(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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Consider the initial condition where", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox[" f[x]=1", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[", if ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["|x|", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" is less than or equal to 1, and ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["f[x]=0", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[" if ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["|x|", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ " is greater than 1. 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The integral can be evaluated symbolically by using", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox[" Integrate", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[":\n", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[CellGroupData[{Cell[TextData["Integrate[Sin[x]/x,{x,0,Infinity}]"], "Input", AspectRatioFixed->True], Cell[TextData["General::intinit: Loading integration packages."], "Message", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData["\<\ Pi/2\ \>", "\<\ Pi -- 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "\nAlternatively, the numerical evaluation of (7.66) proceeds by the \ following command sequence. Notice that this procedure is different than \ that described in the book. The commands shown below are much more \ efficient, since the each range of the integral is calculated only once. The \ effects of this difference are more evident in version 2.1 than they were in \ version 1.2.\n"], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->14], Cell[TextData[ "PartialIntegral[i_]:=NIntegrate[Sin[t]/t,\n \ {t,(i-1)^4,i^4},MaxRecursion->20,\n \ WorkingPrecision->20,AccuracyGoal->7]"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "OldValue=0;\n Do[\n NewValue=OldValue+PartialIntegral[i];\n \ OldValue=NewValue;\n Print[N[NewValue,7]],\n {i,1,15}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "0.9460831\n1.631302\n1.561306\n1.570967\n1.572371\n1.570867\n1.570513\n\ 1.5706\n1.57076\n1.57089\n1.57077\n1.57079\n1.57082\n1.57077\n1.57079"], "Print", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox[ "\nTo four decimal places the numerical value of this integral agrees with \ the exact result: ", Evaluatable->False, AspectRatioFixed->True, FontSize->14], StyleBox["Pi/2=1.570796", Evaluatable->False, AspectRatioFixed->True, FontSize->14, FontSlant->"Italic"], StyleBox[ ". Listing the value of the integral with an increasing upper bound shows \ the slow convergence, which is due to the oscillatory nature of the \ integrand.", Evaluatable->False, AspectRatioFixed->True, FontSize->14] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->14]}, Open]], Cell[CellGroupData[{Cell[TextData[ "7.6 Green's Functions in the Presence of Boundaries:\n The Method of \ Images"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Figure 7.7"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "The two panels of this figure are generated by first constructing the \ fundamental solution and then taking the appropriate linear combination to \ satisfy the boundary condition at the origin. The fundamental solution is \ entered as\n", Evaluatable->False, AspectRatioFixed->True, FontSize->14]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["G0[x_,x0_,t_]:=(1/Sqrt[4 Pi t])Exp[-(x-x0)^2/(4 t)]"], "Input", AspectRatioFixed->True], Cell[TextData[StyleBox[ "\nThe Green's function for an absorbing boundary at the origin is then given \ by\n", Evaluatable->False, AspectRatioFixed->True, FontSize->14]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["GA[x_,x0_,t_]:=G0[x,x0,t]-G0[x,-x0,t]"], "Input", AspectRatioFixed->True], Cell[TextData[StyleBox[ "\nThe top panel of Figure 7.7 is obtain as follows:\n", Evaluatable->False, AspectRatioFixed->True, FontSize->14]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Plot[{GA[x,0.5,0.005],GA[x,0.5,0.01],GA[x,0.5,0.05],\n \ GA[x,0.5,0.15]},{x,0,1},PlotRange->{0,4},\n \ AxesLabel->{x,G},Ticks->{Automatic,Range[0,4,1]}]"], "Input", AspectRatioFixed->True], 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