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 11  FOURIER SERIES and the LAPLACE TRANSFORM

a = A = b = B = c = d = s = S = t = T = u = U = v = V = w = W = x = X = y = Y = z = Z = 
                                        0; 
Clear[a,a1,a2,a3,A,A1,A2,b,b1,b2,B1,B2,B,c,d,EqA,EqB,f,f1,f2,f12,F,F1,F2,F12,F3,g,G,h,H,i,int,j,k,L,Ly,L1,L2,Lf,LDf,ℒf,ℒDf,n,PQ,PR,r,res1,res2,s,solset,s1,s2,s3,S,S9,t,table,τ,T,θ,u,U,U1,U2,U7,v,v1,v2,v3,v4,val,V,w,w1,w2,W,x,X,y,y0,y1,Y,Yi,z,z1,z2,Z]; 
Needs["Graphics`Colors`"]; 
Needs["Graphics`ParametricPlot3D`"];

Section 11.1	Fourier Series

Section 11.2  The Dirichlet Problem for the Unit Disk

The Dirichlet problem for the closed unit disk \(\(\(D\)\(:\)\) | \(z | \(\(≤\)\(\ \)\(1\)\)\)\) is to find a real-valued function u(x,y) that is harmonic in the unit disk D and that takes on the boundary values  

	u (cos θ, sin θ) = 
        U (θ)  for   - π < π ≤ π,  

at points  z = (cos θ, sin θ)  on the unit circle  C: | z |  =  
        1.

Theorem 11.7 (The Dirichlet problem in  D), Page 493. If U (t) has period 2  π and has the Fourier series representation 

	U (t) = 
        ((Subscript[a, 0])/2) + (∑_(j = 1)^∞)((Subscript[a, j]) 
                  cos (j t) + (Subscript[b, j]) sin (j t)),  

then the solution to the Dirichlet problem in  the unit disk D  is  

	u (r cos θ, r sin θ)= ((Subscript[a, 0])/2) + (∑_(j = 1)^∞)((Subscript[a, j]) (r^j) 
                cos (j θ) + (Subscript[b, j]) (r^j) 
                sin (j θ)),  

where x + i y  =  
        r (e^(i θ)) denotes a complex number in the closed disk \(\(\(|\)\(z | \(\(≤\)\(\ \)\(1\)\)\)\)\) .

Proof of Theorem 11.7, see text Page 493.

An important method for solving this problem is our next result which is attributed to the French mathematician Siméon Poisson.

Theorem 11.8 (Poisson Integral Formula for the Unit Disk), Page 494. Let u (x, y) be a function that is harmonic in a simply connected domain that contains the closed unit disk \(\(\(D\)\(:\)\) | \(z | \(\(≤\)\(\ \)\(1\)\)\)\).  If u (x, y) takes on the boundary values  

			u (r cos θ, r sin θ)=U(θ) for -π<θ≤π,  

then  u (x, y) has the integral representation  

			\(u[r\ cos\ θ, 
          r\ sin\ θ] = ∫\_\(-∞\)\%∞\(\(\((1\  - \ r\^2)\) 
                  U[t]\)\/\(1\  + \ r\^2\  - \ 
                  2  r\ Cos[t\  - \ θ]\)\) dt\),   

which is valid for  \(\(\(|\)\(z | \(\(<\)\(\ \)\(1\)\)\)\)\).

Proof of Theorem 11.8, see text Page 494.

Example 11.3,  Page 496.  Find the function u(x,y) that is harmonic in the unit disk D : |z|< 1 and takes on the boundary values  \(u \((cos\ θ, 
            sin\ θ)\) = \(U \((θ)\) = θ\/2\ \ \ for\ \ \  - π < θ < π\)\).

Solution 11.3.

From Example 11.1, Page 399, we have the Fourier series expansion  \(U[θ] = ∑\+\(k = 1\)\%∞\((\(-1\))\)\^\(k - 1\)\/k\ Sin[k\ θ]\).

Using formula (3) for the solution of the Dirichlet problem, to obtain the boundary value approximation  (Subscript[U, 7])[r, θ].

Clear[k, r, θ, U7]; 
U7[
          r_, θ_]  =  (∑_(k = 1)^7)((((-1))^(k + 1))/k) (r^k) Sin[k θ]; 
\(\(Print[\*""\!\(U\_7\)[r,θ] = "", 
        U7[r, θ]];\)\ \)

Plot the boundary curve   (Subscript[U, 7])[1, θ].

