I'd like to know if I have got the following ideas right:
1) $f(r,\theta,t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty a_{nk}J_n(j_{nk}r)\exp[in\theta-j^2_{nk}t]$ subjected to initial condition $f_0(r,\theta)$ is just $\int\limits_0^1 f_0(r,0)J_n(j_{nk}r)rdr \over{1\over2}J^2_{n+1}(j_{nk})$, where $J_n$ is the $n$th order Bessel function and $j_{nk}$ its $k$th zero;
2) In order that $F(r,\theta,t)={f(r,\theta,t)\over f_0(r,\theta)}$ is purely a function of time, we must have $f_0(r,\theta )$ to have the form $a_{nk}J_n(j_{nk}r)\exp[in\theta]$ i.e. only a single term so that all the time-independent stuff cancels;
3) If $f_0(r,\theta)=F_0(r,\theta)$ for all $r<1$, then... what is the form of $f(r,\theta,t)$? (I really need enlightenment on this bit. I am not even sure that I fully understand what $F_0(r,\theta)$ is.)
Thanks.