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DLMF: §25.4 Reflection Formulas ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions

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25 Zeta and Related Functions Riemann Zeta Function 25.3 Graphics 25.5 Integral Representations

§25.4 Reflection Formulas

ⓘ

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		$\zeta\left(1-s\right)=2(2\pi)^{-s}\cos\left(\tfrac{1}{2}\pi s\right)\Gamma% \left(s\right)\zeta\left(s\right),$
ⓘ 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 $s$ : 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		$\zeta\left(s\right)=2(2\pi)^{s-1}\sin\left(\tfrac{1}{2}\pi s\right)\Gamma\left% (1-s\right)\zeta\left(1-s\right).$
ⓘ 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 $s$ : 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		$\xi\left(s\right)=\xi\left(1-s\right),$
ⓘ Symbols: $\xi\left(\NVar{s}\right)$ : Riemann’s $\xi$ -function and $s$ : 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		$\xi\left(s\right)=\tfrac{1}{2}s(s-1)\Gamma\left(\tfrac{1}{2}s\right)\pi^{-s/2}% \zeta\left(s\right).$
ⓘ 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 $s$ : 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		$(-1)^{k}{\zeta}^{(k)}\left(1-s\right)=\frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{% r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\left(\Re% \left(c^{k-m}\right)\cos\left(\tfrac{1}{2}\pi s\right)+\Im\left(c^{k-m}\right)% \sin\left(\tfrac{1}{2}\pi s\right)\right){\Gamma}^{(r)}\left(s\right){\zeta}^{% (m-r)}\left(s\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\Im$ : imaginary part, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer, $m$ : nonnegative integer, $s$ : complex variable and $c$ Keywords: derivative Source: Apostol (1985a, (6), p. 224) Permalink: http://dlmf.nist.gov/25.4.E5 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25

where

25.4.6		$c\equiv-\ln\left(2\pi\right)-\tfrac{1}{2}\pi\mathrm{i}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\equiv$ : equals by definition, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function and $c$ Keywords: definition Source: Apostol (1985a, p. 223) Referenced by: §25.6(ii) Permalink: http://dlmf.nist.gov/25.4.E6 Encodings: TeX, pMML, png See also: Annotations for §25.4 and Ch.25

25.3 Graphics 25.5 Integral Representations

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