TITLE: Fourier transform of sigmoid function
QUESTION [8 upvotes]: I am wondering if there exists a closed form formula for the Fourier transform of the sigmoid function $f(x) = \frac{e^{x}}{(1 + e^{x}}$. More specifically I would need to calculate $F(w) = \int_{-\infty}^{\infty}(f(x)  e^{-iwx} dx)$. This can be expressed as   $F(w) = \int_{-\infty}^{\infty}(\frac{e^{x(1 - iw)}}{(1 + e^{x})}dx$. How would I go from here, or would there be a better way of finding the Fourier transform of the Sigmoid function?
I have tried obtaining a result using the in-built $fourier$ matlab function, it did however not find a solution.

REPLY [2 votes]: The transform exists in the distributional sense. Subtracting $H(x)$ leaves a function for which the transform can be computed directly, and we obtain
$$\left( \frac {e^x} {1 + e^x}, e^{-i w x} \right) =
\pi \delta(w) - i \pi \operatorname{csch} \pi w,$$
where $\operatorname{csch}$ is understood in the sense of the principal value:
$$(\operatorname{csch}, \phi) =
\operatorname{v.\!p.} \int_{-\infty}^\infty \phi(x) \operatorname{csch} x \,dx.$$