Teaching Complex Analysis as a Lab- Type ( flipped ) Course with a Focus on Geometric Interpretations using Mathematica

Transcription

1 Teaching Complex Analysis as a Lab- Type ( flipped ) Course with a Focus on Geometric Interpretations using Mathematica Bill Kinney, Bethel University, St. Paul, MN

2 2 KinneyComplexAnalysisLabCourse.nb Complex Arithmetic Products of Complex Numbers Manipulate Grid Show ListPlot pt1, pt2, PlotStyle Red, PointSize.05, ListPlot Re pt1 1 I pt1 2 pt2 1 I pt2 2, Im pt1 1 I pt1 2 pt2 1 I pt2 2, PlotStyle Magenta, PointSize.05, Graphics Thick, Blue, Line 0, 0, pt1, 0, 0, pt2, Graphics Thick, Green, Line 0, 0, Re pt1 1 I pt1 2 pt2 1 I pt2 2, Im pt1 1 I pt1 2 pt2 1 I pt2 2, PlotRange 5, AspectRatio 1, AxesOrigin 0, 0, AxesLabel "real", "imaginary", ImageSize Medium,,, Column Row Style Text " z 1 ", Large, " ", Item NumberForm Style N Norm pt1, Large, 6, 4, Frame True, Row Style Text " z 2 ", Large, " ", Item NumberForm Style N Norm pt2, Large, 6, 4, Frame True, Row Style Text " z 1 z 2 ", Large, " ", Item NumberForm Style N Norm Re pt1 1 I pt1 2 pt2 1 I pt2 2, Im pt1 1 I pt1 2 pt2 1 I pt2 2, Large, 6, 4, Frame True, Row Style Text "Arg z 1 degrees ", Large, " ", Item NumberForm Style N ArcTan pt1 1, pt Π, Large, 6, 4, Frame True, Row Style Text "Arg z 2 degrees ", Large, " ", Item NumberForm Style N ArcTan pt2 1, pt Π, Large, 6, 4, Frame True, Row Style Text "Arg z 1 z 2 degrees ", Large, " ", Item NumberForm Style N 180 Π ArcTan Re pt1 1 I pt1 2 pt2 1 I pt2 2, Im pt1 1 I pt1 2 pt2 1 I pt2 2, Large, 6, 4, Frame True, pt1, 1, 1, Locator, pt2, 1, 1, Locator

3 KinneyComplexAnalysisLabCourse.nb 3 Multiplication by Θ cos Θ sin Θ...can also be thought of in terms of the iteration of the linear transformation defined by T x cos Θ sin Θ y sin Θ cos Θ x y A Θ_ : Cos Θ Sin Θ Sin Θ Cos Θ ; T Θ_ x_, y_ : A Θ. x, y ; Manipulate Show ListPlot NestList T Θ, pt, n, PlotStyle Red, PointSize.02, PlotRange 2, AxesOrigin 0, 0, AspectRatio Automatic, AxesLabel "real", "imaginary", n, 0, 100, 1, Θ, Π 2, 0, 2 Π, pt, 1, 0, Locator, LabelStyle Large

4 4 KinneyComplexAnalysisLabCourse.nb Roots Manipulate Show Graphics Darker Green, Circle, ListPlot Table Arg pt 1 I pt 2 n 2 Π Abs pt 1 pt 2 ^ 1 m Re E^ I m Arg pt 1 I pt 2 n 2 Π Im E^ I, n, 0, 11, m AspectRatio Automatic, PlotStyle Red, PointSize.03, ListPlot pt, PlotStyle Black, PointSize.04, Table Graphics Thick, Blue, Line 0, 0, Abs pt 1 pt 2 ^ 1 m Arg pt 1 I pt 2 n 2 Π Re E^ I m Arg pt 1 I pt 2 n 2 Π Im E^ I m,, n, 0, 11, Table Graphics Thick, Dashed, Magenta, Line Abs pt 1 pt 2 ^ 1 m Arg pt 1 I pt 2 n 2 Π Re E^ I m, Arg pt 1 I pt 2 n 2 Π Im E^ I m, Abs pt 1 pt 2 ^ Arg pt 1 I pt 2 n 1 2 Π 1 m Re E^ I m, Arg pt 1 I pt 2 n 1 2 Π Im E^ I m, n, 0, 11, PlotRange 2, 2, 2, 2, Axes True, AxesLabel "real", "imaginary", pt, 1, 0, Locator, m, 2, 12, 1, LabelStyle Large Orbits of Points under the iteration of f z z 2, x_, y_ : x^2 y^2, 2 x y ; Manipulate Show Graphics Thick, Blue, Circle, ListPlot NestList, pt, n, PlotStyle Red, PointSize.05, PlotRange 2, Axes True, AxesOrigin 0, 0, TicksStyle 15, AspectRatio Automatic, AxesLabel "real", "imaginary", n, 0, 10, 1, pt, 3 5, 4 5, Locator, LabelStyle Large

5 KinneyComplexAnalysisLabCourse.nb 5 Successive Images of Disks under the iteration of f z z 2 x_, y_ : x^2 y^2, 2 x y ; Manipulate Show ParametricPlot NestList, x, y, n. x r Cos t c 1, y r Sin t c 2, t, 0, 2 Π, PlotStyle Thick, Red, ParametricPlot.5 Cos t,.5 Sin t, Cos t, Sin t, t, 0, 2 Π, PlotStyle Thick, Blue, Thick, Magenta, PlotRange 2, AxesOrigin 0, 0, AxesLabel "real", "imaginary", n, 0, 5, 1, r,.1,.01,.2, c,.6,.6, Locator, LabelStyle Large

6 6 KinneyComplexAnalysisLabCourse.nb Mandelbrot Set and Julia Sets (from the Wolfram Demonstrations Project) Julia Sets and the Mandelbrot Set

