https://drmf-beta.wmflabs.org/index.php?title=Polylogarithms&feed=atom&action=history
Polylogarithms - Revision history
2024-03-29T11:36:58Z
Revision history for this page on the wiki
MediaWiki 1.42.0-wmf.12
https://drmf-beta.wmflabs.org/index.php?title=Polylogarithms&diff=2215&oldid=prev
imported>SeedBot: DRMF
2017-03-05T22:34:41Z
<p>DRMF</p>
<p><b>New page</b></p><div>{{DISPLAYTITLE:Polylogarithms}}<br />
<div id="drmf_head"><br />
<div id="alignleft"> << [[Hurwitz Zeta Function|Hurwitz Zeta Function]] </div><br />
<div id="aligncenter"> [[Zeta_and_Related_Functions#Polylogarithms|Polylogarithms]] </div><br />
<div id="alignright"> [[Periodic Zeta Function|Periodic Zeta Function]] >> </div><br />
</div><br />
<br />
== Polylogarithms ==<br />
<br />
== Dilogarithms ==<br />
<br />
<math id="DLMF:25.12:E1">{\displaystyle <br />
\Dilogarithm@{z} = \sum_{n=1}^\infty \frac{z^n}{n^2}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle |z| \leq 1}</math></div><br /><br />
<math id="DLMF:25.12:E2">{\displaystyle <br />
\Dilogarithm@{z} = -\int_0^z t^{-1} \ln@{1-t} \diff{t}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle z \in \Complex \setminus (1,\infty)}</math></div><br /><br />
<math id="DLMF:25.12:E3">{\displaystyle <br />
\Dilogarithm@{z} + \Dilogarithm@{\frac{z}{z-1}}<br />
= - \frac{1}{2} (\ln@{1-z})^2<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle z \in \Complex \setminus [1,\infty)}</math></div><br /><br />
<math id="DLMF:25.12:E4">{\displaystyle <br />
\Dilogarithm@{z} + \Dilogarithm@{\frac{1}{z}}<br />
= - \frac{1}{6} \cpi^2 -\frac{1}{2} (\ln@{-z})^2<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle z \in \Complex \setminus [0,\infty)}</math></div><br /><br />
<math id="DLMF:25.12:E5">{\displaystyle <br />
\Dilogarithm@{z^m}<br />
= m \sum_{k=0}^{m-1} \Dilogarithm@{z \expe^{2 \cpi \iunit k/m}}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle m = 1,2,3,\dots}</math> &<br /> <math>{\displaystyle \{z\in\Complex:|z|<1\}\setminus[0,1)}</math></div><br /><br />
<math id="DLMF:25.12:E6">{\displaystyle <br />
\Dilogarithm@{x} + \Dilogarithm@{1-x} = \frac{1}{6} \cpi^2 -<br />
(\ln x) \ln@{1-x}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle 0 < x < 1}</math></div><br /><br />
<math id="DLMF:25.12:E7">{\displaystyle <br />
\Dilogarithm@{\expe^{\iunit\theta}}<br />
= \sum_{n=1}^\infty \frac{\cos@{n \theta}}{n^2}<br />
+ \iunit \sum_{n=1}^\infty \frac{\sin@{n \theta}}{n^2}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle 0 \leq \theta \leq 2\cpi}</math></div><br /><br />
<math id="DLMF:25.12:E8">{\displaystyle <br />
\sum_{n \hiderel{=} 1}^\infty \frac{\cos@{n \theta}}{n^2}<br />
= \frac{\cpi^2}{6} - \frac{\cpi \theta}{2} + \frac{\theta^2}{4}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle 0 \leq \theta \leq 2\cpi}</math></div><br /><br />
<math id="DLMF:25.12:E9">{\displaystyle <br />
\sum_{n \hiderel{=} 1}^\infty \frac{\sin@{n \theta}}{n^2}<br />
= - \int_0^\theta \ln@{2 \sin@{\tfrac{1}{2} x}} \diff{x}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle 0 \leq \theta \leq 2\cpi}</math></div><br /><br />
