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December 15th, 2015, 05:19 AM   #1
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Series of Bessel functions

Hello!
I stuck with my physics problem. I have a solution in the form of $\displaystyle \sum_{n=0}^{\infty }\frac{J_n(x)}{H_n(x)}$, where x is grate than 1 but fixed, thus it's not possible to use asymptotic form for Bessel functions with large or small x. However, I made some numeric calculations and found that terms of series start to decrease very quickly if n > x, so it's sufficient to take only first [x] terms. Another point is that I know the answer (from physical sense and also from numeric calculations), and the answer is approximately x/2.
I need help to obtain the answer by direct calculation this series. Any suggestions?
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