7 KinneyComplexAnalysisLabCourse.nb 7 JuliaModified@c_, opts D := Module@ 8invImage, reducedinvimage, pointssofar, res<, res = Resolution. 8opts<. 8Resolution 60<; invimage := H N@81, - 1< Floor@res Sqrt@ð - cdd resd & ð Flatten L &; reducedinvimage@points_d := Module@8newPoints<, newpoints = Complement@invImage@pointsD, pointssofard; pointssofar = Union@newPoints, pointssofard; newpointsd; pointssofar = Nest@invImage, 81<, 5D; FixedPoint@reducedInvImage, pointssofard; ListPlot@8Re@ðD, Im@ðD< & pointssofar, AspectRatio Automatic, ImageSize 250, Axes False, PlotRange , 1.8<, 8-1.8, 1.8<<, PlotStyle 8Black, AbsolutePointSize@TinyD< DD; boundarypic = ; H* The image of the Mandelbrot set was generated with the following code. *L H* The image was inserted into the initialization code to speed up *L H* the start up of the demonstration. *L H* MandelFunction = CompileA88c,_Complex<<, LengthAFixedPointListAð2 +c&,0,100, SameTest HAbs@ðD>2&LEEE; mandeldata = Table@MandelFunction@x+I*yD, 8y,-1.3,1.3, <,8x,-2,0.6, <D; kernel = 8 81,1,1<, 81,-8,1<, 81,1,1< <; convolveddata = Abs@ListConvolve@kernel, mandeldatadd; boundarypic = ArrayPlotAconvolvedData, Frame False, DataRange 88-2,0.6<,8-1.3,1.3<<, ColorFunction IGrayLevelAH1-ðL100 E&M, ImageSize 250E; *L

8 8 KinneyComplexAnalysisLabCourse.nb Manipulate Deploy Grid boundarypic, Dynamic JuliaModified pt 1 pt 2 I, PlotRange 1.8, 1.8, 1.8, 1.8, Text Dynamic If pt 2 0, Row Style "c", Italic, " ", ToString NumberForm pt 1, 5, 4, NumberPadding " ", "0", " ", ToString NumberForm pt 2, 5, 4, NumberPadding " ", "0", " ", Row Style "c", Italic, " ", ToString NumberForm pt 1, 5, 4, NumberPadding " ", "0", " ", ToString NumberForm pt 2, 5, 4, NumberPadding " ", "0", " ", SpanFromLeft, pt, 0, 0, 2, 1.3, 0.6, 1.3, Locator, SaveDefinitions True THIS NOTEBOOK IS THE SOURCE CODE FROM "Julia Sets and the Mandelbrot Set" from The Wolfram Demonstrations Project Contributed by: Mark McClure A full-function Wolfram Mathematica 6 system is required to edit this notebook. GET WOLFRAM MATHEMATICA 6»

9 KinneyComplexAnalysisLabCourse.nb 9 Open Sets in the Complex Plane Re z 0 is an open set Manipulate Show RegionPlot x 0, x, 5, 5, y, 5, 5, Frame False, Axes True, AxesLabel "real", "imaginary", RegionPlot x z0 1 ^2 y z0 2 ^2 z0 1 2 ^2, x, 5, 5, y, 5, 5, PlotStyle Yellow, PlotPoints 100, Graphics Thick, Dashed, Line 0, 10, 0, 10, ImageSize Large, z0, 2, 1, Locator z 1 is an open set Manipulate Show RegionPlot x^2 y^2 1, x, 1.5, 1.5, y, 1.5, 1.5, Frame False, Axes True, PerformanceGoal "Quality", Graphics Thick, Dashed, Circle 0, 0, 1 Abs z0 1 I z0 2 Graphics Thick, Dashed, Circle z0,, 2 1 Abs z0 1 I z0 2 RegionPlot x z0 1 ^2 y z0 2 ^2 2 x, 5, 5, y, 5, 5, PlotStyle Yellow, PlotPoints 100, PerformanceGoal "Quality", AxesLabel "real", "imaginary", ^2, ImageSize Large, z0,.5,.5, Locator

10 10 KinneyComplexAnalysisLabCourse.nb Stereographic Projection onto Riemann Sphere p1 0, 1, 0 ; p2 6, 8, 0 ; p3 3 10, 2 5, 0 ; p4 1, 1, 0 ; 2 p p_, q_, r_ : p^2 q^2 1, 2 q p^2 q^2 1, p^2 q^2 1 p^2 q^2 1 ; P1 p1 ; P2 p2 ; P3 p3 ; P4 p4 ; axes ParametricPlot3D t, 0, 0, 0, t, 0, 0, 0, t, t, 100, 100, PlotStyle Thick, Black ; equator ParametricPlot3D Cos t, Sin t, 0, t, 0, 2 Pi, PlotStyle Thick, Magenta ; Manipulate Show axes, equator, ContourPlot3D x^2 y^2 z^2 1, x, 2, 2, y, 2, 2, z, 1, 1, BoxRatios 2, 2, 1, ContourStyle Opacity.2, Mesh None, PerformanceGoal "Quality", ContourPlot3D z 0, x, 2, 2, y, 2, 2, z, 1, 1, BoxRatios 2, 2, 1, ContourStyle Opacity.1, Mesh None, ListPointPlot3D p1, p2, p3, p4, PlotStyle Red, PointSize.02, ListPointPlot3D P1, P2, P3, P4, PlotStyle Blue, PointSize.02, ParametricPlot3D p1 1 t t P1, p2 1 t t P2, p3 1 t t P3, p4 1 t t P4, t, 0, T, PlotStyle Thickness.007, Green, PlotRange 2, 2, 2, 2, 1, 1, AxesLabel x 1, x 2, x 3, ViewPoint 4, 1, 1, ImageSize Large, T,.0001, 1, LabelStyle Large

