§25.4 Reflection Formulas

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Keywords:: Riemann zeta function, Riemann’s $\xi$ -function, reflection formulas
Notes:: See Apostol (1976, Section 12.8) and Apostol (1985a).
Referenced by:: 3rd item
Permalink:: http://dlmf.nist.gov/25.4
See also:: Annotations for Ch.25

For $s\neq 0,1$ ,

25.4.1 $\zeta\left(1-s\right)=2(2\pi)^{-s}\cos\left(\tfrac{1}{2}\pi s\right)\Gamma% \left(s\right)\zeta\left(s\right),$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function and : complex variable
Keywords:: reflection formula
Source:: Apostol (1976, (12), p. 259)
Referenced by:: §25.10(i), §5.17
Permalink:: http://dlmf.nist.gov/25.4.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §25.4 and Ch.25

25.4.2 $\zeta\left(s\right)=2(2\pi)^{s-1}\sin\left(\tfrac{1}{2}\pi s\right)\Gamma\left% (1-s\right)\zeta\left(1-s\right).$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function and : complex variable
Keywords:: reflection formula
Source:: Apostol (1976, (13), p. 259)
A&S Ref:: 23.2.6
Permalink:: http://dlmf.nist.gov/25.4.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §25.4 and Ch.25

Equivalently,

25.4.3 $\xi\left(s\right)=\xi\left(1-s\right),$

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Symbols:: $\xi\left(\NVar{s}\right)$ : Riemann’s $\xi$ -function and : complex variable
Keywords:: reflection formula
Source:: Apostol (1976, p. 260)
Permalink:: http://dlmf.nist.gov/25.4.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §25.4 and Ch.25

where $\xi\left(s\right)$ is Riemann’s $\xi$ -function, defined by:

25.4.4 $\xi\left(s\right)=\tfrac{1}{2}s(s-1)\Gamma\left(\tfrac{1}{2}s\right)\pi^{-s/2}% \zeta\left(s\right).$

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Defines:: $\xi\left(\NVar{s}\right)$ : Riemann’s $\xi$ -function
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter and : complex variable
Keywords:: definition
Source:: Apostol (1976, p. 260)
Permalink:: http://dlmf.nist.gov/25.4.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §25.4 and Ch.25

For $s\neq 0,1$ and $k=1,2,3,\dots$ ,

25.4.5 $(-1)^{k}{\zeta}^{(k)}\left(1-s\right)=\frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{% r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\left(\Re% \left(c^{k-m}\right)\cos\left(\tfrac{1}{2}\pi s\right)+\Im\left(c^{k-m}\right)% \sin\left(\tfrac{1}{2}\pi s\right)\right){\Gamma}^{(r)}\left(s\right){\zeta}^{% (m-r)}\left(s\right),$

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