Is the Definite Integral of e^(i(u*cos(x)+v*sin(x)) Known?

[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.

 

[pmsrw3]

Yup!

http://www.wolframalpha.com/input/?...sin(x)))+dx+)from+0+to+2pi&fp=1&incTime=true"

In case the link doesn't work, go to www.wolframalpha.com and type "integrate (exp(i(u cos(x) + v sin(x))) dx )from 0 to 2pi". You may have to click "Try again with more time" after the first attempt.

 

[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.