How do I calculate the Fourier integral with non-zero phase shift?

[arcmed]

Hello, I want to calculate the integral

\int^{2\pi}_{0} \exp(i(k t + \cos(t+\delta)))dt

where k and \delta are integer and real numbers, respectivily.

I know with \delta=0 the result is given in terms of Bessel functions, but I don't know what to do if \delta\neq0.

Any help would be appreciate, thanks in advance.

ACM

 

[tiny-tim]

Welcome to PF!

Hi arcmed ! Welcome to PF!

(have a delta: δ )

Hint: substitute u = t + δ.

 

[arcmed]

Hello tiny-tim, thanks to take time for answer.

I had already considered this substitution, but in that case, the integral limits change from 0 and 2\pi to \delta and \delta+2\pi, respectivily. So, again, I can't use the fact the original integral with \delta=0 can be given in terms of Bessel functions.

ACM

 

[Ben Niehoff]

The function

e^{i \cos \theta + i m \theta

is periodic with period 2\pi. Therefore the integral over any period ought to be the same.

 

[arcmed]

Hello Ben, thank you, you are right.
I had never participated in a forum, but now I realize it is an useful tool, sometimes one can forget very obvious things.
Again, thanks to tiny-tim also.
ACM