Fourier transform of $f(x)\cos(ax)$

Question

I am trying to calculate the Fourier transform of the function $f(x)\cos(ax)$. I tried to take it by parts, putting $\cos(ax)$ inside d$x$, but I get terrible integrals and it seems to me that this is not the right way to do it. Is there any easier method to calculate it?

Answer 1

Irrelevant of whatever constants you may have out front (perhaps $1/{\sqrt{2\pi}}$),

$$\begin{align}\mathcal F[f(x)\cos ax]&=\int f(x) \cos(ax)e^{-ikx}\,dx\\&=\frac12\int f(x)\left(e^{-i(k-a)x}+e^{-i(k+a)x}\right)\,dx\\&=\frac12\left(\tilde f(k-a)+\tilde f(k+a)\right)\end{align}$$