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There are many similarities, such as the standard differentiation \ formulas. However, there are some surprises, and in this chapter we will \ encounter on of the hallmarks distinguishing complex functions from their \ real counterparts.\n\n\tIt is possible for a function defined on the real \ numbers to be differentiable everywhere and yet not be expressible as a power \ series (see Exercise 20 at the end of ", ButtonBox["Section 7.2", ButtonData:>{"ca0702.nb", None}, ButtonStyle->"Hyperlink"], "). In the complex case, however, things are much simpler! It turns out \ that if a complex function is analytic in the disk ", Cell[BoxData[ \(\(D\_r\) \((\[Alpha])\)\)]], ", its ", ButtonBox["Taylor", ButtonData:>{ URL[ "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Taylor.\ html"], None}, ButtonStyle->"Hyperlink"], " series about ", Cell[BoxData[ \(\[Alpha]\)]], " will converge to the function at every point in this disk. Thus, \ analytic functions are locally nothing more than \"glorified polynomials.\"\n\ \t\n\t\tWe shall also see that complex functions are the key to unlocking \ many of the mysteries encountered when power series are first introduced in a \ calculus course. We begin by discussing an important property associated \ with power series - uniform convergence.\n" }], "Text"], Cell[TextData[{ "\tRecall that if we have a function f(z) defined on a set T, the \ sequence of functions ", Cell[BoxData[ \({\(S\_n\) \((z)\)}\)]], " ", StyleBox["converges", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " to the function f at the point ", Cell[BoxData[ \(z\_0\ \[Epsilon]\ T\)]], " provided that ", Cell[BoxData[ \(lim\+\(n\ \[Rule] \ \[Infinity]\)\ S \((z\_0)\)\ = \ f \((z\_0)\)\)]], ". Thus, for the particular point ", Cell[BoxData[ \(z\_0\)]], ", this means that for every ", Cell[BoxData[ \(\[Epsilon] > 0\)]], ", there exists a positive integer ", Cell[BoxData[ \(N\_\(\(\[Element]\)\(\(,\)\(z\_0\)\)\)\)]], " (which depends on both ", Cell[BoxData[ \(\[Epsilon]\ \ and\ \ z\_0\)]], ") such that\n\n\t\tif ", Cell[BoxData[ \(n \[GreaterEqual] N\_\(\(\[Element]\)\(\(,\)\(z\_0\)\)\)\)]], ", then ", Cell[BoxData[ \(\(\(|\)\(S \((z\_0)\)\ - \ f \((z\_0)\)\)\(|\)\(\(<\)\(\ \)\(\[Epsilon]\)\)\)\)]], ". \n\t\nIf ", Cell[BoxData[ \(\(S\_n\) \((z)\)\)]], " is the ", Cell[BoxData[ \(n\^th\)]], " partial sum of the infinite series ", Cell[BoxData[ \(\[Sum]\+\(k = o\)\%\[Infinity]\( c\_k\) \((z - \[Alpha])\)\^k\)]], ", this statement becomes\n\n\t\tif ", Cell[BoxData[ \(n \[GreaterEqual] N\_\(\(\[Element]\)\(\(,\)\(z\_0\)\)\)\)]], ", then ", Cell[BoxData[ \(\(\(|\)\(\[Sum]\+\(k = o\)\%\(n - 1\)\(c\_k\) \((z\_0 - \ \[Alpha])\)\^k\ - \ f \((z\_0)\)\)\(|\)\(\(<\)\(\ \)\(\[Epsilon]\)\)\)\)]], ". \n\n\tIt is important to stress that for a given value of ", Cell[BoxData[ \(\[Epsilon]\)]], ", the integer ", Cell[BoxData[ \(N\_\(\(\[Element]\)\(\(,\)\(z\_0\)\)\)\)]], " we need to satisfy the inequality will often depend on our choice of \ point ", Cell[BoxData[ \(z\_0\)]], ". This is not the case if the sequence ", Cell[BoxData[ \({\(S\_n\) \((z)\)}\)]], " converges", " uniformly.", " For a ", StyleBox["uniformly convergent sequence", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ", it is possible to fine an integer N (which depends only on ", Cell[BoxData[ \(\[Epsilon]\)]], ") that guarantees that ", Cell[BoxData[ \(\(\(|\)\(S \((z\_0)\)\ - \ f \((z\_0)\)\)\(|\)\(\(<\)\(\ \)\(\[Epsilon]\)\)\)\)]], " no matter which value ", Cell[BoxData[ \(z\_0\ \[Epsilon]\ T\)]], " we pick. In other words, if n is large enough, the function ", Cell[BoxData[ \(\(S\_n\) \((z)\)\)]], " is ", StyleBox["uniformly close", FontColor->RGBColor[0, 0, 1]], " to f(z). Formally, we have the following definition. " }], "Text"], Cell[TextData[{ StyleBox["\nDefinition 7.1 (", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ButtonBox["Uniform Convergence", ButtonData:>{ URL[ "http://mathworld.wolfram.com/UniformConvergence.html"], None}, ButtonStyle->"Hyperlink"], StyleBox["), Page 262.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " The sequence ", Cell[BoxData[ \({\(S\_n\) \((z)\)}\)]], " converges uniformly to f(z) on the set T if for every ", Cell[BoxData[ \(\[Epsilon] > 0\)]], ", there exists a positive integer ", Cell[BoxData[ \(N\_\[Epsilon]\)]], " (which depends only on ", Cell[BoxData[ \(\[Epsilon]\)]], ") such that \n\n\t\tif ", Cell[BoxData[ \(n \[GreaterEqual] N\)]], ", then ", Cell[BoxData[ \(\(\(|\)\(\(S\_n\) \((z)\) - f \((z)\)\)\(|\)\(\(<\)\(\[Element]\)\)\)\)]], " for all ", Cell[BoxData[ \(z\ \[Epsilon]\ T\)]], ".\n\nIf in the preceding, ", Cell[BoxData[ \(\(S\_n\) \((z)\)\)]], " is the ", Cell[BoxData[ \(n\^th\)]], " partial sum of the infinite series ", Cell[BoxData[ \(\[Sum]\+\(k = o\)\%\[Infinity]\( c\_k\) \((z - \[Alpha])\)\^k\)]], ", we say that the series ", Cell[BoxData[ \(\[Sum]\+\(k = o\)\%\[Infinity]\( c\_k\) \((z - \[Alpha])\)\^k\)]], " converges uniformly to f(z) on the set T." }], "Text"], Cell[TextData[{ StyleBox["\nExample 7.1, Page 262.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " The sequence ", Cell[BoxData[ \({\(S\_n\) \((z)\)} = {\[ExponentialE]\^z + 1\/n}\)]], " converges uniformly to ", Cell[BoxData[ \(\(\(\ \)\(\[ExponentialE]\^z\)\)\)]], " on the entire complex plane because for any ", Cell[BoxData[ \(\[Epsilon] > 0\)]], ", statement (7-2) is satisfied for all z for ", Cell[BoxData[ \(n \[GreaterEqual] N\_\[Epsilon]\)]], ", where ", Cell[BoxData[ \(N\_\[Epsilon]\)]], " is any integer greater than ", Cell[BoxData[ \(1\/\[Epsilon]\)]], ". We leave the details of showing this as an exercise." }], "Text"], Cell[TextData[{ "\n\tA good example of a sequence of functions that does not converge \ uniformly is the sequence of partial sums comprising the geometric series. \ Recall that the geometric series has ", Cell[BoxData[ \(\(S\_n\) \((z)\) \(\[Sum]\+\(k = o\)\%n z\^k\)\)]], " converging to ", Cell[BoxData[ \(f \((z)\) = 1\/\(1\ - \ z\)\)]], " for all ", Cell[BoxData[ \(z \[Element] \(D\_1\) \((0)\)\)]], ". Because the real numbers are a subset of the complex numbers, we can \ show statement (7-2) is not satisfied by demonstrating it does not hold when \ we restrict our attention to the real numbers. In that context, ", Cell[BoxData[ \(\(D\_1\) \((0)\)\)]], " becomes the open interval (-1,1), and the inequality, ", Cell[BoxData[ \(\(\(|\)\(\(S\_n\) \((z)\) - f \((z)\)\)\(|\)\(\(<\)\(\[Element]\)\)\)\)]], ", becomes ", Cell[BoxData[ \(\(\(|\)\(\(S\_n\) \((x)\) - f \((x)\)\)\(|\)\(\(<\)\(\[Element]\)\)\)\)]], ", which for real variables is equivalent to ", Cell[BoxData[ \(\(\(f \((x)\)\)\(-\)\) \[Element] \(\(<\)\(\(S\_n\) \ \((x)\)\)\(<\)\(\(f \((x)\)\)\(+\)\)\) \[Element] \)]], ". If Statement (7-2) were to be satisfied, then given ", Cell[BoxData[ \(\[Epsilon] > 0\)]], ", ", Cell[BoxData[ \(\(S\_n\) \((x)\)\)]], "would be within an ", Cell[BoxData[ \(\[Epsilon]\)]], "-bandwidth of f(x) for all x in the interval (-1,1) provided n were large \ enough. However, there is an ", Cell[BoxData[ \(\[Epsilon]\)]], " such that no matter how large n is, we can find ", Cell[BoxData[ \(x\_0 \[Element] \((\(-1\), 1)\)\)]], " such that ", Cell[BoxData[ \(\(S\_n\) \((x\_0)\)\)]], " lies outside this bandwidth. In other words, this illustrates the \ negation of Statement (7-2), which in technical terms is the following.\n\n\t\ There exists ", Cell[BoxData[ \(\[Epsilon] > 0\)]], ", such that for all positive integers N, \n\tthere is some ", Cell[BoxData[ \(n \[GreaterEqual] N\)]], " and some ", Cell[BoxData[ \(z\_0 \[Element] T\)]], " \n\tsuch that ", Cell[BoxData[ \(\(\(|\)\(\(S\_n\) \((z\_0)\) - f \((z\_0)\)\)\(|\)\(\(\[GreaterEqual]\)\(\[Element]\)\)\)\)]], ". \n\nIn the exercises, we ask you to use this latter statement to show \ that the partial sums of the geometric series do not converge uniformly to ", Cell[BoxData[ \(f \((z)\) = 1\/\(1\ - \ z\)\)]], " for ", Cell[BoxData[ \(z \[Element] \(D\_1\) \((0)\)\)]], ". " }], "Text"], Cell[TextData[{ "\n\tA useful procedure known as the ", ButtonBox["Weierstrass", ButtonData:>{ URL[ "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Weierstrass.\ html"], None}, ButtonStyle->"Hyperlink"], " M-test, which can help determine whether an infinite series is uniformly \ convergent. " }], "Text"], Cell[TextData[{ StyleBox["\nTheorem 7.1, Page 263. (", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ButtonBox["Weierstrass M-Test", ButtonData:>{ URL[ "http://mathworld.wolfram.com/WeierstrassM-Test.html"], None}, ButtonStyle->"Hyperlink"], StyleBox[")", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " Suppose the infinite series ", Cell[BoxData[ \(\[Sum]\+\(k = 0\)\%\[Infinity]\( u\_k\) \((z)\)\)]], " has the property that for each k, ", Cell[BoxData[ \(\(\(|\)\(\(u\_k\) \((z)\)\)\(|\)\(\(\[LessEqual]\)\(\ \ \)\(M\_k\)\)\)\)]], " for all ", Cell[BoxData[ \(z\ \[Epsilon]\ T\)]], ". If ", Cell[BoxData[ \(\[Sum]\+\(k = 0\)\%\[Infinity] M\_k\)]], " converges, then ", Cell[BoxData[ \(\[Sum]\+\(k = 0\)\%\[Infinity]\( u\_k\) \((z)\)\)]], " converges uniformly on T." }], "Text"], Cell[TextData[StyleBox["Proof of Theorem 7.1, see text Page 263.", FontWeight->"Bold", FontColor->RGBColor[0, 1, 1]]], "Text"], Cell["\<\ \tTheorem 7.2 gives an interesting application of the Weierstrass \ M-test.\ \>", "Text"], Cell[TextData[{ StyleBox["\nTheorem 7.2, Page 264.