0 0 2 Mshowa p .002 w .21429 0 m .21429 .00625 L s P [(10)] .21429 0 0 2 Mshowa p .002 w .40476 0 m .40476 .00625 L s P [(20)] .40476 0 0 2 Mshowa p .002 w .59524 0 m .59524 .00625 L s P [(30)] .59524 0 0 2 Mshowa p .002 w .78571 0 m .78571 .00625 L s P [(40)] .78571 0 0 2 Mshowa p .002 w .97619 0 m .97619 .00625 L s P [(50)] .97619 0 0 2 Mshowa p .001 w .0619 0 m .0619 .00375 L s P p .001 w .1 0 m .1 .00375 L s P p .001 w .1381 0 m .1381 .00375 L s P p .001 w .17619 0 m .17619 .00375 L s P p .001 w .25238 0 m .25238 .00375 L s P p .001 w .29048 0 m .29048 .00375 L s P p .001 w .32857 0 m .32857 .00375 L s P p .001 w .36667 0 m .36667 .00375 L s P p .001 w .44286 0 m .44286 .00375 L s P p .001 w .48095 0 m .48095 .00375 L s P p .001 w .51905 0 m .51905 .00375 L s P p .001 w .55714 0 m .55714 .00375 L s P p .001 w .63333 0 m .63333 .00375 L s P p .001 w .67143 0 m .67143 .00375 L s P p .001 w .70952 0 m .70952 .00375 L s P p .001 w .74762 0 m .74762 .00375 L s P p .001 w .82381 0 m .82381 .00375 L s P p .001 w .8619 0 m .8619 .00375 L s P p .001 w .9 0 m .9 .00375 L s P p .001 w .9381 0 m .9381 .00375 L s P [(n)] 1.025 0 -1 0 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w .02381 .09393 m .03006 .09393 L s P [(0.25)] .01131 .09393 1 0 Mshowa p .002 w .02381 .18786 m .03006 .18786 L s P [(0.5)] .01131 .18786 1 0 Mshowa p .002 w .02381 .28179 m .03006 .28179 L s P [(0.75)] .01131 .28179 1 0 Mshowa p .002 w .02381 .37572 m .03006 .37572 L s P [(1)] .01131 .37572 1 0 Mshowa p .002 w .02381 .46965 m .03006 .46965 L s P [(1.25)] .01131 .46965 1 0 Mshowa p .002 w .02381 .56358 m .03006 .56358 L s P [(1.5)] .01131 .56358 1 0 Mshowa p .001 w .02381 .01879 m .02756 .01879 L s P p .001 w .02381 .03757 m .02756 .03757 L s P p .001 w .02381 .05636 m .02756 .05636 L s P p .001 w .02381 .07514 m .02756 .07514 L s P p .001 w .02381 .11272 m .02756 .11272 L s P p .001 w .02381 .1315 m .02756 .1315 L s P p .001 w .02381 .15029 m .02756 .15029 L s P p .001 w .02381 .16907 m .02756 .16907 L s P p .001 w .02381 .20665 m .02756 .20665 L s P p .001 w .02381 .22543 m .02756 .22543 L s P p .001 w .02381 .24422 m .02756 .24422 L s P p .001 w .02381 .263 m .02756 .263 L s P p .001 w .02381 .30058 m .02756 .30058 L s P p .001 w .02381 .31936 m .02756 .31936 L s P p .001 w .02381 .33815 m .02756 .33815 L s P p .001 w .02381 .35693 m .02756 .35693 L s P p .001 w .02381 .39451 m .02756 .39451 L s P p .001 w .02381 .41329 m .02756 .41329 L s P p .001 w .02381 .43208 m .02756 .43208 L s P p .001 w .02381 .45086 m .02756 .45086 L s P p .001 w .02381 .48844 m .02756 .48844 L s P p .001 w .02381 .50722 m .02756 .50722 L s P p .001 w .02381 .52601 m .02756 .52601 L s P p .001 w .02381 .54479 m .02756 .54479 L s P p .001 w .02381 .58237 m .02756 .58237 L s P p .001 w .02381 .60115 m .02756 .60115 L s P [(f)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p .008 w .04286 .37572 Mdot .0619 .46965 Mdot .08095 .5114 Mdot .1 .53488 Mdot .11905 .54991 Mdot .1381 .56034 Mdot .15714 .56801 Mdot .17619 .57388 Mdot .19524 .57852 Mdot .21429 .58228 Mdot .23333 .58538 Mdot .25238 .58799 Mdot .27143 .59022 Mdot .29048 .59213 Mdot .30952 .5938 Mdot .32857 .59527 Mdot .34762 .59657 Mdot .36667 .59773 Mdot .38571 .59877 Mdot .40476 .59971 Mdot .42381 .60056 Mdot .44286 .60134 Mdot .4619 .60205 Mdot .48095 .6027 Mdot .5 .6033 Mdot .51905 .60386 Mdot .5381 .60437 Mdot .55714 .60485 Mdot .57619 .6053 Mdot .59524 .60572 Mdot .61429 .60611 Mdot .63333 .60647 Mdot .65238 .60682 Mdot .67143 .60714 Mdot .69048 .60745 Mdot .70952 .60774 Mdot .72857 .60802 Mdot .74762 .60828 Mdot .76667 .60852 Mdot .78571 .60876 Mdot .80476 .60898 Mdot .82381 .60919 Mdot .84286 .6094 Mdot .8619 .60959 Mdot .88095 .60978 Mdot .9 .60995 Mdot .91905 .61012 Mdot .9381 .61029 Mdot .95714 .61044 Mdot .97619 .61059 Mdot P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Problem 7(c) :[font = input; Cclosed; preserveAspect; startGroup] Clear[f,b,FCoefficients,PartialSum] f[x]=1; b[n_]=(2/Pi)Simplify[Integrate[f[x]Sin[n x],{x,0,Pi}]/.{ Sin[n Pi]->0,Cos[n Pi]->(-1)^n}] (2/Pi)Integrate[(f[x])^2,{x,0,Pi}] :[font = output; output; inactive; preserveAspect] (2*(1 - (-1)^n))/(n*Pi) ;[o] n 2 (1 - (-1) ) ------------- n Pi :[font = output; output; inactive; preserveAspect; endGroup] 2 ;[o] 2 :[font = input; preserveAspect] Clear[FCoefficients,PartialSum] FCoefficients[N_]:=Table[1/(2n+1)^2,{n,0,N}]; PartialSum[N_]:=Apply[Plus,FCoefficients[N]] :[font = input; Cclosed; preserveAspect; startGroup] MatrixForm[Table[{n,N[Pi^2/8],N[PartialSum[n]]}, {n,1,100,10}]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{1, 1.23370055013617, 1.111111111111111}, {11, 1.23370055013617, 1.212879243985775}, {21, 1.23370055013617, 1.222338868908939}, {31, 1.23370055013617, 1.225888685701906}, {41, 1.23370055013617, 1.227748450325156}, {51, 1.23370055013617, 1.228893005975365}, {61, 1.23370055013617, 1.22966837947824}, {71, 1.23370055013617, 1.230228383726509}, {81, 1.23370055013617, 1.230651807431205}, {91, 1.23370055013617, 1.23098318558512}}] ;[o] 1 1.2337 1.11111 11 1.2337 1.21288 21 1.2337 1.22234 31 1.2337 1.22589 41 1.2337 1.22775 51 1.2337 1.22889 61 1.2337 1.22967 71 1.2337 1.23023 81 1.2337 1.23065 91 1.2337 1.23098 :[font = input; Cclosed; preserveAspect; startGroup] {1000,N[Pi^2/8],N[PartialSum[1000]]} :[font = output; output; inactive; preserveAspect; endGroup] {1000, 1.23370055013617, 1.2334507999071905} ;[o] {1000, 1.2337, 1.2334507999071905} :[font = text; inactive; preserveAspect] The convergence here is seen to be somewhat faster than that in Problem 7(b). :[font = input; Cclosed; preserveAspect; startGroup] PS=Table[PartialSum[n],{n,1,50}]; ListPlot[PS,PlotRange->{0,Pi^2/8},AxesLabel->{n,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0190476 6.20706e-18 0.500959 [ [(0)] .02381 0 0 2 Msboxa [(10)] .21429 0 0 2 Msboxa [(20)] .40476 0 0 2 Msboxa [(30)] .59524 0 0 2 Msboxa [(40)] .78571 0 0 2 Msboxa [(50)] .97619 0 0 2 Msboxa [(n)] 1.025 0 -1 0 Msboxa [(0.2)] .01131 .10019 1 0 Msboxa [(0.4)] .01131 .20038 1 0 Msboxa [(0.6)] .01131 .30058 1 0 Msboxa [(0.8)] .01131 .40077 1 0 Msboxa [(1)] .01131 .50096 1 0 Msboxa [(1.2)] .01131 .60115 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 0 m .02381 .00625 L s P [(0)] .02381 0 0 2 Mshowa p .002 w .21429 0 m .21429 .00625 L s P [(10)] .21429 0 0 2 Mshowa p .002 w .40476 0 m .40476 .00625 L s P [(20)] .40476 0 0 2 Mshowa p .002 w .59524 0 m .59524 .00625 L s P [(30)] .59524 0 0 2 Mshowa p .002 w .78571 0 m .78571 .00625 L s P [(40)] .78571 0 0 2 Mshowa p .002 w .97619 0 m .97619 .00625 L s P [(50)] .97619 0 0 2 Mshowa p .001 w .0619 0 m .0619 .00375 L s P p .001 w .1 0 m .1 .00375 L s P p .001 w .1381 0 m .1381 .00375 L s P p .001 w .17619 0 m .17619 .00375 L s P p .001 w .25238 0 m .25238 .00375 L s P p .001 w .29048 0 m .29048 .00375 L s P p .001 w .32857 0 m .32857 .00375 L s P p .001 w .36667 0 m .36667 .00375 L s P p .001 w .44286 0 m .44286 .00375 L s P p .001 w .48095 0 m .48095 .00375 L s P p .001 w .51905 0 m .51905 .00375 L s P p .001 w .55714 0 m .55714 .00375 L s P p .001 w .63333 0 m .63333 .00375 L s P p .001 w .67143 0 m .67143 .00375 L s P p .001 w .70952 0 m .70952 .00375 L s P p .001 w .74762 0 m .74762 .00375 L s P p .001 w .82381 0 m .82381 .00375 L s P p .001 w .8619 0 m .8619 .00375 L s P p .001 w .9 0 m .9 .00375 L s P p .001 w .9381 0 m .9381 .00375 L s P [(n)] 1.025 0 -1 0 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w .02381 .10019 m .03006 .10019 L s P [(0.2)] .01131 .10019 1 0 Mshowa p .002 w .02381 .20038 m .03006 .20038 L s P [(0.4)] .01131 .20038 1 0 Mshowa p .002 w .02381 .30058 m .03006 .30058 L s P [(0.6)] .01131 .30058 1 0 Mshowa p .002 w .02381 .40077 m .03006 .40077 L s P [(0.8)] .01131 .40077 1 0 Mshowa p .002 w .02381 .50096 m .03006 .50096 L s P [(1)] .01131 .50096 1 0 Mshowa p .002 w .02381 .60115 m .03006 .60115 L s P [(1.2)] .01131 .60115 1 0 Mshowa p .001 w .02381 .02004 m .02756 .02004 L s P p .001 w .02381 .04008 m .02756 .04008 L s P p .001 w .02381 .06012 m .02756 .06012 L s P p .001 w .02381 .08015 m .02756 .08015 L s P p .001 w .02381 .12023 m .02756 .12023 L s P p .001 w .02381 .14027 m .02756 .14027 L s P p .001 w .02381 .16031 m .02756 .16031 L s P p .001 w .02381 .18035 m .02756 .18035 L s P p .001 w .02381 .22042 m .02756 .22042 L s P p .001 w .02381 .24046 m .02756 .24046 L s P p .001 w .02381 .2605 m .02756 .2605 L s P p .001 w .02381 .28054 m .02756 .28054 L s P p .001 w .02381 .32061 m .02756 .32061 L s P p .001 w .02381 .34065 m .02756 .34065 L s P p .001 w .02381 .36069 m .02756 .36069 L s P p .001 w .02381 .38073 m .02756 .38073 L s P p .001 w .02381 .42081 m .02756 .42081 L s P p .001 w .02381 .44084 m .02756 .44084 L s P p .001 w .02381 .46088 m .02756 .46088 L s P p .001 w .02381 .48092 m .02756 .48092 L s P p .001 w .02381 .521 m .02756 .521 L s P p .001 w .02381 .54104 m .02756 .54104 L s P p .001 w .02381 .56107 m .02756 .56107 L s P p .001 w .02381 .58111 m .02756 .58111 L s P [(f)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p .008 w .04286 .55662 Mdot .0619 .57666 Mdot .08095 .58688 Mdot .1 .59307 Mdot .11905 .59721 Mdot .1381 .60017 Mdot .15714 .6024 Mdot .17619 .60413 Mdot .19524 .60552 Mdot .21429 .60666 Mdot .23333 .6076 Mdot .25238 .6084 Mdot .27143 .60909 Mdot .29048 .60969 Mdot .30952 .61021 Mdot .32857 .61067 Mdot .34762 .61108 Mdot .36667 .61144 Mdot .38571 .61177 Mdot .40476 .61207 Mdot .42381 .61234 Mdot .44286 .61259 Mdot .4619 .61282 Mdot .48095 .61303 Mdot .5 .61322 Mdot .51905 .6134 Mdot .5381 .61356 Mdot .55714 .61372 Mdot .57619 .61386 Mdot .59524 .61399 Mdot .61429 .61412 Mdot .63333 .61424 Mdot .65238 .61435 Mdot .67143 .61446 Mdot .69048 .61456 Mdot .70952 .61465 Mdot .72857 .61474 Mdot .74762 .61482 Mdot .76667 .6149 Mdot .78571 .61498 Mdot .80476 .61505 Mdot .82381 .61512 Mdot .84286 .61519 Mdot .8619 .61525 Mdot .88095 .61531 Mdot .9 .61537 Mdot .91905 .61542 Mdot .9381 .61548 Mdot .95714 .61553 Mdot .97619 .61558 Mdot P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = section; inactive; Cclosed; preserveAspect; startGroup] Problem 8 :[font = text; inactive; preserveAspect] The Fourier series representation of f[x]=x was derived in Ptoblem 3(b). The partial sums at x=Pi are thus calculated from :[font = input; preserveAspect] Clear[f,b,FCoefficients,PartialSum] f[x]=x; b[n_]=(2/Pi)Integrate[f[x]Sin[n x],{x,0,Pi}]; FCoefficients[x_,N_]:=Table[b[n]Sin[n x],{n,1,N}]; PartialSum[x_,N_]:=Apply[Plus,FCoefficients[x,N]] :[font = text; inactive; preserveAspect] These partial sums can be used to display the behavior of the Fourier series near the discontinuity at x=Pi, as in Fig. 4.5: :[font = input; Cclosed; preserveAspect; startGroup] Plot[{x,PartialSum[x,20],PartialSum[x,40],PartialSum[x,500]}, {x,2.9,Pi},PlotRange->{2.6,3.7},AxesLabel->{x,Sn}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}, {Thickness[0.005]},{Thickness[0.005]}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -11.4083 3.94209 -1.46081 0.561849 [ [(2.9)] .02381 .22474 0 2 Msboxa [(2.95)] .22091 .22474 0 2 Msboxa [(3.05)] .61512 .22474 0 2 Msboxa [(3.1)] .81223 .22474 0 2 Msboxa [(x)] 1.025 .22474 -1 0 Msboxa [(2.6)] .40552 0 1 0 Msboxa [(2.8)] .40552 .11237 1 0 Msboxa [(3.2)] .40552 .33711 1 0 Msboxa [(3.4)] .40552 .44948 1 0 Msboxa [(3.6)] .40552 .56185 1 0 Msboxa [(Sn)] .41802 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .22474 m .02381 .23099 L s P [(2.9)] .02381 .22474 0 2 Mshowa p .002 w .22091 .22474 m .22091 .23099 L s P [(2.95)] .22091 .22474 0 2 Mshowa p .002 w .61512 .22474 m .61512 .23099 L s P [(3.05)] .61512 .22474 0 2 Mshowa p .002 w .81223 .22474 m .81223 .23099 L s P [(3.1)] .81223 .22474 0 2 Mshowa p .001 w .06323 .22474 m .06323 .22849 L s P p .001 w .10265 .22474 m .10265 .22849 L s P p .001 w .14207 .22474 m .14207 .22849 L s P p .001 w .18149 .22474 m .18149 .22849 L s 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.91915 .19394 L .92039 .1751 L .92163 .15859 L .92287 .14494 L .92411 .13464 L .92535 .12811 L .92659 .12571 L .92783 .12772 L .92907 .13432 L .93031 .14558 L .93155 .16149 L .93403 .20656 L Mistroke .93651 .26704 L .94147 .41532 L .94395 .48987 L .94519 .52377 L .94643 .554 L .94767 .57946 L .94891 .59909 L .95015 .61182 L .95139 .61667 L .95263 .6127 L .95387 .59906 L .95511 .57502 L .95635 .53994 L .95759 .49333 L .95883 .43484 L .96131 .28154 L .96379 .08035 L Mfstroke .9646 0 m .96379 .08035 L s s s P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = text; inactive; preserveAspect] To calculate the jump near x=Pi as the number of terms in the partial sum increases, we generate a list of values the partial sums take near x=Pi and deteremine the maximum value in this list for a large enough number of increments. We first evaluate :[font = input; Cclosed; preserveAspect; startGroup] Integrate[Sin[x]/x,{x,0,Pi}] :[font = output; output; inactive; preserveAspect; endGroup] SinIntegral[Pi] ;[o] SinIntegral[Pi] :[font = text; inactive; preserveAspect] The maximum