With $f(x) = e^{ik_0x} $ and the window function $W(x) = \frac{1}{x^2+a^2}$, with $a\in\mathbb{R}$, the fourier transform of $f(x)$ is given by: $$\begin{align} F(k) &= \int^\infty_{-\infty} e^{-ikx}\,W(x)\,f(x)\,dx \\ &= \int^\infty_{-\infty} e^{-i(k-k_0)x}\, \frac{1}{x^2+a^2} \,dx\end{align}$$
I computed this integral using contour integration, splitting into two cases (i) $k-k_0>0$ and (ii) $k-k_0<0$, and taking appropriate contours in the lower and upper half planes respectively. I obtained: $$ F(k) = \frac{\pi e^{-(k-k_0)a}}{a} \qquad \text{for all k}\in\mathbb{R}$$
My question is this: If I were to compute the fourier spectrum for this, I would get: $$\vert F(k)\vert^2 = \frac{\pi^2}{a^2}e^{-2(k-k_0)a}$$ which does not seem to have a peak. Does this have any physical meaning, or did I do my fourier transform wrongly? Thanks.