## Using Bessel generating function to derive a integral representation of Bessel functi

1. The problem statement, all variables and given/known data

The Bessel function generating function is
$$e^{\frac{t}{2}(z-\frac{1}{z})} = \sum_{n=-\infty}^\infty J_n(t)z^n$$

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$$J_n(t) = \frac{1}{\pi} \int_0^\pi cos(tsin(\vartheta)-n\vartheta)d\vartheta$$

2. Relevant equations

3. The attempt at a solution

So far I have been able to use an analytic function theorem to write

$$J_n(t)=\frac{1}{2\pi i} \oint e^{\frac{t}{2}(z-\frac{1}{z})}z^{-n-1}dz$$
(we are required to use this)
But now I have no idea where to go from here.

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 Recognitions: Homework Help Science Advisor It looks to me like you want to insert a specific contour. Like z=exp(i*theta).
 Thanks can't believe I missed it

## Using Bessel generating function to derive a integral representation of Bessel functi

Using Bessel generating function to derive a integral representation of Bessel function