of the partial sums near x=Pi is then determined from :[font = input; preserveAspect] Gibbs[n_,m_]:= Max[Table[N[PartialSum[x,n]],{x,2.9,Pi,(Pi-2.9)/m}]]; :[font = input; Cclosed; preserveAspect; startGroup] MatrixForm[Table[{m,N[2 SinIntegral[Pi]],Gibbs[20,m]}, {m,100,1000,100}]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{100, 3.703874103964932, 3.553082043611932}, {200, 3.703874103964932, 3.553082043611932}, {300, 3.703874103964932, 3.553082043611932}, {400, 3.703874103964932, 3.553082043611932}, {500, 3.703874103964932, 3.553082043611932}, {600, 3.703874103964932, 3.553082043611932}, {700, 3.703874103964932, 3.553083510650877}, {800, 3.703874103964932, 3.553085153311692}, {900, 3.703874103964932, 3.553086069855425}, {1000, 3.703874103964932, 3.553086575701897}}] ;[o] 100 3.70387 3.55308 200 3.70387 3.55308 300 3.70387 3.55308 400 3.70387 3.55308 500 3.70387 3.55308 600 3.70387 3.55308 700 3.70387 3.55308 800 3.70387 3.55309 900 3.70387 3.55309 1000 3.70387 3.55309 :[font = input; Cclosed; preserveAspect; startGroup] MatrixForm[Table[{m,N[2 SinIntegral[Pi]],Gibbs[50,m]}, {m,100,1000,100}]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{100, 3.703874103964932, 3.640868081199415}, {200, 3.703874103964932, 3.642072903758534}, {300, 3.703874103964932, 3.641942364503053}, {400, 3.703874103964932, 3.642072903758534}, {500, 3.703874103964932, 3.642026992400462}, {600, 3.703874103964932, 3.642072903758534}, {700, 3.703874103964932, 3.642050008808254}, {800, 3.703874103964932, 3.642072903758534}, {900, 3.703874103964932, 3.642059369173972}, {1000, 3.703874103964932, 3.642072903758534}}] ;[o] 100 3.70387 3.64087 200 3.70387 3.64207 300 3.70387 3.64194 400 3.70387 3.64207 500 3.70387 3.64203 600 3.70387 3.64207 700 3.70387 3.64205 800 3.70387 3.64207 900 3.70387 3.64206 1000 3.70387 3.64207 :[font = input; Cclosed; preserveAspect; startGroup] MatrixForm[Table[{m,N[2 SinIntegral[Pi]],Gibbs[500,m]}, {m,100,1000,100}]] :[font = output; output; inactive; preserveAspect; endGroup] MatrixForm[{{100, 3.703874103964932, 3.629724231341945}, {200, 3.703874103964932, 3.69324237002454}, {300, 3.703874103964932, 3.695285933180427}, {400, 3.703874103964932, 3.69324237002454}, {500, 3.703874103964932, 3.697592125753005}, {600, 3.703874103964932, 3.695285933180427}, {700, 3.703874103964932, 3.697328511331915}, {800, 3.703874103964932, 3.697199828782699}, {900, 3.703874103964932, 3.696848110145952}, {1000, 3.703874103964932, 3.697592125753005}}] ;[o] 100 3.70387 3.62972 200 3.70387 3.69324 300 3.70387 3.69529 400 3.70387 3.69324 500 3.70387 3.69759 600 3.70387 3.69529 700 3.70387 3.69733 800 3.70387 3.6972 900 3.70387 3.69685 1000 3.70387 3.69759 :[font = input; Cclosed; preserveAspect; startGroup] MatrixForm[Table[{m,N[2 SinIntegral[Pi]],Gibbs[1000,m]}, {m,100,1000,100}]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] MatrixForm[{{100, 3.703874103964932, 3.512075539198599}, {200, 3.703874103964932, 3.633813310013984}, {300, 3.703874103964932, 3.698588292854349}, {400, 3.703874103964932, 3.69614245326931}, {500, 3.703874103964932, 3.682785227746101}, {600, 3.703874103964932, 3.698588292854349}, {700, 3.703874103964932, 3.70040087841328}, {800, 3.703874103964932, 3.69614245326931}, {900, 3.703874103964932, 3.698588292854349}, {1000, 3.703874103964932, 3.700733512889075}}] ;[o] 100 3.70387 3.51208 200 3.70387 3.63381 300 3.70387 3.69859 400 3.70387 3.69614 500 3.70387 3.68279 600 3.70387 3.69859 700 3.70387 3.7004 800 3.70387 3.69614 900 3.70387 3.69859 1000 3.70387 3.70073 :[font = section; inactive; Cclosed; preserveAspect; startGroup] Problem 11 :[font = text; inactive; preserveAspect] We first enter the Dirichlet kernel calculated in Problem 10. For the purposes of integrating this function, the integral is taken inside the summation. :[font = input; preserveAspect] DirichletIntegral[N_]:=(1/(2 Pi))* Sum[Integrate[Exp[I n x],{x,-Pi,Pi}],{n,-N,N}] :[font = text; inactive; preserveAspect] We can now evaluate this integral for various values of N: :[font = input; Cclosed; preserveAspect; startGroup] Do[Print[DirichletIntegral[N]],{N,1,10}] :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 1 :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 1 :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 1 :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 1 :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 1 :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 1 :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 1 :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 1 :[font = print; output; inactive; preserveAspect] No Input Form Generated ;[o] 1 :[font = print; output; inactive; preserveAspect; endGroup] No Input Form Generated ;[o] 1 :[font = text; inactive; preserveAspect] As shown in the solution manual, the value of this integral is unity for integer N because only the n=0 term contributes. To plot the Dirichlet kernel for increasing values of N, the summed form is most expedient: :[font = input; preserveAspect] Dirichlet[N_,x_]:=Sin[(N+(1/2))x]/Sin[x/2] :[font = text; inactive; preserveAspect] The plot of this function over the interval (-Pi,Pi) for increasing N is obtained as follows: :[font = input; Cclosed; preserveAspect; startGroup] Do[Plot[Dirichlet[N,x],{x,-Pi,Pi},PlotRange->All, AxesLabel->{x,"DirichletKernel"[N,x]}],{N,1,101,25}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185; startGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.151576 0.161866 0.147151 [ [(-3)] .04527 .16187 0 2 Msboxa [(-2)] .19685 .16187 0 2 Msboxa [(-1)] .34842 .16187 0 2 Msboxa [(1)] .65158 .16187 0 2 Msboxa [(2)] .80315 .16187 0 2 Msboxa [(3)] .95473 .16187 0 2 Msboxa [(x)] 1.025 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[(20)] .4875 .30221 1 0 Msboxa [(30)] .4875 .39345 1 0 Msboxa [(40)] .4875 .4847 1 0 Msboxa [(50)] .4875 .57595 1 0 Msboxa [(DirichletKernel[26, x])] .5 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .04527 .11972 m .04527 .12597 L s P [(-3)] .04527 .11972 0 2 Mshowa p .002 w .19685 .11972 m .19685 .12597 L s P [(-2)] .19685 .11972 0 2 Mshowa p .002 w .34842 .11972 m .34842 .12597 L s P [(-1)] .34842 .11972 0 2 Mshowa p .002 w .65158 .11972 m .65158 .12597 L s P [(1)] .65158 .11972 0 2 Mshowa p .002 w .80315 .11972 m .80315 .12597 L s P [(2)] .80315 .11972 0 2 Mshowa p .002 w .95473 .11972 m .95473 .12597 L s P [(3)] .95473 .11972 0 2 Mshowa p .001 w .07559 .11972 m .07559 .12347 L s P p .001 w .1059 .11972 m .1059 .12347 L s P p .001 