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Formula:DLMF:25.14:E5 - Revision history
2024-03-28T09:42:44Z
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imported>SeedBot: DRMF
2017-03-05T22:34:49Z
<p>DRMF</p>
<p><b>New page</b></p><div>{{DISPLAYTITLE:Formula:DLMF:25.14:E5}}<br />
<div id="drmf_head"><br />
<div id="alignleft"> << [[Formula:DLMF:25.14:E4|Formula:DLMF:25.14:E4]] </div><br />
<div id="aligncenter"> [[Lerch's_Transcendent#DLMF:25.14:E5|formula in Lerch's Transcendent]] </div><br />
<div id="alignright"> [[Formula:DLMF:25.14:E6|Formula:DLMF:25.14:E6]] >> </div><br />
</div><br />
<br />
<br /><div align="center"><math>{\displaystyle <br />
\LerchPhi@{z}{s}{a}<br />
= \frac{1}{\EulerGamma@{s}}<br />
\int_0^\infty \frac{x^{s-1} \expe^{-ax}}{1-z\expe^{-x}} \diff{x}<br />
}</math></div><br />
<br />
== Constraint(s) ==<br />
<br />
<div align="left"><math>{\displaystyle \realpart{s} > 0}</math> &<br /> <math>{\displaystyle \realpart{a} >0}</math> &<br /><br />
<math>{\displaystyle z \in \Complex \setminus [1,\infty)}</math></div><br /><br />
<br />
== Proof ==<br />
<br />
We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.<br />
<br />
== Symbols List ==<br />
<br />
& : logical and<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/25.14#E1 <math>{\displaystyle \Phi}</math>]</span> : Lerch's transcendent : [http://dlmf.nist.gov/25.14#E1 http://dlmf.nist.gov/25.14#E1]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/5.2#E1 <math>{\displaystyle \Gamma}</math>]</span> : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/1.4#iv <math>{\displaystyle \int}</math>]</span> : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/4.2.E11 <math>{\displaystyle \mathrm{e}}</math>]</span> : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/1.4#iv <math>{\displaystyle \mathrm{d}^nx}</math>]</span> : differential : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/1.9#E2 <math>{\displaystyle \Re {z}}</math>]</span> : real part : [http://dlmf.nist.gov/1.9#E2 http://dlmf.nist.gov/1.9#E2]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9 <math>{\displaystyle \in}</math>]</span> : element of : [http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9 http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r1 <math>{\displaystyle \mathbb{C}}</math>]</span> : set of complex numbers : [http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r1 http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r1]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r19 <math>{\displaystyle \setminus}</math>]</span> : set subtraction : [http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r19 http://dlmf.nist.gov/front/introduction#Sx4.p2.t1.r19]<br />
<br /><br />
<br />
== Bibliography==<br />
<br />
<span class="plainlinks">[HTTP://DLMF.NIST.GOV/25.14#E5 Equation (5), Section 25.14]</span> of [[Bibliography#DLMF|'''DLMF''']].<br />
<br />
== URL links ==<br />
<br />
We ask users to provide relevant URL links in this space.<br />
<br />
<br /><div id="drmf_foot"><br />
<div id="alignleft"> << [[Formula:DLMF:25.14:E4|Formula:DLMF:25.14:E4]] </div><br />
<div id="aligncenter"> [[Lerch's_Transcendent#DLMF:25.14:E5|formula in Lerch's Transcendent]] </div><br />
<div id="alignright"> [[Formula:DLMF:25.14:E6|Formula:DLMF:25.14:E6]] >> </div><br />
</div></div>
imported>SeedBot