§25.5 Integral Representations

Throughout this subsection $s\ne 1$.

25.5.1		$\zeta \left(s\right)$	$={\displaystyle \frac{1}{\mathrm{\Gamma }\left(s\right)}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{x^{s-1}}{{\mathrm{e}}^x-1}}dx,$
$\mathrm{\Re }s>1$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Apostol (1976, p. 251); with ${\mathrm{e}}^{-x}=(1-{\mathrm{e}}^{-x})/({\mathrm{e}}^x-1)$ A&S Ref: 23.2.7 Referenced by: §25.12(ii), (25.5.2), (25.5.6), §25.5(i) Permalink: http://dlmf.nist.gov/25.5.E1 Encodings: TeX, pMML, png See also: Annotations for §25.5(i), §25.5 and Ch.25
25.5.2		$\zeta \left(s\right)$	$={\displaystyle \frac{1}{\mathrm{\Gamma }\left(s+1\right)}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{{\mathrm{e}}^x{x}^s}{{({\mathrm{e}}^x-1)}^2}}dx,$
$\mathrm{\Re }s>1$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Derivable from (25.5.1) by integration by parts. Referenced by: §25.5(i) Permalink: http://dlmf.nist.gov/25.5.E2 Encodings: TeX, pMML, png See also: Annotations for §25.5(i), §25.5 and Ch.25

25.5.3		$\zeta \left(s\right)$	$={\displaystyle \frac{1}{(1-2^{1-s})\mathrm{\Gamma }\left(s\right)}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{x^{s-1}}{{\mathrm{e}}^x+1}}dx,$
$\mathrm{\Re }s>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Erdélyi et al. (1953a, (5), p. 32) A&S Ref: 23.2.8 Referenced by: (25.5.4), §25.5(i) Permalink: http://dlmf.nist.gov/25.5.E3 Encodings: TeX, pMML, png See also: Annotations for §25.5(i), §25.5 and Ch.25
25.5.4		$\zeta \left(s\right)$	$={\displaystyle \frac{1}{(1-2^{1-s})\mathrm{\Gamma }\left(s+1\right)}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{{\mathrm{e}}^x{x}^s}{{({\mathrm{e}}^x+1)}^2}}dx,$
$\mathrm{\Re }s>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Derivable from (25.5.3) by integration by parts. Referenced by: §25.5(i) Permalink: http://dlmf.nist.gov/25.5.E4 Encodings: TeX, pMML, png See also: Annotations for §25.5(i), §25.5 and Ch.25

25.5.5		$$\zeta \left(s\right)=-s{\int }_0^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor -\frac{1}{2}}{x^{s+1}}dx,$$
.
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$, $\lfloor x\rfloor$: floor of $x$, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Titchmarsh (1986b, (2.1.6), p. 15) Referenced by: §25.5(i) Permalink: http://dlmf.nist.gov/25.5.E5 Encodings: TeX, pMML, png See also: Annotations for §25.5(i), §25.5 and Ch.25

25.5.6		$$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{{\mathrm{e}}^x}dx,$$
$\mathrm{\Re }s>-1$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Magnus et al. (1966, p. 20) Source: Derivable from (25.5.1) by first replacing $x^{s-1}\mapsto x^{s-1}(1+{\mathrm{e}}^{-x}-{\mathrm{e}}^{-x})$, using the identity ${\mathrm{e}}^{-x}=(1-{\mathrm{e}}^{-x})/({\mathrm{e}}^x-1)$, and replacing ${({\mathrm{e}}^x-1)}^{-1}\mapsto ({({\mathrm{e}}^x-1)}^{-1}-1/x+1/2-1/2+1/x)$ in the resulting integrand with (5.2.1), (5.5.1). Referenced by: §25.11(vii), (25.5.7), §25.5(i) Permalink: http://dlmf.nist.gov/25.5.E6 Encodings: TeX, pMML, png See also: Annotations for §25.5(i), §25.5 and Ch.25

25.5.7		$$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum _{m=1}^n\frac{B_{2m}}{(2m)!}{\left(s\right)}_{2m-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}-\sum _{m=1}^n\frac{B_{2m}}{(2m)!}{x}^{2m-1}\right)\frac{x^{s-1}}{{\mathrm{e}}^x}dx,$$
$\mathrm{\Re }s>-(2n+1)$, $n=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Derivable from (25.5.6) by adding and subtracting ${\sum }_{m=1}^n{B}_{2m}{x}^{2m-1}/(2m)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that ${({\mathrm{e}}^x-1)}^{-1}-x^{-1}+1/2-{\sum }_{m=1}^n{B}_{2m}{x}^{2m-1}/(2m)!=O\left(x^{2n+1}\right)$ as $x\to 0$, demonstrating the region of convergence. Referenced by: §25.11(vii), §25.5(i), Erratum (V1.0.9) for Chapters 7, 25 Permalink: http://dlmf.nist.gov/25.5.E7 Encodings: TeX, pMML, png Notational Change (effective with 1.0.9): We have rewritten the original summation ${\sum }_{m=1}^n\frac{B_{2m}}{(2m)!}\frac{\mathrm{\Gamma }\left(s+2m-1\right)}{\mathrm{\Gamma }\left(s\right)}$ more concisely as ${\sum }_{m=1}^n\frac{B_{2m}}{(2m)!}{\left(s\right)}_{2m-1}$ using the Pochhammer symbol. See also: Annotations for §25.5(i), §25.5 and Ch.25

