Catalan's constant

Catalan's constant K, which occasionally appears in estimates in combinatorics, is defined by

$\Kappa = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} = \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + ...$

or equivalently

$K = -\int_{0}^{1} \frac{\ln(t)}{1 + t^2} \mbox{ d} t.$

along with

$K = \frac{1}{2}\int_0^1 \mathrm{K}(x)\,dx$

$K = \int_0^1 \frac{\tan^{-1}x}{x}dx$

where K(x) is a complete elliptic integral of the first kind, and has nothing to do with the constant itself.

Uses

K appears in combinatorics, as well as in values of the second polygamma function, also called the trigamma function, at fractional arguments:

$\psi_{1}\left(\frac{1}{4}\right) = \pi^2 + 8K$

$\psi_{1}\left(\frac{3}{4}\right) = \pi^2 - 8K$

Its numerical value is approximately

K = .915 965 594 177 219 015 054 603 514 932 384 110 774 ...

It also appears in connection with the hyperbolic secant distribution.

It is not known whether K is rational or irrational.