Plot[
        U7[1, θ], {θ, -π, π}, 
        PlotRange -> {{-π, π}, {-1.7, 1.7}}, 
        PlotStyle -> Magenta, 
        AxesLabel -> {"θ", "s"}, 
        Ticks -> {Range[-π, π, (π/2)], 
            Range[((-π)/2), (π/2), (π/2)]}]; 
\(\(Print[\*""\!\(U\_7\)[1,θ] = "", 
        U7[1, θ]];\)\ \)

Plot the harmonic function  s = (Subscript[U, 7])[r, θ].

CylindricalPlot3D[
        U7[r, θ], {r, 0, 
          1}, {θ, -π, π}, 
PlotPoints -> {11, 37}, 
        ViewPoint -> {-3, -2, 
            2}, 
PlotRange -> {{-1, 1}, {-1, 
              1}, {((-π)/2), (π/2)}}, 
AxesLabel -> {x, y, s}, 
Ticks -> {Range[-1, 1, 1], 
            Range[-1, 1, 1], Range[((-π)/2), (π/2), (π/2)]}, 
        LightSources -> {{{0.5, 0, 1}, Red}, {{1, 1, 0.5}, 
              Green}, {{0, 1, 0.5}, Blue}}]; 
\(\(Print[\*""\!\(U\_7\)[r,θ] = "", 
        U7[r, θ]];\)\ \)

Second, investigate the infinite sum  U[r, θ].

Clear[k, r, θ, U]; 
U[
          r_, θ_]  =  (∑_(k = 1)^∞)((((-1))^(k + 1))/k) (r^k) Sin[
              k θ]; 
Print["U[r,θ] = ", 
        U[r, θ]];

Plot the boundary curve  U[1, θ].

\(
\(Plot[
        U[1, θ], {θ, \(-3\) π, 3  π}, 

        PlotRange -> {{\(-3\) π, 
              3  π}, {\(-π\)\/2, π\/2}}, 
        AspectRatio -> 1\/2, PlotStyle -> Magenta, 
        AxesLabel -> {"θ", "s"}, 
        Ticks -> {Range[\(-2\) π, 2  π, π], 
            Range[\(-π\)\/2, π\/2, π\/2]}];\)\ \)
Print["U[1,θ] = ", 
        U[1, θ]];

Plot the harmonic function  s = U[r, θ].

CylindricalPlot3D[
        U[r, θ], {r, 0, 
          1}, {θ, -π, π}, 
PlotPoints -> {11, 37}, 
        ViewPoint -> {-3, -2, 
            2}, 
PlotRange -> {{-1, 1}, {-1, 
              1}, {((-π)/2), (π/2)}}, 
AxesLabel -> {x, y, s}, 
Ticks -> {Range[-1, 1, 1], 
            Range[-1, 1, 1], Range[((-π)/2), (π/2), (π/2)]}, 
        LightSources -> {{{0.5, 0, 1}, Red}, {{1, 1, 0.5}, 
              Green}, {{0, 1, 0.5}, Blue}}]; 
Print["U[r,θ] = ", 
        U[r, θ]];

Library Research Experience for Undergraduates

Project I. Write a report on the Dirichlet problem and include some applications.

1. Flanigan, Francis J., (1973), ''Some Half-Plane Dirichlet Problems: A Bare Hands Approach (in Classroom Notes),'' Am. Math. M., Vol. 80, No. 1. (Jan., 1973), pp. 59-61, Jstor.   

2. Garding, Lars, (1979), ''The Dirichlet Problem,'' Math. Intell., V. 2, No. 1, pp. 43-53.

3. Goulet, John, (1983), ''The Dirichlet Problem: A Mathematical Development,'' Pi Mu Epsilon J., V. 7, No. 8, pp. 502-511.

4. Lange, Ridgley, and Robert A. Walsh (1985), ''A Heuristic for the Poisson Integral for the Half Plane and some Caveats (in The Teaching of Math.),'' Am. Math. M., Vol. 92, No. 5. (May, 1985), pp. 356-358, Jstor.   

5. Minda, David, ''The Dirichlet Problem for a Disk (in Notes),'' Am. Math. M., Vol. 97, No. 3. (Mar., 1990), pp. 220-223, Jstor.   

6. Netuka, Ivan, ''The Dirichlet Problem for Harmonic Functions,'' Am. Math. M., Vol. 87, No. 8. (Oct., 1980), pp. 621-628, Jstor.

Section 11.2  Exercises for The Dirichlet Problem for the Unit Disk
See textbook page 497.

Section 11.3	Vibrations in Mechanical Systems

Section 11.4	The Fourier Transform

Section 11.5	The Laplace Transform

Section 11.6	Laplace Transforms of Derivatives and Integrals

Section 11.7	Shifting Theorems and the Step Function

Section 11.8	Multiplication and Division by  t

Section 11.9	Inverting the Laplace Transform

Section 11.10	Convolution

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