Does an analytic solution exist for this integral

[christianjb]

(not a HW problem- I'm writing a paper)

Looking for an analytic solution for the following- if one exists

$$\frac{1}{2\pi}\int_0^{2\pi}e^{-cos^2(\theta)}d\theta$$

 

[Avodyne]

$$\frac{1}{2\pi}\int_0^{2\pi}e^{-2a \cos^2(\theta)}d\theta<br /> =e^{-a}I_0(a)$$
where $I_0(a)$ is a modified Bessel function, and a=1/2 in your case. But, as far as I know, $I_0(1/2)$ has no particular value in terms of other constants.

 

[christianjb]

Thanks. Is there a good web-reference for this? I'll try and track it down on Mathworld.

 

[Avodyne]

http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html

 

[christianjb]

Thanks, I found it.