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 $$\delta$$$$\neq$$0.

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