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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.

Euler–Mascheroni constant.

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

(a12)

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

(a13)

and

(a14)

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

(a15)

References

[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. Hari M. Srivastava (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Catalan_constant&oldid=12431
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098