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

1. Dec 7, 2008

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

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

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

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.

Last edited: Dec 7, 2008
2. Dec 7, 2008

### Dick

Re: Using Bessel generating function to derive a integral representation of Bessel fu

It looks to me like you want to insert a specific contour. Like z=exp(i*theta).

3. Dec 7, 2008

Re: Using Bessel generating function to derive a integral representation of Bessel fu

Thanks can't believe I missed it

4. Aug 26, 2011

### duke oeba

Re: Using Bessel generating function to derive a integral representation of Bessel fu

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