11 KinneyComplexAnalysisLabCourse.nb 11 2 p p_, q_, r_ : p^2 q^2 1, 2 q p^2 q^2 1, p^2 q^2 1 p^2 q^2 1 ; Manipulate Grid Show axes, equator, ContourPlot3D x^2 y^2 z^2 1, x, 2, 2, y, 2, 2, z, 1, 1, BoxRatios 2, 2, 1, ContourStyle Opacity.2, Mesh None, PerformanceGoal "Quality", ContourPlot3D z 0, x, 2.5, 2.5, y, 2.5, 2.5, z, 1, 1, BoxRatios 2, 2, 1, ContourStyle Opacity.1, Mesh None, ListPointPlot3D Append pt, 0, PlotStyle Red, PointSize.02, ListPointPlot3D Append pt, 0, PlotStyle Blue, PointSize.02, ParametricPlot3D Append pt, 0 1 t t Append pt, 0, t, 0, 1, PlotStyle Thickness.007, Green, PlotRange 2.5, 2.5, 2.5, 2.5, 1.5, 1.5, AxesLabel x 1, x 2, x 3, ViewPoint 4, 1, 1, ImageSize Large, Show ListPlot pt, PlotStyle Red, PointSize.08, Graphics Magenta, Circle, PlotRange 2.5, 2.5, 2.5, 2.5, AxesLabel "real", "imaginary", AspectRatio Automatic, pt, 2, 0, Locator

12 12 KinneyComplexAnalysisLabCourse.nb Cauchy-Riemann Equations u v, x y u y v x f z_ : Sin z ; ComplexExpand f x I y u x_, y_ : Cosh y Sin x ; v x_, y_ : Cos x Sinh y ; Manipulate Grid Show ContourPlot u x, y, x, 3.5, 3.5, y, 3.5, 3.5, Contours 40, PerformanceGoal "Quality", ListPlot pt, PlotStyle Red, PointSize.04, Frame False, Axes True, PlotLabel "Contour Map of u x,y Re sin x y ", AxesLabel "real", "imaginary", ImageSize Medium, Show ContourPlot v x, y, x, 3.5, 3.5, y, 3.5, 3.5, Contours 40, PerformanceGoal "Quality", ListPlot pt, PlotStyle Red, PointSize.04, Frame False, Axes True, PlotLabel "Contour Map of v x,y Im sin x y ", AxesLabel "real", "imaginary", ImageSize Medium, Show ContourPlot u x, y, x, pt 1.25, pt 1.25, y, pt 2.25, pt 2.25, PerformanceGoal "Quality", ListPlot pt, PlotStyle Red, PointSize.04, Graphics Thick, Blue, Arrow pt, pt.05, 0, Graphics Thick, Green, Arrow pt, pt 0,.05, Frame False, Axes True, AxesLabel "real", "imaginary", ImageSize Medium, Show ContourPlot v x, y, x, pt 1.25, pt 1.25, y, pt 2.25, pt 2.25, PerformanceGoal "Quality", ListPlot pt, PlotStyle Red, PointSize.04, Graphics Thick, Green, Arrow pt, pt.05, 0, Graphics Thick, Blue, Arrow pt, pt 0,.05, Frame False, Axes True, AxesLabel "real", "imaginary", ImageSize Medium, pt, 1, 1, Locator

13 KinneyComplexAnalysisLabCourse.nb 13 u x_, y_ : Cosh y Sin x ; v x_, y_ : Cos x Sinh y ; Grid Show Plot3D u x, y, x, 3.5, 3.5, y, 3.5, 3.5, PlotLabel "3D plot of u x,y Re sin x y ", AxesLabel "real", "imaginary", "u x,y ", ImageSize Medium, Show Plot3D v x, y, x, 3.5, 3.5, y, 3.5, 3.5, Axes True, PlotLabel "3D plot of v x,y Im sin x y ", AxesLabel "real", "imaginary", "v x,y ", ImageSize Medium

14 14 KinneyComplexAnalysisLabCourse.nb Conformal Mappings Images of Disks and (Filled-in) Squares under f z z 2 x_, y_ : x^2 y^2, 2 x y ; Manipulate Grid Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", ParametricPlot Ε 1 Ρ Cos t z0 1, Ε 1 Ρ Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, AxesLabel "x", "y", PlotLabel "z plane", ImageSize Medium, Frame False, Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", ParametricPlot Ε 1 Ρ Cos t z0 1, Ε 1 Ρ Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, Frame False, AxesLabel "u", "v", PlotLabel "w plane", ImageSize Medium, Ε, 1, 5, Ρ, 0, 1, z0, 0, 0, Locator

15 KinneyComplexAnalysisLabCourse.nb 15 x_, y_ : x^2 y^2, 2 x y ; Manipulate Grid Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", ParametricPlot Ε Cos t z0 1, 5 Ε Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, 5 ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 30, 30, 30, 30, AxesOrigin 0, 0, AxesLabel "x", "y", PlotLabel "z plane", ImageSize Medium, Frame False, Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", ParametricPlot Ε Cos t z0 1, Ε Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 30, 30, 30, 30, AxesOrigin 0, 0, Frame False, AxesLabel "u", "v", PlotLabel "w plane", ImageSize Medium, Ε, 2, 5, z0, 3, 2, Locator x_, y_ : x^2 y^2, 2 x y ; Manipulate Grid Show ParametricPlot x z0 1, y z0 2, x, Ε, Ε, y, Ε, Ε, PerformanceGoal "Quality", ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, AxesLabel "x", "y", PlotLabel "z plane", ImageSize Medium, Frame False, Show ParametricPlot x z0 1, y z0 2, x, Ε, Ε, y, Ε, Ε, PerformanceGoal "Quality", ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, Frame False, AxesLabel "u", "v", PlotLabel "w plane", ImageSize Medium, Ε, 1, 5, z0, 0, 0, Locator

16 16 KinneyComplexAnalysisLabCourse.nb Images under f z z 3 x_, y_ : x^3 3 x y^2, 3 x^2 y y^3 ; Manipulate Grid Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", ParametricPlot Ε 1 Ρ Cos t z0 1, Ε 1 Ρ Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, AxesLabel "x", "y", PlotLabel "z plane", ImageSize Medium, Frame False, Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", ParametricPlot Ε 1 Ρ Cos t z0 1, Ε 1 Ρ Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, Frame False, AxesLabel "u", "v", PlotLabel "w plane", ImageSize Medium, Ε,.5, 5, Ρ, 0, 1, z0, 1, 0, Locator x_, y_ : x^3 3 x y^2, 3 x^2 y y^3 ; Manipulate Grid Show ParametricPlot x z0 1, y z0 2, x, Ε, Ε, y, Ε, Ε, PerformanceGoal "Quality", ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, AxesLabel "x", "y", PlotLabel "z plane", ImageSize Medium, Frame False, Show ParametricPlot x z0 1, y z0 2, x, Ε, Ε, y, Ε, Ε, PerformanceGoal "Quality", ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, Frame False, AxesLabel "u", "v", PlotLabel "w plane", ImageSize Medium, Ε,.5, 5, z0, 1, 0, Locator

