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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://encyclopediaofmath.org/index.php?title=Catalan_constant&oldid=38979
This article was adapted from an original article by Hari M. Srivastava (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
