Finding Fourier Transforms

Question

I am currently trying to find the Fourier Transforms of the following functions:

$$f(x) =\begin{cases} x^ne^{-ax} & x>0\\ 0 & x\leq 0 \end{cases}, \>\>\>\>\>\> f(x)=\begin{cases} x^ne^{-ax}\cos(x) & x>0\\ 0 & x\leq 0 \end{cases}$$

Now I believe the first function I can find by just integrating by parts $n$ times, but the second function I won't be able to use Integration by Parts. How do I compute the second function? Which method do I use?

Answer 1

Use that $\cos(x) = \frac12 (e^{ix} + e^{-ix})$ and merge $e^{\pm ix}$ with the factor $e^{-i\xi x}$ (or what you have) in the Fourier transform integral.

Answer 2

Hint: Denote $\hat{f}(\xi) = \mathcal{F}(f)(\xi)$, the Fourier transform which you seem to know how to compute. Then, you can use the Euler's identity: $$e^{ix}+e^{-ix} = 2\cos x$$ to deduce that $$\mathcal F\{{f(x)\cos(ax)}\}(\xi) = \dfrac{\mathcal{F}(f)(\xi-\frac{a}{2\pi})+\mathcal{F}(f)(\xi+\frac{a}{2\pi})}{2}$$