# Is this a known function ?

1. Jul 11, 2011

### mnb96

Hello,
while attempting to solve a problem, I came up with the following integral:

$$\int_0^{2\pi}e^{i\left( u\cos(x)+v\sin(x) \right)}dx$$

where u,v are two real constants.
I don't know how to solve this definite integral and I am wondering if this formula is already known, and if it pops up in other contexts.

Thanks.

2. Jul 11, 2011

### pmsrw3

Last edited by a moderator: Apr 26, 2017
3. Jul 12, 2011

### mnb96

Thanks a lot!
It seems the definite integral of that expression is given by a 0-order 'modified Bessel function'. Interesting.

I trust the answer from that site but I am wondering now if it would be difficult to prove that result.