Namespaces
Variants
Actions

Difference between revisions of "Catalan constant"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (link)
m (AUTOMATIC EDIT (latexlist): Replaced 32 formulas out of 32 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
Line 1: Line 1:
Named after its inventor, E.Ch. Catalan (1814–1894), the Catalan constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300401.png" /> (which is denoted also by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300402.png" />) is defined by
+
<!--This article has been texified automatically. Since there was no Nroff source code for this article,  
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
<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/c1300403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
Out of 32 formulas, 32 were replaced by TEX code.-->
  
<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/c1300404.png" /></td> </tr></table>
+
{{TEX|semi-auto}}{{TEX|done}}
 +
Named after its inventor, E.Ch. Catalan (1814–1894), the Catalan constant $G$ (which is denoted also by $\lambda$) is defined by
  
If, in terms of the Digamma (or Psi) function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c1300405.png" />, defined by
+
\begin{equation} \tag{a1} G : = \sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { ( 2 k + 1 ) ^ { 2 } } \cong \end{equation}
  
<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/c1300406.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation*} \cong 0.915965594177219015 \ldots . \end{equation*}
 +
 
 +
If, in terms of the Digamma (or Psi) function $\psi ( z )$, defined by
 +
 
 +
\begin{equation} \tag{a2} \psi ( z ) : = \frac { d } { d z } \{ \operatorname { log } \Gamma ( z ) \} = \frac { \Gamma ^ { \prime } ( z ) } { \Gamma ( z ) } \end{equation}
  
 
or
 
or
  
<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/c1300407.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { log } \Gamma ( z ) = \int _ { 1 } ^ { z } \psi ( t ) d t, \end{equation*}
  
 
one puts
 
one puts
  
<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/c1300408.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\begin{equation} \tag{a3} \beta ( z ) : = \frac { 1 } { 2 } \left[ \psi \left( \frac { 1 } { 2 } z + \frac { 1 } { 2 } \right) - \psi \left( \frac { 1 } { 2 } z \right) \right] = \end{equation}
  
<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/c1300409.png" /></td> </tr></table>
+
\begin{equation*} = \sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { z + k }, \end{equation*}
  
 
where
 
where
  
<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/c13004010.png" /></td> </tr></table>
+
\begin{equation*} z \in \mathbf{C} \backslash \mathbf{Z} _ { 0 }^- , \quad \mathbf{Z} _ { 0 } ^ { - } : = \{ 0 , - 1 , - 2 , \ldots \}, \end{equation*}
  
 
then
 
then
  
<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/c13004011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\begin{equation} \tag{a4} G = - \frac { 1 } { 4 } \beta ^ { \prime } \left( \frac { 1 } { 2 } \right) \end{equation}
  
which provides a relationship between the Catalan constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004012.png" /> and the Digamma function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004013.png" />.
+
which provides a relationship between the Catalan constant $G$ and the Digamma function $\psi ( z )$.
  
The Catalan constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004014.png" /> is related also to other functions, such as the Clausen function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004015.png" />, defined by
+
The Catalan constant $G$ is related also to other functions, such as the Clausen function $\operatorname{Cl} _ { 2 } ( z )$, defined 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/c13004016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
+
\begin{equation} \tag{a5}  \operatorname {Cl} _ { 2 } ( z ) : = - \int _ { 0 } ^ { z } \operatorname { log } \left| 2 \operatorname { sin } \left( \frac { 1 } { 2 } t \right) \right| d t = \end{equation}
  
<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>
+
\begin{equation*} = \sum _ { k = 1 } ^ { \infty } \frac { \operatorname { sin } ( k z ) } { k ^ { 2 } }, \end{equation*}
  
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]] $\zeta ( s , a )$, which is defined, when $\operatorname { Re } s &gt; 1$, 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>
+
\begin{equation} \tag{a6} \zeta ( s , a ) : = \sum _ { k = 0 } ^ { \infty } \frac { 1 } { ( k + a ) ^ { s } }, \end{equation}
  
<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/c13004021.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { Re } s &gt; 1 , a \in \mathbf{C} \backslash \mathbf{Z} ^{ - } _ { 0 }. \end{equation*}
  