w .13622 .11972 m .13622 .12347 L s P p .001 w .16653 .11972 m .16653 .12347 L s P p .001 w .22716 .11972 m .22716 .12347 L s P p .001 w .25748 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functions for the same values of N as in Problem 3. We first define the Dirichlet kernel derived in Problem 10: :[font = input; preserveAspect] Dirichlet[N_,x_]:=Sin[(N+(1/2))x]/Sin[x/2] :[font = subsection; inactive; preserveAspect; startGroup] Problem 12(a) :[font = text; inactive; preserveAspect] To insure that we obtain the same results as in Problem 3 from the results of Problem 10, we must construct the odd extensions of each of the functions over the interval (-Pi,Pi). For the function in Problem 3(a), we have :[font = text; inactive; preserveAspect] The integral representation in Problem 10 is therefore calculated from :[font = input; preserveAspect] f[x_]:=x(Pi-x) F[N_,x_]:=(1/(2 Pi))(Integrate[-f[-s] Dirichlet[N,{x-s}], {s,-Pi,0}]+Integrate[f[s] Dirichlet[N,x-s], {s,0,Pi}]) :[font = input; preserveAspect; startGroup] Simplify[F[1,x]] :[font = output; output; inactive; preserveAspect; endGroup] {(8*Sin[x])/Pi} ;[o] 8 Sin[x] {--------} Pi :[font = input; preserveAspect; startGroup] Plot[{f[x],%},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=x(Pi-x), n=1", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.231144 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=x\(Pi-x\), n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.5)] .01131 .13029 1 0 Msboxa [(1)] .01131 .24586 1 0 Msboxa [(1.5)] .01131 .36143 1 0 Msboxa [(2)] .01131 .477 1 0 Msboxa [(2.5)] .01131 .59258 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.5)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(1)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(1.5)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(2)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(2.5)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(3)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(f[x]=x\(Pi-x\), n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .02381 .13029 m .03006 .13029 L s P [(0.5)] .01131 .13029 1 0 Mshowa p .002 w .02381 .24586 m .03006 .24586 L s P [(1)] .01131 .24586 1 0 Mshowa p .002 w .02381 .36143 m .03006 .36143 L s P [(1.5)] .01131 .36143 1 0 Mshowa p .002 w .02381 .477 m .03006 .477 L s P [(2)] .01131 .477 1 0 Mshowa p .002 w .02381 .59258 m .03006 .59258 L s P [(2.5)] .01131 .59258 1 0 Mshowa p .001 w .02381 .03783 m .02756 .03783 L s P p .001 w .02381 .06094 m .02756 .06094 L s P p .001 w .02381 .08406 m .02756 .08406 L s P p .001 w .02381 .10717 m .02756 .10717 L s P p .001 w .02381 .1534 m .02756 .1534 L s P p .001 w .02381 .17652 m .02756 .17652 L s P p .001 w .02381 .19963 m .02756 .19963 L s P p .001 w .02381 .22274 m .02756 .22274 L s P p .001 w .02381 .26897 m .02756 .26897 L s P p .001 w .02381 .29209 m .02756 .29209 L s P p .001 w .02381 .3152 m .02756 .3152 L s P p .001 w .02381 .33832 m .02756 .33832 L s P p .001 w .02381 .38455 m .02756 .38455 L s P p .001 w .02381 .40766 m .02756 .40766 L s P p .001 w .02381 .43077 m .02756 .43077 L s P p .001 w .02381 .45389 m .02756 .45389 L s P p .001 w .02381 .50012 m .02756 .50012 L s P p .001 w .02381 .52323 m .02756 .52323 L s P p .001 w .02381 .54635 m .02756 .54635 L s P p .001 w .02381 .56946 m .02756 .56946 L s P p .001 w .02381 .61569 m .02756 .61569 L s P [(f)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P p p p .0075 w .02381 .01472 m .06349 .10581 L .10317 .18898 L .14286 .26423 L .18254 .33156 L .22222 .39097 L .2619 .44246 L .30159 .48603 L .34127 .52167 L .38095 .5494 L .40079 .56029 L .42063 .5692 L .44048 .57613 L .4504 .57885 L .46032 .58108 L .47024 .58281 L .4752 .58349 L .48016 .58405 L .48512 .58448 L .4876 .58465 L .49008 .58479 L .49256 .5849 L .4938 .58494 L .49504 .58498 L .49628 .58501 L .49752 .58503 L .49876 .58504 L .5 .58504 L .50124 .58504 L .50248 .58503 L .50372 .58501 L .50496 .58498 L .5062 .58494 L .50744 .5849 L .50992 .58479 L .5124 .58465 L .51488 .58448 L .51984 .58405 L .5248 .58349 L .52976 .58281 L .53968 .58108 L .5496 .57885 L .55952 .57613 L .57937 .5692 L .59921 .56029 L .61905 .5494 L .65873 .52167 L .69841 .48603 L .7381 .44246 L .77778 .39097 L Mistroke .81746 .33156 L .85714 .26423 L .89683 .18898 L .93651 .10581 L .97619 .01472 L Mfstroke P P p p .005 w .02381 .01472 m .06349 .09154 L .10317 .16706 L .14286 .23996 L .18254 .30902 L .22222 .37303 L .2619 .43092 L .30159 .48169 L .34127 .52446 L .38095 .55851 L .40079 .57208 L .42063 .58326 L .44048 .59201 L .4504 .59546 L .46032 .59828 L .47024 .60048 L .4752 .60135 L .48016 .60206 L .48512 .60261 L .4876 .60283 L .49008 .603 L .49256 .60314 L .4938 .6032 L .49504 .60324 L .49628 .60327 L .49752 .6033 L .49876 .60331 L .5 .60332 L .50124 .60331 L .50248 .6033 L .50372 .60327 L .50496 .60324 L .5062 .6032 L .50744 .60314 L .50992 .603 L .5124 .60283 L .51488 .60261 L .51984 .60206 L .5248 .60135 L .52976 .60048 L .53968 .59828 L .5496 .59546 L .55952 .59201 L .57937 .58326 L .61905 .55851 L .65873 .52446 L .69841 .48169 L .7381 .43092 L .77778 .37303 L .81746 .30902 L Mistroke .85714 .23996 L .89683 .16706 L .93651 .09154 L .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Simplify[F[3,x]] Plot[{f[x],%},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=x(Pi-x), n=3", AxesLabel->{x,f}] :[font = output; output; inactive; preserveAspect] {(8*(27*Sin[x] + Sin[3*x]))/(27*Pi)} ;[o] 8 (27 Sin[x] + Sin[3 x]) {------------------------} 27 Pi :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.238552 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=x\(Pi-x\), n=3)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.5)] .01131 .13399 1 0 Msboxa [(1)] .01131 .25327 1 0 Msboxa [(1.5)] .01131 .37254 1 0 Msboxa [(2)] .01131 .49182 1 0 Msboxa [(2.5)] .01131 .6111 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.5)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(1)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(1.5)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(2)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(2.5)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(3)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(f[x]=x\(Pi-x\), n=3)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .02381 .13399 m .03006 .13399 L s P [(0.5)] .01131 .13399 1 0 Mshowa p .002 w .02381 .25327 m .03006 .25327 L s P [(1)] .01131 .25327 1 0 Mshowa p .002 w .02381 .37254 m .03006 .37254 L s P [(1.5)] .01131 .37254 1 0 Mshowa p .002 w .02381 .49182 m .03006 .49182 L s P [(2)] .01131 .49182 1 0 Mshowa p .002 w .02381 .6111 m .03006 .6111 L s P [(2.5)] .01131 .6111 1 0 Mshowa p .001 w .02381 .03857 m .02756 .03857 L s P p .001 w .02381 .06243 m .02756 .06243 L s P p .001 w .02381 .08628 m .02756 .08628 L s P p .001 w .02381 .11014 m .02756 .11014 L s P p .001 w .02381 .15785 m .02756 .15785 L s P p .001 w .02381 .1817 m .02756 .1817 L s P p .001 w .02381 .20556 m .02756 .20556 L s P p .001 w .02381 .22941 m .02756 .22941 L s P p .001 w .02381 .27712 m .02756 .27712 L s P p .001 w .02381 .30098 m .02756 .30098 L s P p .001 w .02381 .32483 m .02756 .32483 L s P p .001 w .02381 .34869 m .02756 .34869 L s P p .001 w .02381 .3964 m .02756 .3964 L s P p .001 w .02381 .42025 m .02756 .42025 L s P p .001 w .02381 .44411 m .02756 .44411 L s P p .001 w .02381 .46796 m .02756 .46796 L s P p .001 w .02381 .51567 m .02756 .51567 L s P p .001 w .02381 .53953 m .02756 .53953 L s P p .001 w .02381 .56339 m .02756 .56339 L s P p .001 w .02381 .58724 m .02756 .58724 L s P [(f)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P p p p .0075 w .02381 .01472 m .06349 .10873 L .10317 .19457 L .14286 .27223 L .18254 .34172 L .22222 .40303 L .2619 .45617 L .30159 .50113 L .34127 .53792 L .38095 .56653 L .40079 .57777 L .42063 .58697 L .44048 .59412 L .4504 .59693 L .46032 .59923 L .47024 .60102 L .4752 .60172 L .48016 .6023 L .48512 .60274 L .4876 .60292 L .49008 .60306 L .49256 .60318 L .4938 .60322 L .49504 .60326 L .49628 .60328 L .49752 .6033 L .49876 .60331 L .5 .60332 L .50124 .60331 L .50248 .6033 L .50372 .60328 L .50496 .60326 L .5062 .60322 L .50744 .60318 L .50992 .60306 L .5124 .60292 L .51488 .60274 L .51984 .6023 L .5248 .60172 L .52976 .60102 L .53968 .59923 L .5496 .59693 L .55952 .59412 L .57937 .58697 L .59921 .57777 L .61905 .56653 L .65873 .53792 L .69841 .50113 L .7381 .45617 L .77778 .40303 L Mistroke .81746 .34172 L .85714 .27223 L .89683 .19457 L .93651 .10873 L .97619 .01472 L Mfstroke P P p p .005 w .02381 .01472 m .06349 .10262 L .10317 .18785 L .14286 .26797 L .18254 .34095 L .22222 .4053 L .2619 .46017 L .30159 .50526 L .34127 .5408 L .36111 .55515 L .38095 .56733 L .40079 .57745 L .42063 .58558 L .44048 .5918 L .4504 .59423 L .46032 .5962 L .47024 .59773 L .4752 .59833 L .48016 .59882 L .48512 .5992 L .4876 .59935 L .49008 .59947 L .49256 .59956 L .4938 .5996 L .49504 .59963 L .49628 .59965 L .49752 .59967 L .49876 .59968 L .5 .59968 L .50124 .59968 L .50248 .59967 L .50372 .59965 L .50496 .59963 L .5062 .5996 L .50744 .59956 L .50992 .59947 L .5124 .59935 L .51488 .5992 L .51984 .59882 L .5248 .59833 L .52976 .59773 L .53968 .5962 L .5496 .59423 L .55952 .5918 L .57937 .58558 L .59921 .57745 L .61905 .56733 L .65873 .5408 L .69841 .50526 L .7381 .46017 L Mistroke .77778 .4053 L .81746 .34095 L .85714 .26797 L .89683 .18785 L .93651 .10262 L .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = text; inactive; preserveAspect; endGroup] These results should be compared with those in Problem 3. It should be clear that performing the integral representation produces the partial sums of the Fourier representation of the function. :[font = subsection; inactive; preserveAspect; startGroup] Problem 12(b) :[font = text; inactive; preserveAspect] The function f[x]=x is already an odd function, so the construction of an odd extension is not required. :[font = input; preserveAspect] f[x_]:=x F[N_,x_]:=(1/(2 Pi))Integrate[f[s] Dirichlet[N,{x-s}], {s,-Pi,Pi}] :[font = input; preserveAspect; startGroup] Simplify[F[1,x]] Plot[{f[x],%},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=x, n=1", AxesLabel->{x,f}] :[font = output; output; inactive; preserveAspect] {2*Sin[x]} ;[o] {2 Sin[x]} :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.187358 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=x, n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.5)] .01131 .10839 1 0 Msboxa [(1)] .01131 .20207 1 0 Msboxa [(1.5)] .01131 .29575 1 0 Msboxa [(2)] .01131 .38943 1 0 Msboxa [(2.5)] .01131 .48311 1 0 Msboxa [(3)] .01131 .57679 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.5)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(1)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(1.5)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(2)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(2.5)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(3)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(f[x]=x, n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .02381 .10839 m .03006 .10839 L s P [(0.5)] .01131 .10839 1 0 Mshowa p .002 w .02381 .20207 m .03006 .20207 L s P [(1)] .01131 .20207 1 0 Mshowa p .002 w .02381 .29575 m .03006 .29575 L s P [(1.5)] .01131 .29575 1 0 Mshowa p .002 w .02381 .38943 m .03006 .38943 L s P [(2)] .01131 .38943 1 0 Mshowa p .002 w .02381 .48311 m .03006 .48311 L s P [(2.5)] .01131 .48311 1 0 Mshowa p .002 w .02381 .57679 m .03006 .57679 L s P [(3)] .01131 .57679 1 0 Mshowa p .001 w .02381 .03345 m .02756 .03345 L s P p .001 w .02381 .05219 m .02756 .05219 L s P p .001 w .02381 .07092 m .02756 .07092 L s P p .001 w .02381 .08966 m .02756 .08966 L s P p .001 w .02381 .12713 m .02756 .12713 L s P p .001 w .02381 .14587 m .02756 .14587 L s P p .001 w .02381 .1646 m .02756 .1646 L s P p .001 w .02381 .18334 m .02756 .18334 L s P p .001 w .02381 .22081 m .02756 .22081 L s P p .001 w .02381 .23955 m .02756 .23955 L s P p .001 w .02381 .25828 m .02756 .25828 L s P p .001 w .02381 .27702 m .02756 .27702 L s P p .001 w .02381 .31449 m .02756 .31449 L s P p .001 w .02381 .33322 m .02756 .33322 L s P p .001 w .02381 .35196 m .02756 .35196 L s P p .001 w .02381 .3707 m .02756 .3707 L s P p .001 w .02381 .40817 m .02756 .40817 L s P p .001 w .02381 .4269 m .02756 .4269 L s P p .001 w .02381 .44564 m .02756 .44564 L s P p .001 w .02381 .46438 m .02756 .46438 L s P p .001 w .02381 .50185 m .02756 .50185 L s P p .001 w .02381 .52058 m .02756 .52058 L s P p .001 w .02381 .53932 m .02756 .53932 L s P p .001 w .02381 .55805 m .02756 .55805 L s P p .001 w .02381 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.38943 L .50124 .38943 L .50248 .38942 L .50372 .3894 L .50496 .38938 L .5062 .38935 L .50744 .38932 L .50992 .38923 L .5124 .38912 L .51488 .38898 L .51984 .38863 L .5248 .38818 L .52976 .38763 L .53968 .38623 L .5496 .38443 L .55952 .38223 L .57937 .37666 L .61905 .36091 L .65873 .33923 L .69841 .312 L .7381 .27968 L .77778 .24283 L .81746 .20207 L Mistroke .85714 .15811 L .89683 .1117 L .93651 .06363 L .