17 KinneyComplexAnalysisLabCourse.nb 17 f z 1 z x x_, y_ : x^2 y^2, y x^2 y^2 ; Manipulate Grid Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", ParametricPlot Ε 1 Ρ Cos t z0 1, Ε 1 Ρ Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, AxesLabel "x", "y", PlotLabel "z plane", ImageSize Medium, Frame False, Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", PlotPoints 60, ParametricPlot Ε 1 Ρ Cos t z0 1, Ε 1 Ρ Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, Frame False, AxesLabel "u", "v", PlotLabel "w plane", ImageSize Medium, Ε, 1, 5, Ρ, 0, 1, z0, 2, 1, Locator x x_, y_ : x^2 y^2, y x^2 y^2 ; Manipulate Grid Show ParametricPlot x z0 1, y z0 2, x, Ε, Ε, y, Ε, Ε, PerformanceGoal "Quality", ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, AxesLabel "x", "y", PlotLabel "z plane", ImageSize Medium, Frame False, Show ParametricPlot x z0 1, y z0 2, x, Ε, Ε, y, Ε, Ε, PerformanceGoal "Quality", PlotPoints 60, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, Frame False, AxesLabel "u", "v", PlotLabel "w plane", ImageSize Medium, Ε,.75, 5, z0, 1, 0, Locator

18 18 KinneyComplexAnalysisLabCourse.nb f z z x_, y_ : E^ x Cos y, E^ x Sin y ; Manipulate Grid Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", ParametricPlot Ε 1 Ρ Cos t z0 1, Ε 1 Ρ Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, AxesLabel "x", "y", PlotLabel "z plane", ImageSize Medium, Frame False, Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", PlotPoints 30, ParametricPlot Ε 1 Ρ Cos t z0 1, Ε 1 Ρ Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, Frame False, AxesLabel "u", "v", PlotLabel "w plane", ImageSize Medium, Ε, 1, 5, Ρ, 0, 1, z0, 0, 0, Locator x_, y_ : E^ x Cos y, E^ x Sin y ; Manipulate Grid Show ParametricPlot x z0 1, y z0 2, x, Ε, Ε, y, Ε, Ε, PerformanceGoal "Quality", ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, AxesLabel "x", "y", PlotLabel "z plane", ImageSize Medium, Frame False, Show ParametricPlot x z0 1, y z0 2, x, Ε, Ε, y, Ε, Ε, PerformanceGoal "Quality", PlotPoints 30, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, Frame False, AxesLabel "u", "v", PlotLabel "w plane", ImageSize Medium, Ε,.75, 5, z0, 1, 0, Locator

19 KinneyComplexAnalysisLabCourse.nb 19 f z cos z x_, y_ : Cos x Cosh y, Sin x Sinh y ; Manipulate Grid Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", ParametricPlot Ε 1 Ρ Cos t z0 1, Ε 1 Ρ Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, AxesLabel "x", "y", PlotLabel "z plane", ImageSize Medium, Frame False, Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", PlotPoints 30, ParametricPlot Ε 1 Ρ Cos t z0 1, Ε 1 Ρ Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, Frame False, AxesLabel "u", "v", PlotLabel "w plane", ImageSize Medium, Ε, 1, 5, Ρ, 0, 1, z0, 0, 0, Locator x_, y_ : Cos x Cosh y, Sin x Sinh y ; Manipulate Grid Show ParametricPlot x z0 1, y z0 2, x, Ε, Ε, y, Ε, Ε, PerformanceGoal "Quality", ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, AxesLabel "x", "y", PlotLabel "z plane", ImageSize Medium, Frame False, Show ParametricPlot x z0 1, y z0 2, x, Ε, Ε, y, Ε, Ε, PerformanceGoal "Quality", PlotPoints 30, ListPlot z0, PlotStyle Black, PointSize.03, PlotRange 10, 10, 10, 10, AxesOrigin 0, 0, Frame False, AxesLabel "u", "v", PlotLabel "w plane", ImageSize Medium, Ε,.75, 5, z0, 1, 0, Locator f z sin z, with zeros and critical points marked u x_, y_ : Cosh y Sin x ; v x_, y_ : Cos x Sinh y ; x_, y_ : u x, y, v x, y ; Manipulate Grid Show ParametricPlot z0 Ε x, y, x, 0, 1, y, 0, 1, PlotStyle Red, PerformanceGoal "Quality", ListPlot z0, PlotStyle Red, PointSize.03, ListPlot Pi 2, 0, 3 Pi 2, 0, Pi 2, 0, 3 Pi 2, 0, PlotStyle Green, PointSize.02, ListPlot 0, 0, Pi, 0, Pi, 0, PlotStyle Blue, PointSize.015, PlotRange 6, ImageSize Medium, Frame False, Axes True, AxesLabel "x", "y", Show ParametricPlot z0 Ε x, y, x, 0, 1, y, 0, 1, PlotStyle Red, PerformanceGoal "Quality", ListPlot z0, PlotStyle Red, PointSize.03, PlotRange 6, ImageSize Medium, Frame False, Axes True, AxesLabel "u", "v", z0, 0,.5, Locator, Ε, 2,.001

