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Difference between revisions of "Catalan constant"

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004017.png" /></td> </tr></table>
  
and the Hurwitz zeta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004018.png" />, which is defined, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004019.png" />, by
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and the [[Hurwitz zeta function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004018.png" />, which is defined, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004019.png" />, by
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>

Revision as of 19:52, 14 June 2016

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