25.5.8		$\zeta \left(s\right)$	$={\displaystyle \frac{1}{2(1-2^{-s})\mathrm{\Gamma }\left(s\right)}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{x^{s-1}}{\sinh x}}dx,$
$\mathrm{\Re }s>1$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Erdélyi et al. (1953a, (6), p. 32) Permalink: http://dlmf.nist.gov/25.5.E8 Encodings: TeX, pMML, png See also: Annotations for §25.5(i), §25.5 and Ch.25
25.5.9		$\zeta \left(s\right)$	$={\displaystyle \frac{2^{s-1}}{\mathrm{\Gamma }\left(s+1\right)}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{x^s}{{(\sinh x)}^2}}dx,$
$\mathrm{\Re }s>1$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Erdélyi et al. (1953a, (7), p. 32) Permalink: http://dlmf.nist.gov/25.5.E9 Encodings: TeX, pMML, png See also: Annotations for §25.5(i), §25.5 and Ch.25

25.5.10		$$\zeta \left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}{\int }_0^{\mathrm{\infty }}\frac{\cos \left(s\arctan x\right)}{{(1+x^2)}^{s/2}\cosh \left(\frac{1}{2}\pi x\right)}dx.$$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\arctan z$: arctangent function, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Lindelöf (1905, (6), p. 98); with $x=2t$ Referenced by: §25.5(i) Permalink: http://dlmf.nist.gov/25.5.E10 Encodings: TeX, pMML, png See also: Annotations for §25.5(i), §25.5 and Ch.25

25.5.11		$$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(s\arctan x\right)}{{(1+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx.$$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\sin z$: sine function, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Lindelöf (1905, (3), p. 97) Permalink: http://dlmf.nist.gov/25.5.E11 Encodings: TeX, pMML, png See also: Annotations for §25.5(i), §25.5 and Ch.25

25.5.12		$$\zeta \left(s\right)=\frac{2^{s-1}}{s-1}-2^s{\int }_0^{\mathrm{\infty }}\frac{\sin \left(s\arctan x\right)}{{(1+x^2)}^{s/2}({\mathrm{e}}^{\pi x}+1)}dx.$$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\sin z$: sine function, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Lindelöf (1905, (4), p. 98); with $x=2t$ Referenced by: §25.5(i) Permalink: http://dlmf.nist.gov/25.5.E12 Encodings: TeX, pMML, png See also: Annotations for §25.5(i), §25.5 and Ch.25

25.5.13		$$\zeta \left(s\right)=\frac{{\pi }^{s/2}}{s(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s\right)}+\frac{{\pi }^{s/2}}{\mathrm{\Gamma }\left(\frac{1}{2}s\right)}{\int }_1^{\mathrm{\infty }}\left(x^{s/2}+x^{(1-s)/2}\right)\frac{\omega (x)}{x}dx,$$
$s\ne 1$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Titchmarsh (1986b, p. 22) Referenced by: §25.5(ii) Permalink: http://dlmf.nist.gov/25.5.E13 Encodings: TeX, pMML, png See also: Annotations for §25.5(ii), §25.5 and Ch.25

where

25.5.14		$$\omega (x)\equiv \sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2\pi x}=\frac{1}{2}\left({\theta }_3\left(0\right|\mathrm{i}x)-1\right).$$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $x$: real variable Keywords: definition Source: Titchmarsh (1986b, p. 22) Permalink: http://dlmf.nist.gov/25.5.E14 Encodings: TeX, pMML, png See also: Annotations for §25.5(ii), §25.5 and Ch.25

For ${\theta }_3$ see §20.2(i). For similar representations involving other theta functions see Erdélyi et al. (1954a, p. 339).

In (25.5.15)–(25.5.19), , $\psi \left(x\right)$ is the digamma function, and $\gamma$ is Euler’s constant (§5.2). (25.5.16) is also valid for , $s\ne 1$.

