COMPLEX ANALYSIS: Mathematica 4.1 Notebooks
(c) John H. Mathews, and
Russell W. Howell, 2002
Complimentary software to accompany our textbook:

COMPLEX ANALYSIS: for Mathematics and Engineering, 
4th Edition,  2001,  ISBN: 0-7637-1425-9
Jones & Bartlett Publishers, Inc.  
40 Tall Pine Drive, Sudbury, MA 01776  
Tele.  (800) 832-0034, FAX:  (508)  443-8000  
E-mail:  mkt@jbpub.com  
Internet:  http://www.jbpub.com/  
This free software is compliments of the authors.
John H. Mathews,   mathews@fullerton.edu 
Russell W. Howell,  howell@westmont.edu

CHAPTER 10  APPLICATIONS of HARMONIC FUNCTIONS

a = A = b = B = c = k = R = T = u = U = v = V = w = W = x = X = y = Y = z = Z = 
                                        0; 
Clear[a, a0, a1, a2, a3, A, b, B, 
        CR, Δx, Δy, eq1, eq2, eqns, f, f1, F, F1, 
        F2, F3, F4, g, G, lines, n, P, Ψ, ϕ, Φ, 
        r, r0, solset, θ, t, T, T1, T2, Q, u, u0, U, U1, v, v0, V, V1, 
        w, w0, W, W1, x, x1, x2, x3, y, z, z1, z2, z3, 
        z4]; 
Needs["Graphics`ComplexMap`"]; 
Needs["Graphics`ParametricPlot3D`"]
Needs["Graphics`Colors`"];

Section 10.1  Preliminaries

In most applications involving harmonic functions it is required to find a Harmonic Function that takes on prescribed values along certain contours.  In presenting the material in this chapter, we assume that you familiar with the material in Section 2.5, Section 3.3, Section 5.1, and Section 5.2. If you aren't, please review it before proceeding.

Example 10.1,  Page 387.  A specific example of the general case. Find a function u(x,y) that is harmonic in the vertical strip 1 < x < 2 and takes on the boundary values

u(1,y) = 6   for all y,
u(2,y) = 1   for all y.

Solution 10.1.

Applications involving similar formulas are found in Example 10.14,  Page 331 and Example 10.18,  Page 343.

Remove[a, b, u, U, x, y]; 
a  =  1;  
b  =  2;  
(Subscript[U, 1])  =  6;  
(Subscript[U, 2])  =  1; 
u[x_, y_]  =  
        (Subscript[U, 1])  +  (((Subscript[U, 2])  -  (Subscript[U, 1]))/(b - a)) (x - 
                a); 
cont = {1, 2, 3, 4, 5, 6}; 
ContourPlot[
        u[x, y], {x, 1, 2}, {y, -1, 
          1}, 
Contours  ->  cont, 
        PlotPoints -> 10, 
AspectRatio -> 2, 
        ColorFunction -> Hue]; 
Print["Contour plot for  u[x,y] = ", 
        u[x, y]]; 
Print[""]; 
Print[u[x, y], " = c"]; 
Print["where c = ", cont];

Example 10.2,  Page 388.  A specific example of the general case. Find a function u(x,y) that is harmonic in the sector 0 < Arg (z) < ((2  π)/3) and takes on the boundary values

u(x,0) = 1   for x > 0,
u(x,y) = 9   for all points on the ray r > 0, θ = ((2  π)/3).

Solution 10.2.

Remove[a, c, u, x, y]; 
a  =  ((2  π)/3);  
(Subscript[c, 1])  =  1;  
(Subscript[c, 2])  =  9; 
u[x_, y_]  =  
        (Subscript[c, 1])  +  (((Subscript[c, 2])  -  (Subscript[c, 1]))/a) ArcTan[x, 
              y]; 
cont = {2, 3, 4, 5, 6, 7, 8, 9}; 
ContourPlot[
        u[x, y], {x, -2, 2}, {y, 0.0001, 
          2}, 
Contours  ->  cont, 
        PlotPoints -> 35, 
AspectRatio -> (1/2), 
        ColorFunction -> Hue]; 
Print["Contour plot for  u[x,y] = ", 
        u[x, y]]; 
Print[""]; 
Print[u[x, y], " = c"]; 
Print["where c = ", cont];

Example 10.3,  Page 389.  A specific example of the general case. Find a function u(x,y) that is harmonic in the annulus  1< | z | < 4  and takes on the boundary values

u(x,y) = 1   when  |z| = 1,
u(x,y) = 6   when  |z| =4.

Solution 10.3.

Applications involving similar formulas are found in Example 10.19,  Page 343.

Remove[R, u, x, y]; 
Clear[k]; 
R  =  4;  
(Subscript[k, 1])  =  1;  
(Subscript[k, 2])  =  6; 
u[x_,  y_]  =  
        (Subscript[k, 1])  +  ((Subscript[k, 2])  -  (Subscript[k, 1])) 
            Log[(Sqrt[(x^2)  +  (y^2)])]/Log[R]; 
cont = {1, 2, 3, 4, 5, 6}; 
ContourPlot[
        u[x, y], {x, -4.2, 4.2}, {y, -4.2, 
          4.2}, 
Contours  -> cont, 
        PlotPoints -> 25, 
AspectRatio -> 1, 
        ColorFunction -> Hue]; 
Print["Contour plot for u[x,y] = ", 
        u[x, y]]; 
Print[""]; 
Print[u[x, y], " = c"]; 
Print["where c = ", cont];

Section 10.2	Invariance of Laplace's Equation and the Dirichlet Problem

Section 10.3	Poisson's Integral Formula for the Upper Half Plane

Section 10.4	Two Dimensional Mathematical Models

Section 10.5	Steady State Temperatures

Section 10.6	Two-Dimensional Electrostatics

Section 10.7	Two-Dimensional Fluid Flow

Section 10.8	The Joukowski Airfoil

Section 10.9	The Schwarz-Christoffel Transformation

Section 10.10	Image of a Fluid Flow

Section 10.11	Sources and Sinks

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