Zeroth order modified Bessel function integral representation

2018-06-19 23:57:52

I'm trying to understand the derivation of:

$$ I_0(x) = \frac{1}{\pi}\int_{0}^{\pi} \exp(x\cos\theta) \, d\theta$$

I'm trying to use this generating function:

$$ \exp\left(\frac{x}{2}(z-z^{-1})\right) = \exp(x\cos\theta) = \sum_{n=-\infty}^\infty I_n(x) \exp (in\varphi)$$

Is this correct? (left side real and right side complex). Then, i'm using:

$$ \int_0^\pi \exp(x\cos\theta) \,d\theta = \frac 1 \pi \int_0^\pi \sum_{n=-\infty}^\infty I_n(x) \exp (in\varphi) \, d\theta$$

But this seems no quite right.

Thanks!