25.5.15		$$\zeta \left(s\right)=\frac{1}{s-1}+\frac{\sin \left(\pi s\right)}{\pi }{\int }_0^{\mathrm{\infty }}(\ln \left(1+x\right)-\psi \left(1+x\right))x^{-s}dx,$$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: de Bruijn (1937, (18), p. 175) Referenced by: §25.5(ii) Permalink: http://dlmf.nist.gov/25.5.E15 Encodings: TeX, pMML, png See also: Annotations for §25.5(ii), §25.5 and Ch.25

25.5.16		$\zeta \left(s\right)$	$={\displaystyle \frac{1}{s-1}}+{\displaystyle \frac{\sin \left(\pi s\right)}{\pi (s-1)}}{\displaystyle {\int }_0^{\mathrm{\infty }}}\left({\displaystyle \frac{1}{1+x}}-{\psi }^{\prime }\left(1+x\right)\right)x^{1-s}dx,$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\sin z$: sine function, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: de Bruijn (1937, (23), p. 178) Referenced by: §25.5(ii) Permalink: http://dlmf.nist.gov/25.5.E16 Encodings: TeX, pMML, png See also: Annotations for §25.5(ii), §25.5 and Ch.25
25.5.17		$\zeta \left(1+s\right)$	$={\displaystyle \frac{\sin \left(\pi s\right)}{\pi }}{\displaystyle {\int }_0^{\mathrm{\infty }}}\left(\gamma +\psi \left(1+x\right)\right)x^{-s-1}dx,$
ⓘ Symbols: $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\sin z$: sine function, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: de Bruijn (1937, (22), p. 177) Permalink: http://dlmf.nist.gov/25.5.E17 Encodings: TeX, pMML, png See also: Annotations for §25.5(ii), §25.5 and Ch.25

25.5.18		$\zeta \left(1+s\right)$	$={\displaystyle \frac{\sin \left(\pi s\right)}{\pi s}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\psi }^{\prime }\left(1+x\right)x^{-s}dx,$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\sin z$: sine function, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: de Bruijn (1937, p. 177) Permalink: http://dlmf.nist.gov/25.5.E18 Encodings: TeX, pMML, png See also: Annotations for §25.5(ii), §25.5 and Ch.25
25.5.19		$\zeta \left(m+s\right)$	$={(-1)}^{m-1}{\displaystyle \frac{\mathrm{\Gamma }\left(s\right)\sin \left(\pi s\right)}{\pi \mathrm{\Gamma }\left(m+s\right)}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\psi }^{(m)}\left(1+x\right)x^{-s}dx,$
$m=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\sin z$: sine function, $m$: nonnegative integer, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: de Bruijn (1937, p. 178) Referenced by: §25.5(ii) Permalink: http://dlmf.nist.gov/25.5.E19 Encodings: TeX, pMML, png See also: Annotations for §25.5(ii), §25.5 and Ch.25

25.5.20		$$\zeta \left(s\right)=\frac{\mathrm{\Gamma }\left(1-s\right)}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}\frac{z^{s-1}}{{\mathrm{e}}^{-z}-1}dz,$$
$s\ne 1,2,\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: complex variable and $z$: complex variable Keywords: contour integral, integral representation Source: Apostol (1976, (6), p. 253) A&S Ref: 23.2.4 (in slightly different form) Referenced by: §25.11(vii) Permalink: http://dlmf.nist.gov/25.5.E20 Encodings: TeX, pMML, png See also: Annotations for §25.5(iii), §25.5 and Ch.25

where the integration contour is a loop around the negative real axis; it starts at $-\mathrm{\infty }$, encircles the origin once in the positive direction without enclosing any of the points $z=\pm 2\pi \mathrm{i}$, $\pm 4\pi \mathrm{i}$, …, and returns to $-\mathrm{\infty }$. Equivalently,

25.5.21		$$\zeta \left(s\right)=\frac{\mathrm{\Gamma }\left(1-s\right)}{2\pi \mathrm{i}(1-2^{1-s})}{\int }_{-\mathrm{\infty }}^{(0+)}\frac{z^{s-1}}{{\mathrm{e}}^{-z}+1}dz,$$
$s\ne 1,2,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: complex variable and $z$: complex variable Keywords: contour integral, integral representation Sources: Erdélyi et al. (1953a, (10), p. 32); with $z=-t$; Magnus et al. (1966, p. 21); with $z=-t$ Referenced by: §25.5(iii) Permalink: http://dlmf.nist.gov/25.5.E21 Encodings: TeX, pMML, png See also: Annotations for §25.5(iii), §25.5 and Ch.25

The contour here is any loop that encircles the origin in the positive direction not enclosing any of the points $\pm \pi \mathrm{i}$, $\pm 3\pi \mathrm{i}$, ….