Difference between revisions of "Catalan constant"
(References: Finch (2003)) |
m (link) |
||
Line 35: | Line 35: | ||
<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> | <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 | + | 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 |
Catalan constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Catalan_constant&oldid=38979