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I think I may have misinterpreted this question, anyhow I am very confused. Here it is in its full glory:

Let $f(r,\theta, t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty a_{nk}J_n(j_{nk}r)e^{in\theta-j_{nk}^2 t}$, defined on the unit disc, where $J_n$ is the $n$th degree Bessel function. Initial conditions $f_0(r,\theta)$ and we have $R(r,\theta,t):= {f(r,\theta,t)\over f_0(r,\theta)}$. We also have $f_0=R_0 \,\,\,\,\forall r<1$.

I have to show that $a_{0k}={2R_0\over j_{0k}J_1(j_{0k})}$ and $a_{nk}=0$ otherwise.

But isn't $R_0=1$? (I am guessing $R_0=R(r,\theta,0)$?) Since isn't $f(r,\theta, t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty a_{nk}J_n(j_{nk}r)e^{in\theta-j_{nk}^2 t}\implies f_0(r,\theta)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty a_{nk}J_n(j_{nk}r)e^{in\theta}$?

Thanks in advance!

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