<br />
== Polylogarithms ==<br />
<br />
<math id="DLMF:25.12:E10">{\displaystyle <br />
\Polylogarithm{s}@{z} = \sum_{n=1}^\infty \frac{z^n}{n^s}<br />
}</math><br />
<div align="right">Constraint(s): real or complex <math>{\displaystyle s}</math> and <math>{\displaystyle z}</math></div><br /><br />
<br />
=== Integral Representation ===<br />
<br />
<math id="DLMF:25.12:E11">{\displaystyle <br />
\Polylogarithm{s}@{z}<br />
= \frac{z}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\expe^x - z} \diff{x}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle \realpart{s} > 0}</math> and <math>{\displaystyle \abs{\ph@{1-z}} < \cpi}</math>, or <math>{\displaystyle \realpart{s} > 1}</math> and <math>{\displaystyle z = 1}</math></div><br /><br />
<math id="DLMF:25.12:E12">{\displaystyle <br />
\Polylogarithm{s}@{z}<br />
= \EulerGamma@{1-s} \left( \ln@@{\frac{1}{z}} \right)^{s-1}<br />
+ \sum_{n=0}^\infty \RiemannZeta@{s-n} \frac{(\ln@@{z})^n}{n!}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle s \neq 1,2,3,\dots}</math> &<br /> <math>{\displaystyle |\ln@@{z}| < 2\cpi}</math></div><br /><br />
<math id="DLMF:25.12:E13">{\displaystyle <br />
\Polylogarithm{s}@{\expe^{2 \cpi \iunit a}}<br />
+ \expe^{\cpi \iunit s} \Polylogarithm{s}@{\expe^{-2 \cpi \iunit a}}<br />
= \frac{(2 \cpi)^s \expe^{\cpi \iunit s/2}}{\EulerGamma@{s}} \HurwitzZeta@{1-s}{a}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle \realpart{s} > 0}</math>, <math>{\displaystyle \imagpart{a} > 0}</math> or <math>{\displaystyle \realpart{s} > 1}</math>, <math>{\displaystyle \imagpart{a} = 0}</math></div><br /><br />
<br />
== Fermi-Dirac and Bose-Einstein Integrals ==<br />
<br />
<math id="DLMF:25.12:E14">{\displaystyle <br />
F_s(x)<br />
= \frac{1}{\EulerGamma@{s+1}}<br />
\int_0^\infty \frac{t^s}{\expe^{t-x}+1} \diff{t}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle s > -1}</math></div><br /><br />
<div align="right">This formula has the name: Fermi-Dirac integral</div><br /><br />
<math id="DLMF:25.12:E15">{\displaystyle <br />
G_s(x)<br />
= \frac{1}{\EulerGamma@{s+1}}<br />
\int_0^\infty \frac{t^s}{\expe^{t-x}-1} \diff{t}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle s > -1}</math>, <math>{\displaystyle x < 0}</math>, or <math>{\displaystyle s > 0}</math>, <math>{\displaystyle x \leq 0}</math></div><br /><br />
<div align="right">This formula has the name: Bose-Einstein integral</div><br /><br />
<math id="DLMF:25.12:E16:SE1">{\displaystyle <br />
F_s(x) = -\Polylogarithm{s+1}@{-\expe^x}<br />
}</math><br /><br />
<math id="DLMF:25.12:E16:SE2">{\displaystyle <br />
G_s(x) = \Polylogarithm{s+1}@{\expe^x}<br />
}</math><br />
<div id="drmf_foot"><br />
<div id="alignleft"> << [[Hurwitz Zeta Function|Hurwitz Zeta Function]] </div><br />
<div id="aligncenter"> [[Zeta_and_Related_Functions#Polylogarithms|Polylogarithms]] </div><br />
<div id="alignright"> [[Periodic Zeta Function|Periodic Zeta Function]] >> </div><br />
</div></div>
imported>SeedBot