Steady state temperature in a hollow cylinder

[gophergirl]

Homework Statement

Using cylindrical coordinates rho,phi, and z, let u([rho,phi) denote steady temperatures in a long hollow cylinder a leq rho leq b, -infinity < z < infinity when the temperatures on the inner surface rho= a are f(phi) and the temperature of the outer surface rho = b is zero.

Derive the temperature formula u(rho,phi) = A_{0}(ln b - ln rho)/(ln b - ln a) + summation n = 1 to infinity (a/rho)^n (b^(2n)- rho^(2n))/(b^(2n)- a^(2n)) (A_{n}cos(n*phi) + B_{n}sin(n*phi), where A_{0}, A_{n}, and B_{n} are the standard Fourier coefficients

Homework Equations

The Attempt at a Solution

I've done plenty of boundary value problems for solid cylinders, but never for a hollow one. My textbook offers no examples of hollow cylinders- the only place it appears in the book at all is this homework problem. I can't seem to get beyond setting up the Cauchy-Euler equation, Rho^2R"(rho) + rhoR'(rho) - lambda*R(rho) and the Sturm-Liouville problem Phi"(phi) + lamda*Phi(phi) + 0 with the periodic boundary conditions Phi(-pi) = Phi(pi) and Phi'(-pi) = Phi'(pi).

I think the eigenvalues are the usual ones: lambda_{0} = 0 and lambda_{n} = n^2. That's all the farther I've gotten

 

[epenguin]

Using double hash symbols turns

Derive the temperature formula u(rho,phi) = A_{0}(ln b - ln rho)/(ln b - ln a) + summation n = 1 to infinity (a/rho)^n (b^(2n)- rho^(2n))/(b^(2n)- a^(2n)) (A_{n}cos(n*phi) + B_{n}sin(n*phi), where A_{0}, A_{n}, and B_{n} are the standard Fourier coefficients

into

Derive the temperature formula $u(rho,phi) = A_{0}(ln b - ln rho)/(ln b - ln a) + summation n = 1 to infinity (a/rho)^n (b^(2n)- rho^(2n))/(b^(2n)- a^(2n)) (A_{n}cos(n*phi) + B_{n}sin(n*phi)$ , where A_{0}, A_{n}, and B_{n} are the standard Fourier coefficients