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " Suppose the power series ", Cell[BoxData[ \(\[Sum]\+\(k = o\)\%\[Infinity]\( c\_k\) \((z - \[Alpha])\)\^k\)]], " has radius of convergence ", Cell[BoxData[ \(\[Rho] > 0\)]], ". Then for each r, (where ", Cell[BoxData[ \(0 < r < \[Rho]\)]], "), the series converges uniformly on the closed disk ", Cell[BoxData[ \(\(\(D\_r\)\&_\) \((\[Alpha])\) = {z : \(\(|\)\(z - \ \[Alpha]\)\(|\)\(\(\[LessEqual]\)\(r\)\)\)}\)]], ". " }], "Text"], Cell[TextData[{ StyleBox["Proof of Theorem 7.2, see text Page 264.", FontWeight->"Bold", FontColor->RGBColor[0, 1, 1]], StyleBox[" ", FontColor->RGBColor[0, 1, 1]] }], "Text"], Cell["\<\ \tAn immediate consequence of this theorem is the following result.\ \>", \ "Text"], Cell[TextData[{ StyleBox["\nCorollary 7.1, Page 265.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " For each r, (where ", Cell[BoxData[ \(0 < r < 1\)]], "), the geometric series converges uniformly on the closed disk ", Cell[BoxData[ \(\(\(D\_r\)\&_\) \((0)\) = {z : \(\(|\)\(z\)\(|\)\(\(\[LessEqual]\)\(r\ \)\)\)}\)]], ". " }], "Text"], Cell["\<\ \tTheorem 7.3 gives important properties of uniformly convergent sequences.\ \ \>", "Text"], Cell[TextData[{ StyleBox["\nTheorem 7.3, Page 265.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " Suppose ", Cell[BoxData[ \({\(S\_n\) \((z)\)}\)]], " is a sequence of continuous functions defined on a set T containing the \ contour C. \nIf ", Cell[BoxData[ \({\(S\_n\) \((z)\)}\)]], " converges uniformly to f(z) on the set T, then ", StyleBox[" \n\n\t", FontColor->RGBColor[0, 0, 1]], StyleBox["(i)", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ RowBox[{"f", StyleBox[\((z)\), FontSize->12]}]]], " is continuous on T, and \n\t\n\t", StyleBox["(ii)", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ \(\(\(lim\+\(k\ \[Rule] \ \[Infinity]\)\ \((\[Integral]\_C\ \(S\_n\) \ \((z)\) \[DifferentialD]z)\)\)\(\ \)\(=\)\(\ \)\)\)]], Cell[BoxData[ \(\[Integral]\_C\((lim\+\(k\ \[Rule] \ \[Infinity]\)\ \(S\_n\) \ \((z)\))\) \[DifferentialD]z\ = \ \[Integral]\_C f \((z)\) \[DifferentialD]z\)]], ". " }], "Text"], Cell[TextData[StyleBox["Proof of Theorem 7.3, see text Page 265.", FontWeight->"Bold", FontColor->RGBColor[0, 1, 1]]], "Text"], Cell[TextData[{ StyleBox["\nCorollary 7.2, Page 266.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " If the series ", Cell[BoxData[ \(\[Sum]\+\(k = o\)\%\[Infinity]\( c\_k\) \((z - \[Alpha])\)\^k\)]], " converges uniformly to ", Cell[BoxData[ RowBox[{"f", StyleBox[\((z)\), FontSize->12]}]]], " on the set T, and C is a contour contained in T, then \n\n\t\t", Cell[BoxData[ \(\(\(\[Sum]\+\(k = o\)\%\[Infinity]\((\[Integral]\_C\( c\_k\) \(\((z - \[Alpha])\)\^k\) \ \[DifferentialD]z)\)\)\(=\)\(\ \)\)\)]], Cell[BoxData[ \(\[Integral]\_C\((lim\+\(k\ \[Rule] \ \[Infinity]\)\ \(S\_n\) \ \((z)\))\) \[DifferentialD]z\ = \ \[Integral]\_C f \((z)\) \[DifferentialD]z\)]], ". \n\tor\n\t\t", Cell[BoxData[ \(\(\(\[Sum]\+\(k = o\)\%\[Infinity]\((\[Integral]\_C\( c\_k\) \(\((z - \[Alpha])\)\^k\) \ \[DifferentialD]z)\)\)\(=\)\(\ \)\)\)]], Cell[BoxData[ \(\[Integral]\_C\((\[Sum]\+\(k = o\)\%\[Infinity]\( c\_k\) \((z - \[Alpha])\)\^k)\) \[DifferentialD]z\ = \ \ \[Integral]\_C f \((z)\) \[DifferentialD]z\)]], ". 