I saw somewhere contains below formula. $${e^{ikr\cos \left( \theta \right)}} = \sum\limits_n {{i^n}{J_n}\left( {kr} \right){e^{ - in\theta }}} $$ I don't know if it is right. Does anyone know how to prove it? Or point to some references on this kind of expansions using bessel function?
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A. Erdélyi et al., Higher Transcendental Functions Vol. II, give in 7.2.4 (27) $$e^{iz \cos \phi} = \sum_{n=-\infty}^\infty i^n e^{in\phi} J_n(z)$$ – gammatester Jun 08 '18 at 11:31
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3https://en.wikipedia.org/wiki/Jacobi%E2%80%93Anger_expansion – Chappers Jun 08 '18 at 11:37
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@Chappers gammatester Thank you so much. Exactly what I want – user15964 Jun 08 '18 at 11:51