 
Thus,
 
Thus,
  
<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/c13004022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
+
\begin{equation} \tag{a7} G = \operatorname{Cl} _ { 2 } ( \frac { 1 } { 2 } \pi ) = - \operatorname{Cl} _ { 2 } \left( \frac { 3 } { 2 } \pi \right) = \end{equation}
  
<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/c13004023.png" /></td> </tr></table>
+
\begin{equation*} = \frac { 1 } { 16 } \left[ \zeta \left( 2 , \frac { 1 } { 4 } \right) - \zeta \left( 2 , \frac { 3 } { 4 } \right) \right]. \end{equation*}
  
 
Since
 
Since
  
<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/c13004024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
+
\begin{equation} \tag{a8} \psi ^ { ( n ) } ( z ) = ( - 1 ) ^ { n + 1 } n ! \zeta ( n + 1 , z ), \end{equation}
  
<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/c13004025.png" /></td> </tr></table>
+
\begin{equation*} n \in \mathbf{N} : = \{ 1,2 , \ldots \} , z \in \mathbf{C} \backslash \mathbf{Z} _ { 0 } ^ { - }, \end{equation*}
  
 
the last expression in (a7) would follow also from (a4) in light of the definition in (a3).
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130040/c13004026.png" />. For example,
+
A fairly large number of integrals and series can be evaluated in terms of the Catalan constant $G$. For example,
  
<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/c13004027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
+
\begin{equation} \tag{a9} \int _ { 0 } ^ { 1 } \frac { t\operatorname { log } ( t ^ { - 1 } \pm t ) } { 1 + t ^ { 4 } } d t = \end{equation}
  
<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/c13004028.png" /></td> </tr></table>
+
\begin{equation*} = \int _ { 1 } ^ { \infty } \frac { t \operatorname { log } ( t \pm t ^ { - 1 } ) } { 1 + t ^ { 4 } } d t = \frac { \pi } { 16 } \operatorname { log } 2 \pm \frac { G } { 4 }, \end{equation*}
  
<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/c13004029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>
+
\begin{equation} \tag{a10} \sum _ { k = 1 } ^ { \infty } \left( \frac { ( 2 k + 1 ) ! } { k ! ( k + 1 ) ! } \right) ^ { 2 } \frac { 2 ^ { - 4 k } } { k } = \end{equation}
  
<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/c13004030.png" /></td> </tr></table>
+
\begin{equation*} = 4 \operatorname { log } 2 + 2 - \frac { 4 } { \pi } ( 2 G + 1 ), \end{equation*}
  
 
and
 
and
  
<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/c13004031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>
+
\begin{equation} \tag{a11} \sum _ { k = 1 } ^ { \infty } \frac { \zeta ( 2 k ) } { k ( 2 k + 1 ) 2 ^ { 4 k } } = \operatorname { log } ( \frac { \pi } { 2 } ) - 1 + \frac { 2 G } { \pi }, \end{equation}
  
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 $\zeta ( s ) = \zeta ( s , 1 )$ denotes the familiar [[Riemann zeta-function|Riemann zeta-function]].
  
 
====References====
 
====References====

Revision as of 16:52, 1 July 2020

Named after its inventor, E.Ch. Catalan (1814–1894), the Catalan constant $G$ (which is denoted also by $\lambda$) is defined by

\begin{equation} \tag{a1} G : = \sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { ( 2 k + 1 ) ^ { 2 } } \cong \end{equation}

\begin{equation*} \cong 0.915965594177219015 \ldots . \end{equation*}

If, in terms of the Digamma (or Psi) function $\psi ( z )$, defined by

\begin{equation} \tag{a2} \psi ( z ) : = \frac { d } { d z } \{ \operatorname { log } \Gamma ( z ) \} = \frac { \Gamma ^ { \prime } ( z ) } { \Gamma ( z ) } \end{equation}

or

\begin{equation*} \operatorname { log } \Gamma ( z ) = \int _ { 1 } ^ { z } \psi ( t ) d t, \end{equation*}