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" }], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution 7.2.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[z];\)\ \), "\[IndentingNewLine]", \(\(Clear[f, n, S17, S, S\[Infinity]];\)\ \), "\n", \(\(f[z_]\ = \ Log[1 - z];\)\ \), "\n", \(\(S[z_]\ = \ Normal[Series[f[z], {z, 0, 17}]];\)\ \), "\n", \(\(Print["\< f[z] = \>", f[z]];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\<\!\(S\_17\)[z] = \>\"", S[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell["\<\ Sum up infinitely many terms to verify that we have things right.\ \>", "Text",\ Evaluatable->False, AspectRatioFixed->False], Cell[BoxData[{ \(\[IndentingNewLine]\(S17[ z_]\ \ \ = \(-\(\[Sum]\+\(n = 1\)\%17\ z\^n\/n\)\);\)\ \), "\n", \(\(S\[Infinity][ z_]\ = \(-\(\[Sum]\+\(n = 1\)\%\[Infinity]\ z\^n\/n\)\);\)\ \), "\n", \(\(Print[\*"\"\<\!\(S\_17\)[z] = \>\"", S17[z]];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\<\!\(S\_\[Infinity]\)[z] = -\!\(\[Sum]\+\(n = 1\)\%\ \[Infinity]\) \!\(z\^n\/n\) = \>\"", S\[Infinity][z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nWe can use ", StyleBox["Mathematica", FontSlant->"Italic"], " to investigate how well the series is \"converging\" for real numbers." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Plot[{f[x], S17[x]}, {x, \(-0.99999\), 0.99999}, \[IndentingNewLine]PlotRange \[Rule] {{\(-1\), 1}, {\(-6\), 1}}, \[IndentingNewLine]Ticks \[Rule] {Range[\(-1\), 1, 0.5], Range[\(-6\), 1, 1]}, PlotStyle \[Rule] {Red, Green}, AxesLabel \[Rule] {"\", "\"}];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\\""];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\\"", S\[Infinity][z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell[TextData[{ "\nWe can use ", StyleBox["Mathematica", FontSlant->"Italic"], " to investigate how well the complex series is \"converging\" in the \ complex plane." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(PolarMap[ S17\ , {0, 0.9, 0.15}, {\(-\[Pi]\), \ \[Pi], \ \[Pi]\/12}, \ \[IndentingNewLine]PlotRange \[Rule] {{\(-2.5\), 0.8}, {\(-1.2\), 1.2}}, AspectRatio \[Rule] 2.4\/3.3, \[IndentingNewLine]Ticks \[Rule] {Range[\(-4\), 4, 1], Range[\(-4\), 4, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {DarkGreen, Green}];\)\ \), "\[IndentingNewLine]", \(Print["\", S17[z], "\<, where |z| < 0.9\>"]; 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Write a report on Taylor series.", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.996109, 0, 0.996109]], "\n\n", StyleBox["1.", FontWeight->"Bold"], " Arcache, Alexander, ''Expansion of Analytic Functions in Infinite Series \ and Infinite Products with Application to Multiple Valued Functions'', Am. \ Math. M., Vol. 72, No. 8. (Oct., 1965), pp. 861-864, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["2.", FontWeight->"Bold"], " Boas, R. P., ''Expansions of Analytic Functions,'' Trans. of the Am. \ Math. Soc., Vol. 48, No. 3. (Nov., 1940), pp. 467-487, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["3.", FontWeight->"Bold"], " Eidswick, Jack