Exploring Bessel Function Generating Function

[maddogtheman]

Homework Statement

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

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

Homework Equations

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.

 

[Dick]

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

 

[maddogtheman]

Thanks can't believe I missed it

 

[duke oeba]

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