one puts

\begin{equation} \tag{a3} \beta ( z ) : = \frac { 1 } { 2 } \left[ \psi \left( \frac { 1 } { 2 } z + \frac { 1 } { 2 } \right) - \psi \left( \frac { 1 } { 2 } z \right) \right] = \end{equation}

\begin{equation*} = \sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { z + k }, \end{equation*}

where

\begin{equation*} z \in \mathbf{C} \backslash \mathbf{Z} _ { 0 }^- , \quad \mathbf{Z} _ { 0 } ^ { - } : = \{ 0 , - 1 , - 2 , \ldots \}, \end{equation*}

then

\begin{equation} \tag{a4} G = - \frac { 1 } { 4 } \beta ^ { \prime } \left( \frac { 1 } { 2 } \right) \end{equation}

which provides a relationship between the Catalan constant $G$ and the Digamma function $\psi ( z )$.

The Catalan constant $G$ is related also to other functions, such as the Clausen function $\operatorname{Cl} _ { 2 } ( z )$, defined by

\begin{equation} \tag{a5} \operatorname {Cl} _ { 2 } ( z ) : = - \int _ { 0 } ^ { z } \operatorname { log } \left| 2 \operatorname { sin } \left( \frac { 1 } { 2 } t \right) \right| d t = \end{equation}

\begin{equation*} = \sum _ { k = 1 } ^ { \infty } \frac { \operatorname { sin } ( k z ) } { k ^ { 2 } }, \end{equation*}

and the Hurwitz zeta function $\zeta ( s , a )$, which is defined, when $\operatorname { Re } s > 1$, by

\begin{equation} \tag{a6} \zeta ( s , a ) : = \sum _ { k = 0 } ^ { \infty } \frac { 1 } { ( k + a ) ^ { s } }, \end{equation}

\begin{equation*} \operatorname { Re } s > 1 , a \in \mathbf{C} \backslash \mathbf{Z} ^{ - } _ { 0 }. \end{equation*}

Thus,

\begin{equation} \tag{a7} G = \operatorname{Cl} _ { 2 } ( \frac { 1 } { 2 } \pi ) = - \operatorname{Cl} _ { 2 } \left( \frac { 3 } { 2 } \pi \right) = \end{equation}

\begin{equation*} = \frac { 1 } { 16 } \left[ \zeta \left( 2 , \frac { 1 } { 4 } \right) - \zeta \left( 2 , \frac { 3 } { 4 } \right) \right]. \end{equation*}

Since

\begin{equation} \tag{a8} \psi ^ { ( n ) } ( z ) = ( - 1 ) ^ { n + 1 } n ! \zeta ( n + 1 , z ), \end{equation}

\begin{equation*} n \in \mathbf{N} : = \{ 1,2 , \ldots \} , z \in \mathbf{C} \backslash \mathbf{Z} _ { 0 } ^ { - }, \end{equation*}

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 $G$. For example,

\begin{equation} \tag{a9} \int _ { 0 } ^ { 1 } \frac { t\operatorname { log } ( t ^ { - 1 } \pm t ) } { 1 + t ^ { 4 } } d t = \end{equation}

\begin{equation*} = \int _ { 1 } ^ { \infty } \frac { t \operatorname { log } ( t \pm t ^ { - 1 } ) } { 1 + t ^ { 4 } } d t = \frac { \pi } { 16 } \operatorname { log } 2 \pm \frac { G } { 4 }, \end{equation*}

\begin{equation} \tag{a10} \sum _ { k = 1 } ^ { \infty } \left( \frac { ( 2 k + 1 ) ! } { k ! ( k + 1 ) ! } \right) ^ { 2 } \frac { 2 ^ { - 4 k } } { k } = \end{equation}

\begin{equation*} = 4 \operatorname { log } 2 + 2 - \frac { 4 } { \pi } ( 2 G + 1 ), \end{equation*}

and

\begin{equation} \tag{a11} \sum _ { k = 1 } ^ { \infty } \frac { \zeta ( 2 k ) } { k ( 2 k + 1 ) 2 ^ { 4 k } } = \operatorname { log } ( \frac { \pi } { 2 } ) - 1 + \frac { 2 G } { \pi }, \end{equation}

where $\zeta ( s ) = \zeta ( s , 1 )$ 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