Catalan's constant
In mathematics, Catalan's constant G, which occasionally appears in estimates in combinatorics, is defined by
where β is the Dirichlet beta function. Its numerical value is approximately (sequence in the OEIS)
It is not known whether G is irrational, let alone transcendental.
Catalan's constant was named after Eugène Charles Catalan.
The similar but apparently more complicated series
can be evaluated exactly and is π3/32.
Some identities involving definite integrals include
If K(t) is a complete elliptic integral of the first kind, then
With the gamma function Γ(x + 1) = x!
The integral
is a known special function, called the inverse tangent integral, and was extensively studied by Srinivasa Ramanujan.
G appears in combinatorics, as well as in values of the second polygamma function, also called the trigamma function, at fractional arguments:
Simon Plouffe gives an infinite collection of identities between the trigamma function, π2 and Catalan's constant; these are expressible as paths on a graph.
In low-dimensional topology, Catalan's constant is a rational multiple of the volume of an ideal hyperbolic octahedron, and therefore of the hyperbolic volume of the complement of the Whitehead link.
...
Wikipedia