# Sommelier of bell-shaped functions

## Q0. Functions are scaled so that $$f(0) = -f''(0) = 1$$.

 0 1 2 3 4 $$\exp \left( -\frac{1}{2} x^2 \right)$$ Gauss dist. $$\frac{1}{\cosh x}$$ Hyperbolic secant $$\frac{1}{\cosh^2 (x / \sqrt{2})} = \tanh' \frac{1}{\sqrt{2}} x$$ Derivative of Fermi dist. $$\frac{1}{2} + \frac{1}{2} \cos \left( \frac{\pi}{2} x \right) \ \mathrm{if}\ |x| \lt 2, \ 0 \ \mathrm{otherwise}.$$ Hann window $$\frac{1}{1 + x^2 / 2} = \arctan' \frac{1}{\sqrt{2}} x$$ Cauchy dist.

Score: .

## Q1. Functions are scaled so that $$\int dx\, f(x) = \int dx\, x^2 f(x) = 1$$.

 0 1 2 3 4 $$\frac{1}{\sqrt{2 \pi}} \exp \left( -\frac{1}{2} x^2 \right)$$ Gauss dist. $$\frac{1}{2} \frac{1}{\cosh(\pi x / 2)}$$ Hyperbolic secant $$\frac{a}{2} \frac{1}{\cosh^2 a x} = \frac{a}{2} \tanh' a x,\ a = \frac{\pi}{2 \sqrt{3}}$$ Derivative of Fermi dist. $$\frac{a}{2 \pi} (1 + \cos a x) \ \mathrm{if}\ |x| \lt \frac{\pi}{a}, \ 0 \ \mathrm{otherwise},\ a = \sqrt{\frac{\pi^2 - 6}{3}}$$ Hann window $$\frac{2}{\pi} \frac{1}{(1 + x^2)^2}$$ ν = 3 Student's t-dist.

Score: .

(c) Yasuhiro Fujii <y-fujii at mimosa-pudica.net>