20 20 KinneyComplexAnalysisLabCourse.nb Calculating areas of images by a change-of-variables formula (involving the Jacobian determinant of the mapping) Jac _ x_, y_ : D u x0, y0, x0. x0 x, y0 y D u x0, y0, y0. x0 x, y0 y Det D v x0, y0, x0. x0 x, y0 y D v x0, y0, y0. x0 x, y0 y ; Area _, Ε_, z0_ : NIntegrate Jac r Cos Θ z0 1, r Sin Θ z0 2 r, r, 0, Ε, Θ, 0, 2 Π ; MappingsAndJacobianDeterminants _, Jac_, x_, xmin_, xmax_, y_, ymin_, ymax_, U_, umin_, umax_, V_, vmin_, vmax_ : Manipulate Grid Style Text "Value of Jacobian Det in Center ", Medium, Red,, Item NumberForm Style Jac z0, FontSize 20, Red, 6, 4,, Style Text "Domain Disk Area ", Medium, Red, Style Text "Image Disk Area based on overlaps as well ", Medium, Blue,, Item NumberForm Style N Π Ε^2, FontSize 20, Red, 6, 4, Item NumberForm Style Area, Ε, z0, FontSize 20, Blue, 6, 4,, Style Text "Derivative Based Estimate for Area of Image ", Medium, Blue,, Item NumberForm Style N Jac z0 Π Ε^2, FontSize 20, Red, 6, 4,, Show ContourPlot Jac x, y, x, xmin, xmax, y, ymin, ymax, Contours 30, PerformanceGoal "Quality", ContourLabels Automatic, ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", PlotStyle Green, ParametricPlot Ε Ρ Cos t z0 1, Ε Ρ Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Red, PointSize.03, PlotRange xmin, xmax, ymin, ymax, AxesOrigin 0, 0, Axes True, AxesLabel "x", "y", PlotLabel "z plane", TicksStyle 15, ImageSize Medium, Frame False, Show ParametricPlot r Cos t z0 1, r Sin t z0 2, t, 0, 2 Π, r, 0, Ε, PerformanceGoal "Quality", PlotPoints 30, PlotStyle Green, ParametricPlot Ε Ρ Cos t z0 1, Ε Ρ Sin t z0 2, t, 0, 2 Π, PlotStyle Thick, Blue, ListPlot z0, PlotStyle Red, PointSize.03, PlotRange umin, umax, vmin, vmax, AxesOrigin 0, 0, Frame False, AxesLabel "u", "v", PlotLabel "w plane", TicksStyle 15, ImageSize Medium, Ε, 1,.1, 2, Ρ, 1, 0, z0, 1, 1, Locator, LabelStyle Large ; ComplexExpand x I y ^3 5 x I y ^2 u x_, y_ : 5 x 2 x 3 5 y 2 3 x y 2 ; v x_, y_ : 10 x y 3 x 2 y y 3 ; x_, y_ : u x, y, v x, y ; MappingsAndJacobianDeterminants, Jac, x, 4, 4, y, 4, 4, U, 100, 100, V, 100, 100

21 KinneyComplexAnalysisLabCourse.nb 21

22 22 KinneyComplexAnalysisLabCourse.nb The Derivative as a Local Amplitwist...termi nology credit to Tristan Needham (from his book Visual Complex Analysis ) f z z 2

23 KinneyComplexAnalysisLabCourse.nb 23 x_, y_ : x^2 y^2, 2 x y ; Prime x_, y_ : 2 x, 2 y ; Manipulate Grid Style Text "Local Dilation Factor ", Medium,, Item NumberForm Style N Norm Prime z0, FontSize 20, Red, 6, 4,, Style Text "Local Twist Amount in degrees ", Medium,, Item NumberForm Style N 180 Π ArcTan Prime z0 1, Prime z0 2, FontSize 20, Blue, 6, 4,, Show ParametricPlot z0 x, y, x, 0, Ε, y, 0, Ε, PerformanceGoal "Quality", Graphics Thick, Red, Arrow z0, z0 Ε, 0, Graphics Thick, Blue, Arrow z0, z0 0, Ε, PlotRange 0, 5, 0, 5, ImageSize Medium, Frame False, Axes True, AxesLabel "x", "y", TicksStyle Large, Show ParametricPlot z0 x, y, x, 0, Ε, y, 0, Ε, PerformanceGoal "Quality", Graphics Thick, Red, Arrow z0, z0 Ε, 0, Graphics Thick, Blue, Arrow z0, z0 0, Ε, PlotRange 12, 12, 0, 12, ImageSize Large, Frame False, Axes True, AxesLabel "u", "v", TicksStyle Large, z0, 2, 1, Locator, Ε, 1,.001, LabelStyle Large f z sin z u x_, y_ : Cosh y Sin x ; v x_, y_ : Cos x Sinh y ; x_, y_ : u x, y, v x, y ; Prime x_, y_ : Cos x Cosh y, Sin x Sinh y ; Manipulate Grid Style Text "Local Dilation Factor ", Medium,, Item NumberForm Style N Norm Prime z0, FontSize 20, Red, 6, 4,, Style Text "Local Twist Amount in degrees ", Medium,, Item NumberForm Style N 180 Π ArcTan Prime z0 1, Prime z0 2, FontSize 20, Blue, 6, 4,, Show ParametricPlot z0 x, y, x, 0, Ε, y, 0, Ε, PerformanceGoal "Quality", Graphics Thick, Red, Arrow z0, z0 Ε, 0, Graphics Thick, Blue, Arrow z0, z0 0, Ε, PlotRange 0, 5, 0, 5, ImageSize Medium, Frame False, Axes True, AxesLabel "x", "y", Show ParametricPlot z0 x, y, x, 0, Ε, y, 0, Ε, PerformanceGoal "Quality", Graphics Thick, Red, Arrow z0, z0 Ε, 0, Graphics Thick, Blue, Arrow z0, z0 0, Ε, PlotRange 0, 5, 2.5, 2.5, ImageSize Medium, Frame False, Axes True, AxesLabel "u", "v", z0, 1, 1, Locator, Ε,.5,.001, LabelStyle Large

