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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.


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