= Solution
Separation gives angular modes $e^{im\theta}$ and $-[rR\prime]\prime+m^2R/r=\lambda rR$. Frobenius gives $p=m$ and $a_k=-a_{k-2}/[k(k+2m)]$, producing $J_m$; termwise <differentiation> gives the two stated recurrences. A radial solution is $u=\sum_n[A_n\cos(cj_{0n}t)+B_n\sin(cj_{0n}t)]J_0(j_{0n}r)$. Here $B_n=0$ and
$$
A_n=\frac{2}{J_0\prime(j_{0n})^2}\int_0^1r(1-r)J_0(j_{0n}r)dr,
$$
equivalently $A_n=2[j_{0n}J_0\prime(j_{0n})^2]^{-1}\int_0^1rJ_1(j_{0n}r)dr$ after integration by parts.
Back to article page
