§25.4 Reflection Formulas

Notes:: See Apostol (1976, pp. 224 and 259).
Keywords:: Riemann’s $\xi$ -function, Riemann zeta function
Referenced by:: §25.20
Permalink:: http://dlmf.nist.gov/25.4

For $s\neq 0,1$ ,

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\cos\/}\nolimits z$ : cosine function and $s$ : complex variable
Referenced by:: §25.10(i), §5.17
Permalink:: http://dlmf.nist.gov/25.4.E1
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\sin\/}\nolimits z$ : sine function and $s$ : complex variable
A&S Ref:: 23.2.6
Permalink:: http://dlmf.nist.gov/25.4.E2
Encodings:: TeX, pMML, png

Equivalently,

25.4.3 $\mathop{\xi\/}\nolimits\!\left(s\right)=\mathop{\xi\/}\nolimits\!\left(1-s\right),$

Symbols:: $\mathop{\xi\/}\nolimits\!\left(s\right)$ : Riemann’s $\mathop{\xi\/}\nolimits$ -function and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.4.E3
Encodings:: TeX, pMML, png

where $\mathop{\xi\/}\nolimits\!\left(s\right)$ is Riemann’s $\mathop{\xi\/}\nolimits$ -function, defined by:

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

Defines:: $\mathop{\xi\/}\nolimits\!\left(s\right)$ : Riemann’s $\mathop{\xi\/}\nolimits$ -function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.4.E4
Encodings:: TeX, pMML, png

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

25.4.5 $(-1)^{k}{\mathop{\zeta\/}\nolimits^{{(k)}}}\!\left(1-s\right)=\frac{2}{(2\pi)^{s}}\sum _{{m=0}}^{k}\sum _{{r=0}}^{m}\binom{k}{m}\binom{m}{r}\left(\realpart{(c^{{k-m}})}\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\pi s\right)+\imagpart{(c^{{k-m}})}\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi s\right)\right){\mathop{\Gamma\/}\nolimits^{{(r)}}}\!\left(s\right){\mathop{\zeta\/}\nolimits^{{(m-r)}}}\!\left(s\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\binom{m}{n}$ : binomial coefficient, $\mathop{\cos\/}\nolimits z$ : cosine function, $\imagpart{}$ : imaginary part, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : nonnegative integer, $m$ : nonnegative integer, $s$ : complex variable and $c$
Permalink:: http://dlmf.nist.gov/25.4.E5
Encodings:: TeX, pMML, png

where

25.4.6 $c=-\mathop{\ln\/}\nolimits\!\left(2\pi\right)-\tfrac{1}{2}\pi i.$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $c$
Referenced by:: §25.6(ii)
Permalink:: http://dlmf.nist.gov/25.4.E6
Encodings:: TeX, pMML, png