A., ''Alternatives to Taylor's Theorem in Proving \ Analyticity (in Classroom Notes),'' Am. Math. M., Vol. 82, No. 9. (Nov., \ 1975), pp. 929-931., ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["4.", FontWeight->"Bold"], " .Small, Robert D., ''Modified Convergence of Taylor Series (in Classroom \ Notes),'' Am. Math. M., Vol. 88, No. 6. (Jun. - Jul., 1981), pp. 439-441, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " " }], "Text"], Cell[TextData[{ StyleBox["Project II. Write a report on the topic of Analytic \ Continnuation.", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.996109, 0, 0.996109]], "\n\n", StyleBox["1.", FontWeight->"Bold"], " Bak, Joseph, and Donald J. Newman, (1982), Complex Analysis, N.Y., \ Springer-Verlag.\n\n", StyleBox["2.", FontWeight->"Bold"], " Boas, R. P., (1987), Invitation to Complex Analysis, N.Y., Random House.\n\ \n", StyleBox["3.", FontWeight->"Bold"], " Cunningham, John, (1965), Complex Variable Methods in Science and \ Technology, London, N.Y., Van Nostrand.\n\n", StyleBox["4.", FontWeight->"Bold"], " DePree, John D. and Charles C. Oehring, (1969), Elements of Complex \ Analysis, Reading, Mass., Addison-Wesley Pub. Co.\n\n", StyleBox["5.", FontWeight->"Bold"], " Derrick, William R., (1984), Complex Analysis and Applications, 2nd Ed., \ Belmont, CA, Wadsworth Pub. 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Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{12, 10}, {7, 7}}, LineSpacing->{1, 3}, CounterIncrements->"Text"], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}], Cell[StyleData["Text", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}], Cell[StyleData["Text", "Printout"], CellMargins->{{2, 2}, {6, 6}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], CellMargins->{{12, 10}, {6, 6}}, LineSpacing->{1, 3}, CounterIncrements->"SmallText", FontFamily->"Helvetica", FontSize->9], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, 10}, {8, 8}}, LineSpacing->{1, 5}, FontSize->12], Cell[StyleData["SmallText", "Condensed"], CellMargins->{{8, 10}, {5, 5}}, LineSpacing->{1, 2}, FontSize->9], Cell[StyleData["SmallText", "Printout"], CellMargins->{{2, 2}, {5, 5}}, FontSize->7] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles 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ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{43, Inherited}, {Inherited, 0}}, StyleMenuListing->None], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Condensed"], ImageSize->{175, 175}, ImageMargins->{{38, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Printout"], ImageSize->{250, 250}, ImageMargins->{{30, Inherited}, {Inherited, 0}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->12], Cell[StyleData["CellLabel", "Condensed"], FontSize->9], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", 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If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{1, -5}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]] }, Open ]] }] ] (******************************************************************* Cached data follows. 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