# Homework Help: Laplace's equation on a ring

1. Jan 16, 2012

### A_B

1. The problem statement, all variables and given/known data
Solve Laplace's equation on a ring in the plane with r_1<r<r_2. And arbitrary functions on the edges of the ring.

3. The attempt at a solution

After separation of variables the solution to the radial factor is an often-seen problem. It's eigenvalues are λ=n² the eigenfunctions Θ(θ)~sin(nθ) + cos(nθ).

The differential equation for the radial part becomes

r²R" + rR' - n²R = 0

I don't know how to solve this thing but my book gives solutions:

n=0: r → 1 and r → ln(r)
n>0: r → r^n and r → r^(-n)

In my book the functions ln(r) and r^(-n) are thrown away because they are singular in r=0. But for this problem r=0 is not part of the domain, so I should keep all these solutions.

Now I wonder, since ln(r) and r^(-n) can both be written as a Taylor series on the interval [r_1,r_2], can I throw them away anyways?

If so, the solution will be of the form

u(r,θ) = Ʃ A_n r^n cos(nθ) + Ʃ B_n r^n sin(nθ)

Boundary condtions

u(r_1,θ) = Ʃ A_n r_1^n cos(nθ) + Ʃ B_n r_1^n sin(nθ)
u(r_2,θ) = Ʃ A_n r_2^n cos(nθ) + Ʃ B_n r_2^n sin(nθ)

But this seems to be fixing A_n and B_n twice, since both are Fourier series.
Thanks
A_B