Fourier transform of a function of $\cos(\omega \cdot t)$ I need your help. 
Suppose I have a function $V(x)$. 
Now suppose I vary it with time as $W(x,t) = V(x+\cos(\omega \cdot t)$.
I need to find out What is the fourier transform of $W(x,t)$ with respect to $t$.
In other words I need to derive $\int{W(x,t)e^{i\omega t}dt} = \int{V(x+\cos(\omega\cdot t))e^{i\omega t}dt}$
Really appreciate your help.
 A: Just a possible idea:
If V(x) is a periodic function in x with period $\tau$, take the exponential Fourier series of $V(x)$:
$V(x) = \sum_{n =-\infty}^{\infty}c_n e^{inx}$
where $c_n = \frac{1}{\tau}\int_0^\tau V(x)e^{-inx}$
and plug in $W(x, t) = V(x + \cos(\omega t))$
$W(x,t) = \sum_{n =-\infty}^{\infty}c_n e^{inx}e^{i n\cos(\omega t)}$
Use the general form of the Jacobi-Anger expansion to get:
$W(x,t) = \sum_{n =-\infty}^{\infty}c_n e^{inx}\sum_{k =-\infty}^{\infty}i^kJ_k(n)e^{ik\omega t}$
or in terms of cosines:
$W(x,t) = \sum_{n =-\infty}^{\infty}c_n e^{inx}(J_0(n) + 2\sum_{k =-\infty}^{\infty}i^kJ_k(n)\cos(k \omega t)) = $
$\sum_{n =-\infty}^{\infty}c_n e^{inx}J_0(n) + 2\sum_{n =-\infty}^{\infty}c_n e^{inx}\sum_{k =-\infty}^{\infty}i^kJ_k(n)\cos(k \omega t)$
Where $J_k$ is the Bessel function of order k.
Fourier transform both sides:
$\mathscr{F} W(x,t) = \sum_{n =-\infty}^{\infty}c_n e^{inx}J_0(n)\int_{-\infty}^{\infty} 1 e^{-2\pi i s t}dt + 2\sum_{n =-\infty}^{\infty}c_n e^{inx}\sum_{k =-\infty}^{\infty}i^kJ_k(n)\int_{-\infty}^{\infty}\cos(k \omega t)e^{-2\pi i s t}dt =$ 
$\sum_{n =-\infty}^{\infty}c_n e^{inx}J_0(n)\delta(s) + \sum_{n =-\infty}^{\infty}\sum_{k =-\infty}^{\infty}c_n e^{inx}i^kJ_k(n)((\delta(s - k f_0) + \delta(s + k f_0))$
Where $\omega = 2 \pi f_0.$
