Reflection Formulas: Difference between revisions

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== Reflection Formulas ==
== Reflection Formulas ==


<math id="DLMF:25.4:E1">  
<math>\Riemannzeta@{1-s} = 2 (2\cpi)^{-s} \cos@{\tfrac{1}{2} \cpi s} \EulerGamma@{s} \Riemannzeta@{s}</math>
  \RiemannZeta@{1-s}
  = 2 (2\cpi)^{-s} \cos@{\tfrac{1}{2} \cpi s} \EulerGamma@{s} \RiemannZeta@{s}
</math>
<div align="right">Constraint(s): <math>{\displaystyle s \neq 0,1}</math></div><br />
<div align="right">Constraint(s): <math>{\displaystyle s \neq 0,1}</math></div><br />
<math id="DLMF:25.4:E2">{\displaystyle  
<math id="DLMF:25.4:E2">{\displaystyle
   \RiemannZeta@{s}
   \RiemannZeta@{s}
   = 2 (2\cpi)^{s-1} \sin@{\tfrac{1}{2} \cpi s}
   = 2 (2\cpi)^{s-1} \sin@{\tfrac{1}{2} \cpi s}

Latest revision as of 19:24, 16 December 2019

Reflection Formulas

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{1-s} = 2 (2\cpi)^{-s} \cos@{\tfrac{1}{2} \cpi s} \EulerGamma@{s} \Riemannzeta@{s}}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle s \neq 0,1}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = 2 (2\cpi)^{s-1} \sin@{\tfrac{1}{2} \cpi s} \EulerGamma@{1-s} \RiemannZeta@{1-s} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannXi@{s} = \RiemannXi@{1-s} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannXi@{s} = \tfrac{1}{2} s (s-1) \EulerGamma@{\tfrac{1}{2} s} \cpi^{-s/2} \RiemannZeta@{s} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \opminus^k \RiemannZeta^{(k)}@{1-s} = \frac{2}{(2\cpi)^s} \sum_{m=0}^k \sum_{r=0}^m \binom{k}{m} \binom{m}{r} \left( \realpart{(c^{k-m})} \cos@{\tfrac{1}{2} \cpi s} + \imagpart{(c^{k-m})} \sin@{\tfrac{1}{2} \cpi s} \right) \EulerGamma^{(r)}@{s} \RiemannZeta^{(m-r)}@{s} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\displaystyle c = -\ln@{2\cpi} - \tfrac{1}{2} \cpi \iunit }}}


Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle s \neq 0,1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle k = 1,2,3,\dots}}