Dirichlet problem on unit disc in polar:
$u_{rr} + (1/r) + (1/r^2)u_{\theta\theta} = 0$
$u(1,\theta) = f$
Period in $\theta$ gives
$u(r,\theta) = \sum R_n(r) e^{in\theta}$
Inserted into our PDE gives
$R_n '' + (1/r)R_n'-(n^2/r^2)R_n = 0$ which is an Euler equation. Solutions are:
$R_n(r) = A ln(r) + B : n = 0$
$R_n(r) = Ar^n + Br^{-n} : n \neq 0$
Here's where I get confused. The book just jumps right into the following conclusion:
$R_n(r) = c_nr^{|n|} \rightarrow u = \sum c_n r^{|n|}e^{in\theta}$
I don't get why we're discarding all the negative n:s. Can anyone shed a light?
Take your $R_n(r) = Ar^n + Br^{-n}$. We want this to hold for all possible values on $r$, however we still want our solution to be physically plausible. This means that it can not run away to infinite, i.e. we need to care about when $r=0$. Setting $R_n(r) = c_nr^{|n|}$ fixes this problem, with $c_n = A,B$ depending on $n$.