24 24 KinneyComplexAnalysisLabCourse.nb Harmonic Functions Solving Laplace s Equation on a Domain Φ x_, y_ : Arg x I y 1 Arg x I y 2 ; Grid ContourPlot Φ x, y, x, 4, 4, y, 4, 4, Contours 50, Frame False, Axes True, RegionFunction Function x, y, z, y 0, AxesLabel "x", "y", ImageSize Medium,, Show Plot3D Φ x, y, x, 4, 4, y, 4, 4, PlotStyle Opacity.8, RegionFunction Function x, y, z, y 0, ParametricPlot3D t, 0, 0, 0, t, 0, 0, 0, t, t, 20, 20, PlotStyle Thick, ListPointPlot3D 2, 3, Pi 4, PlotStyle Red, PointSize.03, ViewPoint 2, 1, 1, Axes True, AxesLabel "x", "y", "Φ", ImageSize Medium Illustrating the Maximum/Minimum Principle f z_ : z^4 5 z^3 2 z^2 6 z 1; ComplexExpand f x I y u x_, y_ : 1 6 x 2 x 2 5 x 3 x 4 2 y 2 15 x y 2 6 x 2 y 2 y 4 ; Manipulate Show Plot3D u x, y, x, 4, 4, y, 4, 4, PlotRange 30, 30, PerformanceGoal "Quality", ParametricPlot3D R Cos t pt 1, R Sin t pt 2, u R Cos t pt 1, R Sin t pt 2, t, 0, 2 Π, PlotStyle Thick, Red, ImageSize Large, pt, 0, 0, Locator, R, 1,.5, 3, LabelStyle Large

25 KinneyComplexAnalysisLabCourse.nb 25 Complex Integration...though t of as f z z u, v s v, u s f z x 2 y y 2 u x_, y_ : x 2 y; v x_, y_ : y^2; x t_ : Cos t ; y t_ : Sin t ; 2 Π z t_ : x t, y t ; u x t, y t, v x t, y t. x' t, y' t t 0 2 Π I v x t, y t, u x t, y t. x' t, y' t t 0 2 Π

26 26 KinneyComplexAnalysisLabCourse.nb u x_, y_ : x 2 y; v x_, y_ : y^2; x t_ : Cos t ; y t_ : Sin t ; z t_ : x t, y t ; unegvbackground Show VectorPlot u x, y, v x, y, x, 2, 2, y, 2, 2, VectorStyle Blue, ParametricPlot z t, t, 0, 2 Π, PlotStyle Thick, Red, ListPlot 0, 0, PlotStyle Black, PointSize.03, Flatten Table Graphics Thick, Magenta, Arrow z Θ, z Θ u x Θ, y Θ, v x Θ, y Θ, Θ, 0, 2 Π, 2 Π 40, Frame False, Axes True, AxesLabel "x", "y", PlotLabel "The vector field u, v and ", TicksStyle 15 ; vubackground Show VectorPlot v x, y, u x, y, x, 2, 2, y, 2, 2, VectorStyle Blue, ParametricPlot z t, t, 0, 2 Π, PlotStyle Thick, Red, ListPlot 0, 0, PlotStyle Black, PointSize.03, Flatten Table Graphics Thick, Magenta, Arrow z Θ, z Θ v x Θ, y Θ, u x Θ, y Θ, Θ, 0, 2 Π, 2 Π 40, Frame False, Axes True, AxesLabel "x", "y", PlotLabel "The vector field v,u and ", TicksStyle 15 ; Manipulate Grid Show unegvbackground, Graphics Thick, Black, Arrow z b, z b z' b, ImageSize Medium, Show Plot u x t, y t, v x t, y t.z' t, t, 0, 2 Π, PlotStyle Thick, Magenta, ListPlot b, u x b, y b, v x b, y b.z' b, PlotStyle Black, PointSize.02, Graphics Thick, Black, Dashed, Line 0, 1, 2 Π, 1, ImageSize Medium, AxesLabel "t", " u, v z t ", TicksStyle 15, Show vubackground, Graphics Thick, Black, Arrow z b, z b z' b, ImageSize Medium, Show Plot v x t, y t, u x t, y t.z' t, t, 0, 2 Π, PlotStyle Thick, Magenta, ListPlot b, v x b, y b, u x b, y b.z' b, PlotStyle Black, PointSize.02, Graphics Thick, Black, Dashed, Line 0,.5, 2 Π,.5, ImageSize Medium, AxesLabel "t", " v,u z t ", TicksStyle 15, b, 0, 2 Π, LabelStyle Large

27 KinneyComplexAnalysisLabCourse.nb 27 f z z 2 u x_, y_ : x^2 y^2; v x_, y_ : 2 x y; x t_ : Cos t ; y t_ : Sin t ; z t_ : x t, y t ; unegvbackground Show VectorPlot u x, y, v x, y, x, 2, 2, y, 2, 2, VectorStyle Blue, ParametricPlot z t, t, 0, 2 Π, PlotStyle Thick, Red, ListPlot 0, 0, PlotStyle Black, PointSize.03, Flatten Table Graphics Thick, Magenta, Arrow z Θ, z Θ u x Θ, y Θ, v x Θ, y Θ, Θ, 0, 2 Π, 2 Π 40, Frame False, Axes True, AxesLabel x, y, PlotLabel "The vector field u, v and " ; vubackground Show VectorPlot v x, y, u x, y, x, 2, 2, y, 2, 2, VectorStyle Blue, ParametricPlot z t, t, 0, 2 Π, PlotStyle Thick, Red, ListPlot 0, 0, PlotStyle Black, PointSize.03, Flatten Table Graphics Thick, Magenta, Arrow z Θ, z Θ v x Θ, y Θ, u x Θ, y Θ, Θ, 0, 2 Π, 2 Π 40, Frame False, Axes True, AxesLabel x, y, PlotLabel "The vector field v,u and " ; Manipulate Grid Show unegvbackground, Graphics Thick, Black, Arrow z b, z b z' b, ImageSize Medium, Show Plot u x t, y t, v x t, y t.z' t, t, 0, 2 Π, PlotStyle Thick, Magenta, ListPlot b, u x b, y b, v x b, y b.z' b, PlotStyle Black, PointSize.02, Graphics Thick, Black, Dashed, Line 0, 0, 2 Π, 0, ImageSize Medium, AxesLabel "t", " u, v z t ", Show vubackground, Graphics Thick, Black, Arrow z b, z b z' b, ImageSize Medium, Show Plot v x t, y t, u x t, y t.z' t, t, 0, 2 Π, PlotStyle Thick, Magenta, ListPlot b, v x b, y b, u x b, y b.z' b, PlotStyle Black, PointSize.02, Graphics Thick, Black, Dashed, Line 0, 0, 2 Π, 0, ImageSize Medium, AxesLabel "t", " v,u z t ", b, 0, 2 Π

