Catalan constant

Named after its inventor, E.Ch. Catalan (1814–1894), the Catalan constant (which is denoted also by ) is defined by

(a1)

If, in terms of the Digamma (or Psi) function , defined by

(a2)

or

one puts

(a3)

where

then

(a4)

which provides a relationship between the Catalan constant and the Digamma function .

The Catalan constant is related also to other functions, such as the Clausen function , defined by

(a5)

and the Hurwitz zeta-function , which is defined, when , by

(a6)

Thus,

(a7)

Since

(a8)

the last expression in (a7) would follow also from (a4) in light of the definition in (a3).

A fairly large number of integrals and series can be evaluated in terms of the Catalan constant . For example,

(a9)

(a10)

and

(a11)

where denotes the familiar Riemann zeta-function.