Integral Representations

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Integral Representations

In Terms of Elementary Functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\expe^x-1} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{\EulerGamma@{s+1}} \int_0^\infty \frac{\expe^x x^s}{(\expe^x-1)^2} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{(1 - 2^{1-s}) \EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\expe^x+1} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{(1 - 2^{1-s}) \EulerGamma@{s+1}} \int_0^\infty \frac{\expe^x x^s}{(\expe^x+1)^2} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = -s \int_0^\infty \frac{x-\floor{x}-\frac{1}{2}}{x^{s+1}} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{2} + \frac{1}{s-1} + \frac{1}{\EulerGamma@{s}} \int_0^\infty \left( \frac{1}{\expe^x-1} - \frac{1}{x} + \frac{1}{2} \right) \frac{x^{s-1}}{\expe^x} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{2} + \frac{1}{s-1} + \sum_{m=1}^n \frac{\BernoulliB{2m}}{(2m)!} \frac{\EulerGamma@{s+2m-1}}{\EulerGamma@{s}} + \frac{1}{\EulerGamma@{s}} \int_0^\infty \left( \frac{1}{\expe^x-1} - \frac{1}{x} + \frac{1}{2} - \sum_{m=1}^n \frac{\BernoulliB{2m}}{(2m)!} x^{2m-1} \right) \frac{x^{s-1}}{\expe^x} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{2 (1 - 2^{-s}) \EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\sinh@@{x}} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{2^{s-1}}{\EulerGamma@{s+1}} \int_0^\infty \frac{x^s}{(\sinh@@{x})^2} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{2^{s-1}}{1 - 2^{1-s}} \int_0^\infty \frac{\cos@{s \atan@@{x}}}{(1 + x^2)^{s/2} \cosh@{\frac{1}{2} \cpi x}} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{2} + \frac{1}{s-1} + 2 \int_0^\infty \frac{\sin@{s \atan@@{x}}}{(1 + x^2)^{s/2} (\expe^{2\cpi x} - 1)} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{2^{s-1}}{s-1} - 2^s \int_0^\infty \frac{\sin@{s \atan@@{x}}}{(1 + x^2)^{s/2} (\expe^{\cpi x} + 1)} \diff{x} }}

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


In Terms of Other Functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{\cpi^{s/2}}{s(s-1)\EulerGamma@{\frac{1}{2}s}} + \frac{\cpi^{s/2}}{\EulerGamma@{\frac{1}{2}s}} \* \int_1^\infty \left (x^{s/2} + x^{(1-s)/2} \right) \frac{\omega(x)}{x} \diff{x} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\displaystyle \omega(x) = \sum_{n=1}^\infty \expe^{-n^2 \cpi x} = \frac{1}{2} \left( \JacobiThetaTau{3}@{0}{\iunit x} - 1 \right)}}}


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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{s-1} + \frac{\sin@{\cpi s}}{\cpi} \* \int_0^\infty (\ln@{1+x} - \digamma@{1+x}) x^{-s} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{s-1} + \frac{\sin@{\cpi s}}{\cpi (s-1)} \* \int_0^\infty \left( \frac{1}{1+x} - \digamma'@{1+x} \right) x^{1-s} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{1+s} = \frac{\sin@{\cpi s}}{\cpi} \int_0^\infty \left( \EulerConstant + \digamma@{1+x} \right) x^{-s-1} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{1+s} = \frac{\sin@{\cpi s}}{\cpi s} \int_0^\infty \digamma'@{1+x} x^{-s} \diff{x} }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{m+s} = \opminus^{m-1} \frac{\EulerGamma@{s} \sin@{\cpi s}}{\cpi \EulerGamma@{m+s}} \* \int_0^\infty \digamma^{(m)}@{1+x} x^{-s} \diff{x} }}

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


Contour Integrals

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{\EulerGamma@{1-s}}{2 \cpi \iunit} \int_{-\infty}^{(0+)} \frac{z^{s-1}}{\expe^{-z}-1} \diff{z} }}

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

The integration contour is a loop around the negative real axis; it starts at Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -\infty}} , encircles the origin once in the positive direction without enclosing any of the points

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle z=\pm2\cpi\iunit}} , Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \pm4\cpi\iunit, \ldots,}} and returns to Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -\infty}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{\EulerGamma@{1-s}}{2 \cpi \iunit (1 - 2^{1-s})} \* \int_{-\infty}^{(0+)} \frac{z^{s-1}}{\expe^{-z}+1} \diff{z} }}

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

The contour here is any loop that encircles the origin in the positive direction

not enclosing any of the points Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \pm\cpi\iunit}} , Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \pm3\cpi\iunit, \ldots}}


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