Definitions, Theorems, Examples, Topics for Asmar 2nd edition
Chapter 4: 4.4 only

 4.4 Laplace's Equation in Circular Domains: Dirichlet Problem 
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  u_rr + (1/r) u_r + (1/r^2) u_ww = 0,  where w=theta
  u(a,w)=f(w) [boundary condition]

  PRODUCT SOLUTION 
    u=R(r)W(w)  r=radial variable, w=theta

    THEOREM
      W'' + mu W = 0, W(0)=W(2Pi), W'(0)=W'(2Pi)
      r^2 R'' + r R' + (-mu)R = 0, R bounded
      Separation constant mu to be determined

    THEOREM
      W(w) = linear combination of the Euler solution atoms
             cos(n w), sin(n w)
             n=1,2,3, ...
             w=theta variable
             mu = n^2 required because W(w) is 2Pi-periodic.
             
    THEOREM    
      R(r)=1 for n=0
      R(r) = (r/a)^n for n=1,2,3, ...
      The analysis excludes the unbounded solution which appears
      in the general solution, because R(r) must be bounded.

    THEOREM
      The solution of the Dirichlet problem is the sum n=0 to infinity of 
      the terms (r/a)^n (a_n cos(n w) + b_n sin(n w)). The coefficients are
      the Fourier coefficients of f(w)=u(a,w).

 EXAMPLE 1. Dirichlet problem on a disk, a=1, f(w)=100 on 0<w<Pi,
            f(w)=0 on Pi<w<2Pi. Inserting the Fourier coefficients from 
            the previous THEOREM gives the series answer

                u(r,w) = 50 + (100/Pi)*(sum of terms u[n] for n=1 to infinity)
                u[n] = (1/n)(1-(-1)^n) r^n cos(n w)
 
 EXAMPLE 2. Cartesian coodinate example.
            Re-write the solution of EXAMPLE 1 in the form

                u(r,w) = 50 + (100/Pi)(arctan(v1) + arctan(v2))
                    v1 = y/(1-x), v2 = y/(1+x)

             LEMMA (Dirichlet series identity)
             arctan(v) = Sum of terms r^n sin(n w)/n for n=1 to infinity,
             where v = y sin(w) / (1- r cos(w)).
             PROOF: See exercise 19, section 4.4.
 
 EXAMPLE 3. Isotherm example
 EXAMPLE 4. Robin problem on a wedge 0<r<1. 0<w<alpha.
            Boundary conditions u(r,0)=0, u(r,alpha)=0, 
            u_r(1,w)=-u(1,w)-w