Catalan constant

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


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



one puts





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


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






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,




where denotes the familiar Riemann zeta-function.

Euler–Mascheroni constant.

Another important mathematical constant is the Euler–Mascheroni constant (which is denoted also by ), defined by


It is named after L. Euler (1707–1783) and L. Mascheroni (1750–1800). Indeed, one also has




where an empty sum is interpreted, as usual, to be zero. In terms of the Riemann zeta-function , Euler's classical results state:



[a1] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, "Higher transcendental functions" , I , McGraw-Hill (1953)
[a2] L. Lewin, "Polylogarithms and associated functions" , Elsevier (1981)
[a3] H.M. Srivastava, J. Choi, "Series associated with the zeta and related functions" , Kluwer Acad. Publ. (2001)
How to Cite This Entry:
Catalan constant. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Hari M. Srivastava (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article