28 28 KinneyComplexAnalysisLabCourse.nb f z z 2 u x_, y_ : x 2 y 2 ; v x_, y_ : x 2 y x y x 2 y 2 2 ; x t_ : Cos t ; y t_ : Sin t ; z t_ : x t, y t ; unegvbackground Show VectorPlot u x, y, v x, y, x, 2, 2, y, 2, 2, VectorStyle Blue, VectorScale.03, 1, None, ParametricPlot z t, t, 0, 2 Π, PlotStyle Thick, Red, ListPlot 0, 0, PlotStyle Black, PointSize.03, Flatten Table Graphics Thick, Magenta, Arrow z Θ, z Θ u x Θ, y Θ, v x Θ, y Θ, Θ, 0, 2 Π, 2 Π 40, Frame False, Axes True, AxesLabel x, y, PlotLabel "The vector field u, v and " ; vubackground Show VectorPlot v x, y, u x, y, x, 2, 2, y, 2, 2, VectorStyle Blue, VectorScale.03, 1, None, ParametricPlot z t, t, 0, 2 Π, PlotStyle Thick, Red, ListPlot 0, 0, PlotStyle Black, PointSize.03, Flatten Table Graphics Thick, Magenta, Arrow z Θ, z Θ v x Θ, y Θ, u x Θ, y Θ, Θ, 0, 2 Π, 2 Π 40, Frame False, Axes True, AxesLabel x, y, PlotLabel "The vector field v,u and " ; Manipulate Grid Show unegvbackground, Graphics Thick, Black, Arrow z b, z b z' b, ImageSize Medium, Show Plot u x t, y t, v x t, y t.z' t, t, 0, 2 Π, PlotStyle Thick, Magenta, ListPlot b, u x b, y b, v x b, y b.z' b, PlotStyle Black, PointSize.02, Graphics Thick, Black, Dashed, Line 0, 0, 2 Π, 0, ImageSize Medium, AxesLabel "t", " u, v z t ", Show vubackground, Graphics Thick, Black, Arrow z b, z b z' b, ImageSize Medium, Show Plot v x t, y t, u x t, y t.z' t, t, 0, 2 Π, PlotStyle Thick, Magenta, ListPlot b, v x b, y b, u x b, y b.z' b, PlotStyle Black, PointSize.02, Graphics Thick, Black, Dashed, Line 0, 0, 2 Π, 0, ImageSize Medium, AxesLabel "t", " v,u z t ", b, 0, 2 Π

29 KinneyComplexAnalysisLabCourse.nb 29 f z z 1 u x_, y_ : x y ; v x_, y_ : x 2 2 y x 2 y 2 ; x t_ : Cos t ; y t_ : Sin t ; z t_ : x t, y t ; unegvbackground Show VectorPlot u x, y, v x, y, x, 2, 2, y, 2, 2, VectorStyle Blue, VectorScale.03, 1, None, ParametricPlot z t, t, 0, 2 Π, PlotStyle Thick, Red, ListPlot 0, 0, PlotStyle Black, PointSize.03, Flatten Table Graphics Thick, Magenta, Arrow z Θ, z Θ u x Θ, y Θ, v x Θ, y Θ, Θ, 0, 2 Π, 2 Π 40, Frame False, Axes True, AxesLabel x, y, PlotLabel "The vector field u, v and " ; vubackground Show VectorPlot v x, y, u x, y, x, 2, 2, y, 2, 2, VectorStyle Blue, VectorScale.03, 1, None, ParametricPlot z t, t, 0, 2 Π, PlotStyle Thick, Red, ListPlot 0, 0, PlotStyle Black, PointSize.03, Flatten Table Graphics Thick, Magenta, Arrow z Θ, z Θ v x Θ, y Θ, u x Θ, y Θ, Θ, 0, 2 Π, 2 Π 40, Frame False, Axes True, AxesLabel x, y, PlotLabel "The vector field v,u and " ; Manipulate Grid Show unegvbackground, Graphics Thick, Black, Arrow z b, z b z' b, ImageSize Medium, Show Plot u x t, y t, v x t, y t.z' t, t, 0, 2 Π, PlotStyle Thick, Magenta, ListPlot b, u x b, y b, v x b, y b.z' b, PlotStyle Black, PointSize.02, Graphics Thick, Black, Dashed, Line 0, 0, 2 Π, 0, ImageSize Medium, AxesOrigin 0, 0, AxesLabel "t", " u, v z t ", Show vubackground, Graphics Thick, Black, Arrow z b, z b z' b, ImageSize Medium, Show Plot v x t, y t, u x t, y t.z' t, t, 0, 2 Π, PlotStyle Thick, Magenta, PlotRange 2, 2, ListPlot b, v x b, y b, u x b, y b.z' b, PlotStyle Black, PointSize.02, Graphics Thick, Black, Dashed, Line 0, 1, 2 Π, 1, ImageSize Medium, AxesOrigin 0, 0, AxesLabel "t", " v,u z t ", b, 0, 2 Π

30 30 KinneyComplexAnalysisLabCourse.nb Complex Integration...though t of as f z z u v z F U V f z z 2 F z_ : z^3 3; z t_ : E^ I t ; Grid Show ParametricPlot3D t, 0, 0, 0, t, 0, 0, 0, t, t, 5, 5, PlotStyle Thick, Plot3D Re F x I y, x, 4, 4, y, 4, 4, ParametricPlot3D Re z t, Im z t, Re F z t, t, 0, 2 Π, PlotStyle Thick, Red, ImageSize 500, PlotRange 2, 2, ViewPoint 2, 1, 1, AxesLabel "Re z ", "Im z ", "Re F z ", Show ParametricPlot3D t, 0, 0, 0, t, 0, 0, 0, t, t, 5, 5, PlotStyle Thick, Plot3D Im F x I y, x, 4, 4, y, 4, 4, ParametricPlot3D Re z t, Im z t, Im F z t, t, 0, 2 Π, PlotStyle Thick, Red, ImageSize 500, ViewPoint 2, 1, 1, PlotRange 2, 2, AxesLabel "Re z ", "Im z ", "Im F z "

