Named after its inventor, E.Ch. Catalan (1814–1894), the Catalan constant
(which is denoted also by
) is defined by
![](https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300403.png) | (a1) |
If, in terms of the Digamma (or Psi) function
, defined by
![](https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300406.png) | (a2) |
or
one puts
![](https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300408.png) | (a3) |
where
then
![](https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004011.png) | (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
![](https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004016.png) | (a5) |
and the Hurwitz zeta-function
, which is defined, when
, by
![](https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004020.png) | (a6) |
Thus,
![](https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004022.png) | (a7) |
Since
![](https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004024.png) | (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,
![](https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004027.png) | (a9) |
![](https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004029.png) | (a10) |
and
![](https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004031.png) | (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