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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

(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.

References

[Fi] Steven R. Finch, "Mathematical constants" , Encyclopedia of mathematics and its applications 94, Cambridge University Press (2003) ISBN 0-521-81805-2 Zbl 1054.00001
How to Cite This Entry:
Catalan constant. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Catalan_constant&oldid=35953
This article was adapted from an original article by Hari M. Srivastava (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article