I have this Hankel function, $H_{1}(R_{1}+R_{2})e^{i\cos(a)}$. Would it be possible to simplify this function in terms of $H_{1}(R_{1})$ and $H_{1}(R_{2})$?
2026-04-01 02:00:05.1775008805
Simplification of Hankel functions
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There exists an addition formula for Bessel functions due to Graf. In the case you are interested in this addition formula takes the form $$H^{(1)}_{\nu}\left(R_1+R_2\right)=\sum_{k\in\mathbb{Z}}H^{(1)}_{\nu-k}\left(R_1\right)J_k\left(R_2\right),$$ where it is assumed that $|R_2|<|R_1|$. I'm not sure whether this can be called a simplification, but there seems to be no way to get anything more compact.