How to express exp(kCos(gamma)) using spherical harmonics?

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Hello all, i am looking for a relation to express exp(kCos(gamma)) in spherical harmonics. How can I express it?


thanks
 
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Look in any advanced text in math physics, quantum mechanics or E&M, what you are looking for is the expansion of a plane wave in terms of spherical harmonics and Bessel functions.
 
Thanks for your reply Dr. Transport, I want to use spherical harmonic expansions for exponential functions in a transfer matrix.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

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