31 KinneyComplexAnalysisLabCourse.nb 31 f z 3 z 2 along line segment from 0 to 1 2 F z_ : z^3; z t_ : 1 2 I t; Grid Show ParametricPlot3D t, 0, 0, 0, t, 0, 0, 0, t, t, 20, 20, PlotStyle Thick, Plot3D Re F x I y, x, 2.1, 2.1, y, 2.1, 2.1, ParametricPlot3D Re z t, Im z t, Re F z t, t, 0, 2 Π, PlotStyle Thickness.02, Red, ImageSize 500, PlotRange 2.1, 2.1, 2.1, 2.1, 11, 11, ViewPoint 2, 1, 1, AxesLabel "Re z ", "Im z ", "Re F z ", BoxRatios 1, 1, 1, Show ParametricPlot3D t, 0, 0, 0, t, 0, 0, 0, t, t, 20, 20, PlotStyle Thick, Plot3D Im F x I y, x, 2.1, 2.1, y, 2.1, 2.1, ParametricPlot3D Re z t, Im z t, Im F z t, t, 0, 2 Π, PlotStyle Thickness.02, Red, ImageSize 500, ViewPoint 2, 1, 1, PlotRange 2.1, 2.1, 2.1, 2.1, 2, 2, AxesLabel "Re z ", "Im z ", "Im F z ", BoxRatios 1, 1, 1 f z z 2 F z_ : z^ 1 ; z t_ : E^ I t ; Grid Show ParametricPlot3D t, 0, 0, 0, t, 0, 0, 0, t, t, 5, 5, PlotStyle Thick, Plot3D Re F x I y, x, 4, 4, y, 4, 4, ParametricPlot3D Re z t, Im z t, Re F z t, t, 0, 2 Π, PlotStyle Thick, Red, ImageSize 500, PlotRange 2, 2, ViewPoint 2, 1, 1, AxesLabel "Re z ", "Im z ", "Re F z ", Show ParametricPlot3D t, 0, 0, 0, t, 0, 0, 0, t, t, 5, 5, PlotStyle Thick, Plot3D Im F x I y, x, 4, 4, y, 4, 4, ParametricPlot3D Re z t, Im z t, Im F z t, t, 0, 2 Π, PlotStyle Thick, Red, ImageSize 500, ViewPoint 2, 1, 1, PlotRange 2, 2, AxesLabel "Re z ", "Im z ", "Im F z " f z z 1 F z_ : Log z ; z t_ : E^ I t ; Grid Show ParametricPlot3D t, 0, 0, 0, t, 0, 0, 0, t, t, 5, 5, PlotStyle Thick, Plot3D Re F x I y, x, 1.2, 1.2, y, 1.2, 1.2, ParametricPlot3D Re z t, Im z t, Re F z t, t, 0, 2 Π, PlotStyle Thickness.02, Red, ImageSize 500, PlotRange 1.2, 1.2, 1.2, 1.2, 2, 1, ViewPoint 2, 1, 1, AxesLabel "Re z ", "Im z ", "Re Log z ", BoxRatios 1, 1, 1, Show ParametricPlot3D t, 0, 0, 0, t, 0, 0, 0, t, t, 5, 5, PlotStyle Thick, Plot3D Im F x I y, x, 1.2, 1.2, y, 1.2, 1.2, ParametricPlot3D Re z t, Im z t, Im F z t, t, 0, 2 Π, PlotStyle Thickness.02, Red, ImageSize 500, ViewPoint 2, 1, 1, PlotRange 1.2, 1.2, 1.2, 1.2, Π, Π, AxesLabel "Re z ", "Im z ", "Im Log z ", BoxRatios 1, 1, 1

32 32 KinneyComplexAnalysisLabCourse.nb f z 1 z z 2 f z_ : z I z 2 ; f z z 2 z ArcTan 1 2 z Log 2 z Log 1 z2 z1 t_ : I E^ I t ; z2 t_ : 2 E^ I t ; z3 t_ : 3 E^ I t ; Residue f z, z, I Π Residue f z, z, I Π Residue f z, z, Π Residue f z, z, Π N 2 Π Residue f z, z, I NIntegrate f z1 t z1' t, t, 0, 2 Π

33 KinneyComplexAnalysisLabCourse.nb 33 N 2 Π Residue f z, z, 2 NIntegrate f z2 t z2' t, t, 0, 2 Π NIntegrate f z3 t z3' t, t, 0, 2 Π F z_ : z ArcTan 1 2 z Log 2 z Log 1 z2 ; Grid Show ParametricPlot3D t, 0, 0, 0, t, 0, 0, 0, t, t, 5, 5, PlotStyle Thick, Plot3D Re F x I y, x, 4, 4, y, 4, 4, ParametricPlot3D Re z1 t, Im z1 t, Re F z1 t, Re z2 t, Im z2 t, Re F z2 t, Re z3 t, Im z3 t, Re F z3 t, t, 0, 2 Π, PlotStyle Thickness.02, Red, Thickness.02, Blue, Thickness.02, Darker Green, ImageSize 500, PlotRange 4, 4, 4, 4, 1, 1, ViewPoint 2, 1, 1, AxesLabel "Re z ", "Im z ", "Re F z ", BoxRatios 1, 1, 1, Show ParametricPlot3D t, 0, 0, 0, t, 0, 0, 0, t, t, 5, 5, PlotStyle Thick, Plot3D Im F x I y, x, 4, 4, y, 4, 4, ParametricPlot3D Re z1 t, Im z1 t, Im F z1 t, Re z2 t, Im z2 t, Im F z2 t, Re z3 t, Im z3 t, Im F z3 t, t, 0, 2 Π, PlotStyle Thickness.02, Red, Thickness.02, Blue, Thickness.02, Darker Green, ImageSize 500, PlotRange 4, 4, 4, 4, 1.5, 1.5, ViewPoint 2, 1, 1, AxesLabel "Re z ", "Im z ", "Im F z ", BoxRatios 1, 1, 1