Pairs of elementary Fourier transforms in $L^2$

Question

It is customary to teach Fourier transform on the real line by starting with functions from $L^1$, $L^2$ or the Schwartz space. It is not so easy to illustrate the theory by computing explicit pairs of transforms that are both elementary. The list given in the standard books is short: algebraic fractions, gaussian, exponentials, sinc and that's it.

I was surprised to learn that the Fourier transforms of $1/\cosh(x)$ and $\hbox{atan}(x+1) - \hbox{atan}(x-1)$ can be computed explicitely and are of an elementary nature.

Are you aware of other $L^1$ or $L^2$ examples besides the classical ones?

I use the convention $\hat{f}(\xi) = \int_{-\infty}^\infty f(x) e^{-i\xi x} dx$, so that

$$\widehat{\hbox{atan}(x+1) - \hbox{atan}(x-1)} = 2\pi \ e^{-|x|} {\sin(x)\over x}$$

$$\widehat{1\over \cosh(x)} = {\pi \over \cosh({\pi x\over 2})}$$

EDIT1: I just noticed that $\hbox{atan}(x+1) - \hbox{atan}(x-1) = \hbox{atan}(2/x^2)$ and that the Fourier transform of this function is computed in the classical book of Oberhettinger.

Another explicit pair as suggested in the answer of C. Beenakker. $$ {\hbox{tanh}(x)\over x} = -2 \ln(\hbox{tanh}({\pi |\xi|\over 4})) $$

EDIT2: Nemo in this post gives many examples of Fourier transforms which are self-reciprocal and provides references. Some of them are elementary functions with explicit Fourier transforms. Note that the convention concerning the definition of the transform is different from the one used here.

Answer 1

The following example is interesting

$$\widehat{\sin(x)\over x \sqrt{|x|}} = \sqrt{2\pi} \ \left\{ { \matrix{ \scriptstyle \sqrt{|\xi+1|} + \sqrt{|\xi-1|} & \hbox{ if } |\xi| \leq 1, \cr {2\over \sqrt{|\xi+1|} + \sqrt{|\xi-1|}} & \hbox{ if } |\xi| \geq 1.\cr}} \right.$$

because it is in $L^1$ but not in $L^2$ and its Fourier transform is continuous but with infinite derivative at $\xi = \pm 1$.

Answer 2

Does a Bessel function count as "elementary" ? The Fourier transform of $J_0(x)$ is $2/\sqrt{1-x^2}$ for $|x|<1$ and zero otherwise.

Concerning the combination of arctangents, you can Fourier transform ${\rm arctan}\,x$ itself into $-i\pi e^{-|x|}/x$.

Besides $1/\cosh x$, you can Fourier transform $x/\sinh x$ into $(\pi^2/2)/\cosh^2(\pi x/2)$, and $\tanh x$ into $-i\pi/\sinh(\pi x/2)$.

$$\widehat{1\over \cosh^2(x)} = {\pi \xi\over \sinh({\pi\xi\over 2})}$$