Difference between revisions of "Catalan constant"
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− | and the Hurwitz zeta | + | 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
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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://encyclopediaofmath.org/index.php?title=Catalan_constant&oldid=38979