Given the Helmholtz equation
$$ \begin{cases} \Delta u + k^2 u &= 0 & \text{ in } \Omega = \{(r,\phi) \mid r_0 < r < r_1 \} \\ u(r,\phi) &= \cos(n\phi)& \text{ on } \Gamma_1 = \{(r,\phi) \mid r=r_0 \} \\ \frac{\partial u}{\partial r} &= iku & \text{ on } \Gamma_2 = \{(r,\phi) \mid r=r_0 \} \end{cases} $$
satisfying the Sommerfeld radiation condition.
Using separation of variables ($u(r,\phi) = R(r)\cdot \Phi(\phi)$) I got the solution
$$u(r,\phi) = \cos(n\phi) \frac{H_n^{(1)}(kr)}{H_n^{(1)}(kr_0)} $$
where $H_n^{(1)}$ denotes the Hankel function of the first kind with parameter $n$.
Can anyone confirm this? I'm trying to test a FD-solver for this equation but for that I need the exact solution, and I was unable to find one using maple.