The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel function of order $\ell$ and $P_\ell$ is the Legendre polynomial. I have run into a problem where it makes sense to replace $j_\ell$ in the above formula by $h_\ell$, which is the spherical Hankel function of order $\ell$ (or Neumann function, either would work). I would like to find a simple formula for the LHS if this replacement is made. Is anyone aware of a way of doing this, or any research that has been done on this type of generalization?
2026-03-26 04:38:04.1774499884
I would like to find a generalization of the plane wave expansion to Hankel functions.
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