I hope you're all doing well! I'm having some issues while trying to proof this property: $$\mathscr{F}{(f(x)e^{ax})(k) = \mathscr{F}(k+ia)}$$ I've tried by myself and got this:
$$\mathscr{F}{(f(x)e^{ax})(k) = \frac{1}{\sqrt[]{2\pi}}\int^{+\infty}_{-\infty} f(x) e^{x*(a-ik)} \, dx }$$ but I got stuck in this point. My professor is using this definition of Fourier Transform: $$\mathscr{F}({f(x))(k)} = \frac{1}{\sqrt[]{2\pi}}\int^{+\infty}_{-\infty} f(x) e^{-ikx} dx$$ Can someone help me??
If you're actually given nothing about $f$ then it's impossible to prove anything about $f$.
One might regard "$f\in L^1$" as sort of the default assumption about $f$, since this is a problem about the Fourier transform. But if that's all we assume about $f$ then the problem simply makes no sense; the function $e^{at}f(t)$ need not be integrable, and there's no such thing as $\hat f(k+ia)$, since $\hat f:\mathbb R\to\mathbb C$ and $k+ia\notin\mathbb R$. (If for example $f(t)=\frac1{t^2+1}$ then the integral $\int f(t) e^{at}e^{-ikt}\,dt$ does not exist, and there's no sort of summability or principal-value interpretation that works for that integral either.)
A standard context in which the question makes sense would be if we assume $f$ is integrable with compact support. Suppose so. Then it's not hard to see that $$F(z)=\int f(t)e^{-itz}\,dt$$defines an entire function (a clean way to show this is via Morera's theorem plus Fubini). And since $F$ is an entire function that agrees with $\hat f$ on the real axis it's not unreasonable to say $$F(z)=\hat f(z)\quad(z\in\mathbb C).$$ With this assumption and that convention the problem's trivial: $$\widehat{e^{at}f(t)}(k)=\int e^{at}f(t)e^{-ikt} =\int f(t)e^{-i(k+ia)t}\,dt=\hat f(k+ia).$$