Laplace's Eqn on half an annulus

[Trail_Builder]

Homework Statement

Show that Laplace’s equation ∇^2(φ) = 0 in polar coordinates (r,θ) has solutions proportional to r^(±α)*sin(αθ), r^(±α)*cos(αθ) for any constant α.

Find the function φ satisfying Laplace’s equation in the region a < r < b, 0 < θ < pi, where φ(a,θ) = (sin(θ))^3, φ(b,θ) = φ(r,0) = φ(r,pi) = 0.

Homework Equations

The Attempt at a Solution

Basically, I know how to find the general solution for Laplace's Eqn in polars, by separating variables. However, I can't reconcile this general solution with what the (first bit of the) question says. In particular, why does the question ignore the log(r) term? I know this term would normally disappear by requiring regularly at the origin, but we have half an annulus, so this doesn't apply. This log(r) term is also making the process of determining coefficient a bit of a nightmare, to the point of me thinking I'm missing something.



Any help would be greatly appreciated :) Thanks!

 

[hunt_mat]

Can you separate the variables?

 

[Trail_Builder]

yeah, I separated the variables in the usual way, to get a solution. but the solution has a constant term and a log(r) term, in addition to the "r^(±α)*sin(αθ), r^(±α)*cos(αθ)" terms they mention. and I am not sure why the qu doesn't have all the term I have :S

 

[hunt_mat]

Show your working, I think you may have made a mistake.