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.
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 |
Catalan constant. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Catalan_constant&oldid=38979