let the functions $f_{\alpha}(x)= e^{-\alpha x} H(x), \ \alpha \in \mathbb{N}$ where $H$ is function of Heaviside and $ g_a(x)= \begin{cases} 1 &: |x| < a\\ 0 &:|x|\geq a \end{cases}$.
The question is calculate Fourier of $f_{\alpha}$, Fourier of $g_a$, inverse Fourier of $f_{\alpha}$ and inverse Fourier of $g_a$.
I try this: 1. to calculate Fourier of $f_{\alpha}$: $F f_{\alpha}$ we have $$ F f_{\alpha}(\xi)= \displaystyle\int_0^{+\infty} e^{- i x \xi} e^{-\alpha x} dx = -\dfrac{1}{i \xi + \alpha}. $$ to calculer inverse of Fourier transform, we have $$ \overline{F} f_{\alpha}(x)= \displaystyle\int_0^{+\infty} e^{i x \xi} e^{-\alpha \xi} d \xi = \displaystyle\int_0^{+\infty} e^{\xi(ix - \alpha)} d\xi =\dfrac{1}{ix-\alpha} [e^{\xi(ix-\alpha)}]_0^{+\infty} $$ but isn't converge, i have problem to determine the value of $\overline{F} f_{\alpha}(x)$.
- To calculate Fourier of $g_a$ i find that $F g_a(\xi)=0$ if $\xi \neq 0$ and $Fg_a(0)= 2 a$. And to calculate $\overline{F}g_a(x)$ i find that $\overline{F} g_a(x)= 0$ if $x \neq 0$ and $\overline{F}g_a(0)=2a$. Is it correct? Please.
Thank you in advance to the help.