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Simplify[F[5,x]] Plot[{f[x],%},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=x, n=5", AxesLabel->{x,f}] :[font = output; output; inactive; preserveAspect] {2*Sin[x] - Sin[2*x] + (2*Sin[3*x])/3 - Sin[4*x]/2 + (2*Sin[5*x])/5} ;[o] 2 Sin[3 x] Sin[4 x] {2 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---------- - -------- + 3 2 2 Sin[5 x] Sin[6 x] 2 Sin[7 x] Sin[8 x] ---------- - -------- + ---------- - -------- + 5 3 7 4 2 Sin[9 x] Sin[10 x] ---------- - ---------} 9 5 :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.172413 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=x, n=10)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.5)] .01131 .10092 1 0 Msboxa [(1)] .01131 .18713 1 0 Msboxa [(1.5)] .01131 .27333 1 0 Msboxa [(2)] .01131 .35954 1 0 Msboxa [(2.5)] .01131 .44575 1 0 Msboxa [(3)] .01131 .53195 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa 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L .21974 .11758 L .22098 .11764 L .22222 .11772 L .2247 .11793 L .22718 .11822 L .22966 .11859 L .23214 .11906 L .2371 .12031 L .24206 .12201 L .24702 .12421 L .25198 .12692 L .2619 .13389 L .28175 .15328 L .30159 .17667 L .32143 .19862 L .33135 .20741 L .33631 .21106 L .34127 .21416 L .34623 .2167 L .35119 .21868 L .35367 .21947 L .35615 .22013 L .35863 .22067 L .36111 .22108 L .36235 .22124 L .36359 .22138 L .36483 .22149 L .36607 .22158 L .36731 .22164 L .36855 .22168 L .36979 .22169 L .37103 .22169 L .37227 .22167 L .37351 .22162 L .37599 .22149 L .37723 .2214 L .37847 .2213 L .38095 .22106 L .39087 .21992 L .39583 .2194 L .39831 .21919 L .39955 .21911 L .40079 .21904 L Mistroke .40203 .21899 L .40327 .21896 L .40451 .21894 L .40575 .21895 L .40699 .21898 L .40823 .21903 L .40947 .2191 L .41071 .21921 L .41319 .21949 L .41567 .2199 L .41815 .22043 L .42063 .22109 L .4256 .22283 L .43056 .22517 L .44048 .2317 L .4504 .24063 L .46032 .25165 L .48016 .27742 L .5 .30262 L .50992 .31287 L .51488 .3171 L .51984 .32068 L .5248 .32355 L .52728 .32472 L .52976 .3257 L .53224 .32652 L .53472 .32715 L .53596 .32741 L .5372 .32762 L .53844 .32779 L .53968 .32792 L .54092 .32801 L .54216 .32807 L .5434 .32808 L .54464 .32806 L .54588 .328 L .54712 .32791 L .54836 .32779 L .5496 .32763 L .55952 .32545 L .56448 .3239 L .56944 .32222 L .5744 .32052 L .57937 .31895 L .58433 .31765 L .58681 .31713 L .58805 .31691 L .58929 .31672 L .59053 .31657 L .59177 .31645 L Mistroke .59301 .31636 L .59425 .31631 L .59549 .3163 L .59673 .31632 L .59797 .31639 L .59921 .31651 L .60045 .31666 L .60169 .31687 L .60417 .31741 L .60665 .31815 L .60913 .3191 L .61409 .32161 L .61905 .32499 L .62897 .33432 L .63889 .34683 L .65873 .3784 L .67857 .41111 L .68849 .42465 L .69345 .43019 L .69841 .43477 L .70089 .43667 L .70337 .4383 L .70585 .43965 L .70833 .44072 L .70957 .44115 L .71081 .4415 L .71205 .44179 L .71329 .442 L .71453 .44215 L .71577 .44222 L .71701 .44222 L .71825 .44216 L .71949 .44202 L .72073 .44182 L .72197 .44155 L .72321 .44122 L .72817 .43926 L .73065 .43793 L .73313 .43639 L .7381 .43274 L .75794 .41393 L .76786 .40502 L .77282 .40151 L .7753 .40009 L .77778 .39892 L .77902 .39844 L .78026 .39804 L .7815 .39771 L .78274 .39747 L .78398 .3973 L Mistroke .78522 .39722 L .78646 .39723 L .7877 .39733 L .78894 .39752 L .79018 .39781 L .79266 .39867 L .7939 .39925 L .79514 .39994 L .79762 .40162 L .80258 .40624 L .80754 .41259 L .81746 .43039 L .82738 .45435 L .8373 .48304 L .85714 .54515 L .86706 .57249 L .87202 .58375 L .87698 .59283 L .87946 .59642 L .88194 .59931 L .88318 .60048 L .88442 .60146 L .88566 .60223 L .8869 .60281 L .88814 .60317 L .88938 .60332 L .89062 .60325 L .89187 .60295 L .89311 .60243 L .89435 .60167 L .89559 .60067 L .89683 .59943 L .89931 .59621 L .90179 .59197 L .90675 .58035 L .91171 .56442 L .91667 .54407 L .92659 .49013 L .93651 .41917 L .95635 .23429 L .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Simplify[F[50,x]]; Plot[{f[x],%},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=x, n=50", AxesLabel->{x,f}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.161614 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=x, n=50)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.5)] .01131 .09552 1 0 Msboxa [(1)] .01131 .17633 1 0 Msboxa [(1.5)] .01131 .25714 1 0 Msboxa [(2)] .01131 .33794 1 0 Msboxa [(2.5)] .01131 .41875 1 0 Msboxa [(3)] .01131 .49956 1 0 Msboxa [(3.5)] .01131 .58037 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.5)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(1)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(1.5)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(2)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(2.5)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(3)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(f[x]=x, n=50)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .02381 .09552 m .03006 .09552 L s P [(0.5)] .01131 .09552 1 0 Mshowa p .002 w .02381 .17633 m .03006 .17633 L s P [(1)] .01131 .17633 1 0 Mshowa p .002 w .02381 .25714 m .03006 .25714 L s P [(1.5)] .01131 .25714 1 0 Mshowa p .002 w .02381 .33794 m .03006 .33794 L s P [(2)] .01131 .33794 1 0 Mshowa p .002 w .02381 .41875 m .03006 .41875 L s P [(2.5)] .01131 .41875 1 0 Mshowa p .002 w .02381 .49956 m .03006 .49956 L s P [(3)] .01131 .49956 1 0 Mshowa p .002 w .02381 .58037 m .03006 .58037 L s P [(3.5)] .01131 .58037 1 0 Mshowa p .001 w .02381 .03088 m .02756 .03088 L s P p .001 w .02381 .04704 m .02756 .04704 L s P p .001 w .02381 .0632 m .02756 .0632 L s P p .001 w .02381 .07936 m .02756 .07936 L s P p .001 w .02381 .11168 m .02756 .11168 L s P p .001 w .02381 .12785 m .02756 .12785 L s P p .001 w .02381 .14401 m .02756 .14401 L s P p .001 w .02381 .16017 m .02756 .16017 L s P p .001 w .02381 .19249 m .02756 .19249 L s P p .001 w .02381 .20865 m .02756 .20865 L s P p .001 w .02381 .22481 m .02756 .22481 L s P p .001 w .02381 .24098 m .02756 .24098 L s P p .001 w .02381 .2733 m .02756 .2733 L s P p .001 w .02381 .28946 m .02756 .28946 L s P p .001 w .02381 .30562 m .02756 .30562 L s P p .001 w .02381 .32178 m .02756 .32178 L s P p .001 w .02381 .35411 m .02756 .35411 L s P p .001 w .02381 .37027 m .02756 .37027 L s P p .001 w .02381 .38643 m .02756 .38643 L s P p .001 w .02381 .40259 m .02756 .40259 L s P p .001 w .02381 .43491 m .02756 .43491 L s P p .001 w .02381 .45107 m .02756 .45107 L s P p .001 w .02381 .46724 m .02756 .46724 L s P p .001 w .02381 .4834 m .02756 .4834 L s P p .001 w .02381 .51572 m .02756 .51572 L s P p .001 w .02381 .53188 m .02756 .53188 L s P p .001 w .02381 .54804 m .02756 .54804 L s P p .001 w .02381 .5642 m .02756 .5642 L s P p .001 w .02381 .59653 m .02756 .59653 L s P p .001 w .02381 .61269 m .02756 .61269 L s P [(f)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P p p p .0075 w .02381 .01472 m .06349 .03587 L .10317 .05703 L .14286 .07818 L .18254 .09934 L .22222 .12049 L .2619 .14165 L .30159 .1628 L .34127 .18396 L .38095 .20511 L .42063 .22627 L .46032 .24742 L .5 .26858 L .53968 .28973 L .57937 .31089 L .61905 .33204 L .65873 .3532 L .69841 .37435 L .7381 .39551 L .77778 .41667 L .81746 .43782 L .85714 .45898 L .89683 .48013 L .93651 .50129 L .97619 .52244 L s P P p p .005 w .02381 .01472 m .06349 .03484 L .10317 .05506 L .14286 .07546 L .18254 .09613 L .22222 .11712 L .2619 .13845 L .30159 .16013 L .34127 .18213 L .38095 .20439 L .42063 .22682 L .46032 .24934 L .5 .27181 L .53968 .29411 L .57937 .31611 L .61905 .33767 L .65873 .35868 L .69841 .37901 L .7381 .39852 L .77778 .41703 L .81746 .43419 L .8187 .43246 L .81994 .43095 L .82118 .42976 L .82242 .42898 L .82366 .42866 L .8249 .42886 L .82614 .42962 L .82738 .43092 L .83234 .44097 L .8373 .45473 L .83978 .46076 L .84102 .46317 L .84226 .46506 L .8435 .46637 L .84474 .46705 L .84598 .46711 L .84722 .46655 L .84846 .46541 L .8497 .46376 L .85218 .4593 L .85466 .45409 L .85714 .44924 L .85838 .44728 L .85962 .44581 L .86086 .44491 L .8621 .44467 L .86334 .44515 L .86458 .44636 L .86582 .44831 L Mistroke .86706 .45094 L .87202 .46659 L .8745 .47561 L .87698 .48373 L .87822 .48706 L .87946 .48973 L .8807 .49164 L .88194 .4927 L .88318 .49288 L .88442 .49218 L .88566 .49063 L .8869 .48831 L .88814 .48532 L .88938 .4818 L .89187 .47385 L .89435 .46599 L .89559 .46259 L .89683 .4598 L .89807 .45779 L .89931 .45671 L .90055 .45665 L .90179 .45768 L .90303 .45981 L .90427 .46304 L .90551 .46727 L .90675 .4724 L .91171 .49809 L .91419 .51071 L .91543 .51605 L .91667 .52044 L .91791 .52367 L .91915 .52558 L .92039 .52606 L .92163 .52505 L .92287 .52255 L .92411 .51864 L .92659 .50715 L .92907 .49232 L .93155 .47666 L .93403 .46311 L .93527 .45804 L .93651 .45455 L .93775 .45294 L .93899 .45343 L .94023 .45615 L .94147 .46119 L .94395 .47799 L .94519 .48941 L .94643 .50246 L .94891 .5317 L Mistroke .95139 .56147 L .95387 .58657 L .95511 .59562 L .95635 .60143 L .95759 .60332 L .95883 .60071 L .96007 .59308 L .96131 .58 L .96379 .53639 L .96503 .5056 L .96627 .4689 L .97123 .26923 L .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = subsection; inactive; preserveAspect; startGroup] Problem 12(c) :[font = text; inactive; preserveAspect] The function f[x]=1 does require an odd periodic extension. Thus, :[font = input; preserveAspect] F[N_,x_]:=(1/(2 Pi))(Integrate[-Dirichlet[N,{x-s}], {s,-Pi,0}]+Integrate[Dirichlet[N,{x-s}], {s,0,Pi}]) :[font = input; preserveAspect; startGroup] Simplify[F[1,x]] Plot[{1,%},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=1, n=1", AxesLabel->{x,f}] :[font = output; output; inactive; preserveAspect] {(4*Sin[x])/Pi} ;[o] 4 Sin[x] {--------} Pi :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.462288 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=1, n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.2)] .01131 .10717 1 0 Msboxa [(0.4)] .01131 .19963 1 0 Msboxa [(0.6)] .01131 .29209 1 0 Msboxa [(0.8)] .01131 .38455 1 0 Msboxa [(1)] .01131 .477 1 0 Msboxa [(1.2)] .01131 .56946 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.5)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(1)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(1.5)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(2)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(2.5)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(3)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 .01472 m .14507 .01847 L s P p .001 w .2057 .01472 m .2057 .01847 L s P p .001 w .23602 .01472 m .23602 .01847 L s P p .001 w .26633 .01472 m .26633 .01847 L s P p .001 w .29665 .01472 m .29665 .01847 L s P p .001 w .35728 .01472 m .35728 .01847 L s P p .001 w .38759 .01472 m .38759 .01847 L s P p .001 w .41791 .01472 m .41791 .01847 L s P p .001 w .44822 .01472 m .44822 .01847 L s P p .001 w .50885 .01472 m .50885 .01847 L s P p .001 w .53917 .01472 m .53917 .01847 L s P p .001 w .56948 .01472 m .56948 .01847 L s P p .001 w .5998 .01472 m .5998 .01847 L s P p .001 w .66043 .01472 m .66043 .01847 L s P p .001 w .69074 .01472 m .69074 .01847 L s P p .001 w .72106 .01472 m .72106 .01847 L s P p .001 w .75137 .01472 m .75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(f[x]=1, n=1)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .02381 .10717 m .03006 .10717 L s P [(0.2)] .01131 .10717 1 0 Mshowa p .002 w .02381 .19963 m .03006 .19963 L s P [(0.4)] .01131 .19963 1 0 Mshowa p .002 w .02381 .29209 m .03006 .29209 L s P [(0.6)] .01131 .29209 1 0 Mshowa p .002 w .02381 .38455 m .03006 .38455 L s P [(0.8)] .01131 .38455 1 0 Mshowa p .002 w .02381 .477 m .03006 .477 L s P [(1)] .01131 .477 1 0 Mshowa p .002 w .02381 .56946 m .03006 .56946 L s P [(1.2)] .01131 .56946 1 0 Mshowa p .001 w .02381 .03321 m .02756 .03321 L s P p .001 w .02381 .0517 m .02756 .0517 L s P p .001 w .02381 .07019 m .02756 .07019 L s P p .001 w .02381 .08868 m .02756 .08868 L s P p .001 w .02381 .12566 m .02756 .12566 L s P p .001 w .02381 .14416 m .02756 .14416 L s P p .001 w .02381 .16265 m .02756 .16265 L s P p .001 w .02381 .18114 m .02756 .18114 L s P p .001 w .02381 .21812 m .02756 .21812 L s P p .001 w .02381 .23661 m .02756 .23661 L s P p .001 w .02381 .25511 m .02756 .25511 L s P p .001 w .02381 .2736 m .02756 .2736 L s P p .001 w .02381 .31058 m .02756 .31058 L s P p .001 w .02381 .32907 m .02756 .32907 L s P p .001 w .02381 .34756 m .02756 .34756 L s P p .001 w .02381 .36605 m .02756 .36605 L s P p .001 w .02381 .40304 m .02756 .40304 L s P p .001 w .02381 .42153 m .02756 .42153 L s P p .001 w .02381 .44002 m .02756 .44002 L s P p .001 w .02381 .45851 m .02756 .45851 L s P p .001 w .02381 .49549 m .02756 .49549 L s P p .001 w .02381 .51399 m .02756 .51399 L s P p .001 w .02381 .53248 m .02756 .53248 L s P p .001 w .02381 .55097 m .02756 .55097 L s P p .001 w .02381 .58795 m .02756 .58795 L s P p .001 w .02381 .60644 m .02756 .60644 L s P [(f)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m .02381 .61803 L s P P p p p .0075 w .02381 .477 m .06349 .477 L .10317 .477 L .14286 .477 L .18254 .477 L .22222 .477 L .2619 .477 L .30159 .477 L .34127 .477 L .38095 .477 L .42063 .477 L .46032 .477 L .5 .477 L .53968 .477 L .57937 .477 L .61905 .477 L .65873 .477 L .69841 .477 L .7381 .477 L .77778 .477 L .81746 .477 L .85714 .477 L .89683 .477 L .93651 .477 L .97619 .477 L s P P p p .005 w .02381 .01472 m .06349 .09154 L .10317 .16706 L .14286 .23996 L .18254 .30902 L .22222 .37303 L .2619 .43092 L .30159 .48169 L .34127 .52446 L .38095 .55851 L .40079 .57208 L .42063 .58326 L .44048 .59201 L .4504 .59546 L .46032 .59828 L .47024 .60048 L .4752 .60135 L .48016 .60206 L .48512 .60261 L .4876 .60283 L .49008 .603 L .49256 .60314 L .4938 .6032 L .49504 .60324 L .49628 .60327 L .49752 .6033 L .49876 .60331 L .5 .60332 L .50124 .60331 L .50248 .6033 L .50372 .60327 L .50496 .60324 L .5062 .6032 L .50744 .60314 L .50992 .603 L .5124 .60283 L .51488 .60261 L .51984 .60206 L .5248 .60135 L .52976 .60048 L .53968 .59828 L .5496 .59546 L .55952 .59201 L .57937 .58326 L .61905 .55851 L .65873 .52446 L .69841 .48169 L .7381 .43092 L .77778 .37303 L .81746 .30902 L Mistroke .85714 .23996 L .89683 .16706 L .93651 .09154 L .97619 .01472 L Mfstroke P P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Simplify[F[5,x]] Plot[{1,%},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=1, n=5", AxesLabel->{x,f}] :[font = output; output; inactive; preserveAspect] {(4*(15*Sin[x] + 5*Sin[3*x] + 3*Sin[5*x]))/(15*Pi)} ;[o] 4 (15 Sin[x] + 5 Sin[3 x] + 3 Sin[5 x]) {---------------------------------------} 15 Pi :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 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.75137 .01847 L s P p .001 w .81201 .01472 m .81201 .01847 L s P p .001 w .84232 .01472 m .84232 .01847 L s P p .001 w .87264 .01472 m .87264 .01847 L s P p .001 w .90295 .01472 m .90295 .01847 L s P p .001 w .96358 .01472 m .96358 .01847 L s P p .001 w .9939 .01472 m .9939 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(f[x]=1, n=5)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .02381 .11378 m .03006 .11378 L s P [(0.2)] .01131 .11378 1 0 Mshowa p .002 w .02381 .21284 m .03006 .21284 L s P [(0.4)] .01131 .21284 1 0 Mshowa p .002 w .02381 .3119 m .03006 .3119 L s P [(0.6)] .01131 .3119 1 0 Mshowa p .002 w .02381 .41096 m .03006 .41096 L s P [(0.8)] .01131 .41096 1 0 Mshowa p .002 w .02381 .51002 m .03006 .51002 L s P [(1)] .01131 .51002 1 0 Mshowa p .002 w .02381 .60909 m .03006 .60909 L s P [(1.2)] .01131 .60909 1 0 Mshowa p .001 w .02381 .03453 m .02756 .03453 L s P p .001 w .02381 .05434 m .02756 .05434 L s P p .001 w .02381 .07415 m .02756 .07415 L s P p .001 w .02381 .09396 m .02756 .09396 L s P p .001 w .02381 .13359 m .02756 .13359 L s P p .001 w .02381 .1534 m .02756 .1534 L s P p .001 w .02381 .17321 m .02756 .17321 L s P p .001 w .02381 .19303 m .02756 .19303 L s P p .001 w .02381 .23265 m .02756 .23265 L s P p .001 w .02381 .25246 m .02756 .25246 L s P p .001 w .02381 .27228 m .02756 .27228 L s P p .001 w .02381 .29209 m .02756 .29209 L s P p .001 w .02381 .33171 m .02756 .33171 L s P p .001 w .02381 .35153 m .02756 .35153 L s P p .001 w .02381 .37134 m .02756 .37134 L s P p .001 w .02381 .39115 m .02756 .39115 L s P p .001 w .02381 .43077 m .02756 .43077 L s P p .001 w .02381 .45059 m .02756 .45059 L s P p .001 w .02381 .4704 m .02756 .4704 L s P p .001 w .02381 .49021 m .02756 .49021 L s P p .001 w .02381 .52984 m .02756 .52984 L s P p .001 w .02381 .54965 m .02756 .54965 L s P p .001 w .02381 .56946 m .02756 .56946 L s P p .001 w .02381 .58927 m .02756 .58927 L s P [(f)] .02381 .61803 0 -4 Mshowa p .002 w .02381 0 m 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:[font = input; preserveAspect; startGroup] Simplify[F[10,x]] Plot[{1,%},{x,0,Pi}, PlotStyle->{{Thickness[0.0075]},{Thickness[0.005]}}, PlotRange->All,PlotLabel->"f[x]=1, n=10", AxesLabel->{x,f}] :[font = output; output; inactive; preserveAspect] {(4*(315*Sin[x] + 105*Sin[3*x] + 63*Sin[5*x] + 45*Sin[7*x] + 35*Sin[9*x]))/(315*Pi)} ;[o] {(4 (315 Sin[x] + 105 Sin[3 x] + 63 Sin[5 x] + 45 Sin[7 x] + 35 Sin[9 x])) / (315 Pi)} :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.303152 0.0147151 0.497837 [ [(0.5)] .17539 .01472 0 2 Msboxa [(1)] .32696 .01472 0 2 Msboxa [(1.5)] .47854 .01472 0 2 Msboxa [(2)] .63011 .01472 0 2 Msboxa [(2.5)] .78169 .01472 0 2 Msboxa [(3)] .93327 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(f[x]=1, n=10)] .5 .61803 0 -2 0 0 1 Mouter Mrotsboxa [(0.2)] .01131 .11428 1 0 Msboxa [(0.4)] .01131 .21385 1 0 Msboxa [(0.6)] .01131 .31342 1 0 Msboxa [(0.8)] .01131 .41299 1 0 Msboxa [(1)] .01131 .51255 1 0 Msboxa [(1.2)] .01131 .61212 1 0 Msboxa [(f)] .02381 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .17539 .01472 m .17539 .02097 L s P [(0.5)] .17539 .01472 0 2 Mshowa p .002 w .32696 .01472 m .32696 .02097 L s P [(1)] .32696 .01472 0 2 Mshowa p .002 w .47854 .01472 m .47854 .02097 L s P [(1.5)] .47854 .01472 0 2 Mshowa p .002 w .63011 .01472 m .63011 .02097 L s P [(2)] .63011 .01472 0 2 Mshowa p .002 w .78169 .01472 m .78169 .02097 L s P [(2.5)] .78169 .01472 0 2 Mshowa p .002 w .93327 .01472 m .93327 .02097 L s P [(3)] .93327 .01472 0 2 Mshowa p .001 w .05412 .01472 m .05412 .01847 L s P p .001 w .08444 .01472 m .08444 .01847 L s P p .001 w .11476 .01472 m .11476 .01847 L s P p .001 w .14507 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.01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P [(f[x]=1, n=10)] .5 .61803 0 -2 0 0 1 Mouter Mrotshowa p .002 w .02381 .11428 m .03006 .11428 L s P [(0.2)] .01131 .11428 1 0 Mshowa p .002 w .02381 .21385 m .03006 .21385 L s P [(0.4)] .01131 .21385 1 0 Mshowa p .002 w .02381 .31342 m .03006 .31342 L s P [(0.6)] .01131 .31342 1 0 Mshowa p .002 w .02381 .41299 m .03006 .41299 L s P [(0.8)] .01131 .41299 1 0 Mshowa p .002 w .02381 .51255 m .03006 .51255 L s P [(1)] .01131 .51255 1 0 Mshowa p .002 w .02381 .61212 m .03006 .61212 L s P [(1.2)] .01131 .61212 1 0 Mshowa p .001 w .02381 .03463 m .02756 .03463 L s P p .001 w .02381 .05454 m .02756 .05454 L s P p .001 w .02381 .07446 m .02756 .07446 L s P p .001 w .02381 .09437 m .02756 .09437 L s P p .001 w .02381 .1342 m .02756 .1342 L s P p .001 w .02381 .15411 m .02756 .15411 L s P p .001 w .02381 .17402 m .02756 .17402 L s P p .001 w .02381 .19394 m .02756 .19394 L s P p .001 w .02381 .23376 m .02756 .23376 L s 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