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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> | | <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> |
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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 | + | 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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| <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> |
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| where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004032.png" /> denotes the familiar [[Riemann zeta-function|Riemann zeta-function]]. | | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004032.png" /> denotes the familiar [[Riemann zeta-function|Riemann zeta-function]]. |
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− | ==Euler–Mascheroni constant.==
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− | Another important mathematical constant is the Euler–Mascheroni constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004033.png" /> (which is denoted also by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004034.png" />), defined by
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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/c13004035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a12)</td></tr></table>
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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/c13004036.png" /></td> </tr></table>
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− | It is named after L. Euler (1707–1783) and L. Mascheroni (1750–1800). Indeed, one also has
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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/c13004037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a13)</td></tr></table>
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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/c13004038.png" /></td> </tr></table>
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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/c13004039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a14)</td></tr></table>
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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/c13004040.png" /></td> </tr></table>
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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/c13004041.png" /></td> </tr></table>
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− | where an empty sum is interpreted, as usual, to be zero. In terms of the Riemann zeta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004042.png" />, Euler's classical results state:
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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/c13004043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a15)</td></tr></table>
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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/c13004044.png" /></td> </tr></table>
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| ====References==== | | ====References==== |
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, "Higher transcendental functions" , '''I''' , McGraw-Hill (1953)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L. Lewin, "Polylogarithms and associated functions" , Elsevier (1981)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H.M. Srivastava, J. Choi, "Series associated with the zeta and related functions" , Kluwer Acad. Publ. (2001)</TD></TR></table>
| + | {| |
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| + | |valign="top"|{{Ref|Fi}}||valign="top"| Steven R. Finch, "Mathematical constants" , Encyclopedia of mathematics and its applications '''94''', Cambridge University Press (2003) ISBN 0-521-81805-2 {{ZBL|1054.00001}} |
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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
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