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

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  Images of a Fluid Flow

We have already examined several two-dimensional fluid flows and have discovered that the image of a flow under a conformal transformation is a flow.   The mapping  w = u (x, y) + i v (x, y),  which is obtained by using the Schwarz-Christoffel formula, will allow us to find the streamlines for flows in domains in the w plane that are bounded by straight line segments.

	The first technique is finding the image of a fluid flowing horizontally from left to right across the upper half plane  Im (z) > 0.  The image of the streamline  -∞ < t < ∞,  y = c  will be the streamline given by the parametric equations

	u = u (t, c)  and
v = v (t, c)  for   - ∞ < t < ∞  

and will be oriented in the counterclockwise (positive) sense.  The streamline  u = u (t, 0), v = v (t, 0)  for  -∞ < t < ∞,  is considered to be the boundary wall for a containing vessel for the fluid flow.

Example 10.29,  Page 465.  Show that the mapping  w = (1/π) ((Sqrt[(z^2) - 1]) + 
                Log[z + (Sqrt[(z^2) - 1])]) - i  maps the upper half plane Im[z] > 0  onto the domain in the w-plane that lies above the boundary curve consisting of the rays  u ≤ 0, v = 1  and  u ≥ 0, v = 0 and the segment  u = 0, -1 ≤ v ≤ 0.

Solution 10.29.

Use the Schwarz Christoffel formula, enter the formula  f'[z] = ((A (Sqrt[z  +  1]))/(Sqrt[z  -  1]))  and integrate it to construct  f(z).

Remove[A, B, z]; 
Clear[f, f1, F]; 
f1[z_]  =  ((A (Sqrt[z  +  1]))/(Sqrt[z  -  1])); 
f[z_]  =  
        B  +  ∫((A (Sqrt[z  +  1]))/(Sqrt[z  -  1])) \[DifferentialD]z; 
Print["f'[z] = ", f1[z]]; 
Print[""]; 
Print[" f[z] = B + ∫ ", 
        f1[z], " \[DifferentialD]z"]; 
Print[""]; 
Print[" f[z] = ", f[z]];

This is one, formula for the integral.   However, we will use the following form of the integral to continue the computations.

f1[
          z_]  =  ((A (z  +  1))/(Sqrt[(z^2)  -  1])); 
F[z_]  =  
        B  +  ∫((A (z  +  
                        1))/(Sqrt[(z^2)  -  1])) \[DifferentialD]z; 
Print["f'[z] = ", f1[z]]; 
Print[""]; 
Print[" f[z] = B + ∫ ", 
        f1[z], " \[DifferentialD]z"]; 
Print[""]; 
Print[" f[z] = ", F[z]];

Now solve for the coefficients  A  and  B.

eq1  =  
        F[-1]  ==   0; 
eq2  =   F[1]  ==  -I; 
solset  =  Solve[{eq1, eq2}, {A, B}]; 
f[z_]  =  
        ReplaceAll[F[z], 
          (Subscript[solset, [[1]]])]; 
Print[eq1]; 
Print[eq2]; 
Print[
        (Subscript[solset, [[1]]])]; 
Print[""]; 
\(\(Print[\*""f[z] = B + A (Sqrt[-1 + (z^2)])+ A Log[z+(Sqrt[-1 + (z^2)])]""];\)\ \)
Print[""]; 
Print["f[z] = ", f[z]];

Unfortunately, the branch cuts don't work out right and we will have to do this one by hand.

f[z_]  =  
        ReplaceAll[F[z], {A -> (1/π), B -> 0}]; 
Print["f[z] = ", f[z]];

Using this formula in the book we form the two branches of the function.

Clear[f1, f2]; 
f1[
          z_]  =  (( (Sqrt[((z^2) - 1)]) + Log[z + (Sqrt[(z^2) - 1])])/π); 
f2[
          z_]  =  ((-(Sqrt[((z^2) - 1)]) + Log[z - (Sqrt[(z^2) - 1])])/π); 
\(\(Print[\*""\!\(f\_1\)[z] = "", f1[z]];\)\ \)
\(\(Print[\*""\!\(f\_2\)[z] = "", 
        f2[z]];\)\ \)

Use Mathematica to graph conformal mapping  w = f(z).

g1  =  
        CartesianMap[
          f1, {0, 2, 0.25}, {0.00001, 1.20001, 
            0.2}, 
PlotStyle -> {Magenta, Pink}, 
          DisplayFunction -> Identity];  
g2  =  
        CartesianMap[
          f2, {-6.00001, -0.00001, 0.25}, 
{ 
            0.00001, 1.20001, 0.2}, PlotStyle -> {Magenta, Pink}, 
          DisplayFunction -> Identity];  
Show[g1, g2, Ticks -> {Range[-1, 1, 0.5], Range[0, 1, 0.5]}, 
        DisplayFunction -> $DisplayFunction];  
Print["The image of the upper half plane Im[z] > 0,"]; 
Print["is the domain that lies above the boundary curve \
consisting"]; 
Print["of the rays u≤0,v=1 and u≥0,v=0 and \
the segment u=0,-1≤v≤0,"]; 
Print["where  w = f[z] = ", f[z]];

Here is the function that produces the desired result.  The graph is drawn parametrically in this case.

f[
          z_]  =  ((I (Sqrt[ 1  -  (z^2)])  +  ((I π)/2)  -  I ArcSin[z])/π); 
eqns  =  
        Table[{Re[f[x + I 0.2  k]], 
            Im[f[x + I 0.2  k]]}, {k, 1, 10}]; 
ParametricPlot[Evaluate[eqns], {x, -9, 4}, 
        PlotRange -> {{-1, 2}, {-0.1, 1.6}}, 
        Ticks -> {None, None}, AspectRatio -> (1.8/3), 
        DefaultColor -> Blue]; 
Print["The image of the upper half plane Im[z] > 0,"]; 
Print["under w = f[z] = ", f[z]];

We have shown that the mapping  w = (1/π) ((Sqrt[(z^2) - 1]) + 
                Log[z + (Sqrt[(z^2) - 1])]) - i  maps the upper half plane Im[z] > 0  onto the domain in the w-plane that lies above the boundary curve consisting of the rays  u ≤ 0, v = 1  and  u ≥ 0, v = 0 and the segment  u = 0, -1 ≤ v ≤ 0.

Section 10.10  Exercises for Images of a Fluid Flow
See textbook page 466.

Section 10.11	Sources and Sinks

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