I need to count the Fourier transform of the following function but it does not seem so obvious for me.
$f(x)=(e^{-ab})-1$ for $x\ge0$ and $f(x)=0$ for $x<0$
where: $a=1$ and $b=-1$
I don't know whether I should use the standard method or is there any other efficient way?
The original post has $f$ defined such that
$$f(x)=(e^{-ab}-1)H(x)$$
where $H$ is the Heaviside function defined as
$$H(x) = \begin{cases} 1, & \text{if $x\ge 0$} \\ 0, & \text{if $x<0$} \end{cases} $$
The Fourier-Transform $\hat H(k)$ of the Heaviside function is
$$\hat H(k)\equiv \int_{-\infty}^{\infty}H(x)e^{ikx}dx=\pi \delta(k)+\text{PV}\left(\frac{i}{k}\right)$$
where PV denotes the Cauchy Principal Value and where $\delta$ is the Dirac delta and is a Generalized function or Distribution defined as
$$\int_{-\infty}^{\infty}\delta(x)f(x)dx=f(0)$$
for all test functions $f$.
Thus, we have for $a=1$ and $b=-1$ (as in the original post)
$$\hat f(k)=(e-1)\left(\pi \delta(k)+\text{PV}\left(\frac{i}{k}\right)\right)$$