Formula:DLMF:25.5:E20
Constraint(s)
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}}
Proof
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Symbols List
& : logical and
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \zeta}}
: Riemann zeta function : http://dlmf.nist.gov/25.2#E1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Gamma}}
: Euler's gamma function : http://dlmf.nist.gov/5.2#E1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \pi}}
: ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int}}
: integral : http://dlmf.nist.gov/1.4#iv
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mathrm{e}}}
: the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mathrm{d}^nx}}
: differential : http://dlmf.nist.gov/1.4#iv
Bibliography
Equation (20), Section 25.5 of DLMF.
URL links
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