# 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 $\int{W(x,t)e^{i\omega t}dt} = \int{V(x+\cos(\omega\cdot t))e^{i\omega t}dt}$

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Welcome to MSE! Do you have any thoughts and have tried some things so responders don't rehash those items? Regards – Amzoti Jun 2 '13 at 23:08
Since $\cos(\omega t)$ is periodic in $t$, so is $W(x,t)$. Namely $W(x,t + 2\pi / \omega) = W(x,t)$. This means that function will not be absolutely integrable on the line, as a function of $t$, and the Fourier integral will fail to converge in general. On the other hand, since $W(x,t)$ is periodic with period $2\pi / \omega$, perhaps you want to instead consider its corresponding Fourier series? – Willie Wong Jun 3 '13 at 16:22

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 $\omega = 2 \pi f_0.$

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Thank you for the quick response Birtex! however I fail to understand it.. $V(x)$ is not periodic with time it is a function of $x$ (not $t$) I define a new function $W(x,t) = V(x+\cos(\omega\cdot t))$ and I want to find the fourier transform of $W(x,t)$. I do not understand what do you mean by $V(W(x,t))$... Thanks again! – Grisha Jun 3 '13 at 7:58
@Grisha Sorry, I did this when I was tired and made some mistakes. I have edited my answer. – Bitrex Jun 3 '13 at 15:39