# §25.5 Integral Representations

## §25.5(i) In Terms of Elementary Functions

Throughout this subsection $s\neq 1$.

 25.5.1 $\displaystyle\zeta\left(s\right)$ $\displaystyle=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{e^% {x}-1}\mathrm{d}x,$ $\Re s>1$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\Re$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Erdélyi et al. (1953a, (1.12.4), p. 32) A&S Ref: 23.2.7 Referenced by: §25.12(ii), (25.5.2), (25.5.6) 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 $\displaystyle\zeta\left(s\right)$ $\displaystyle=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{e^{x}x^{s% }}{(e^{x}-1)^{2}}\mathrm{d}x,$ $\Re s>1$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\Re$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Proof sketch: Derivable from (25.5.1) by integration by parts. 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 $\displaystyle\zeta\left(s\right)$ $\displaystyle=\frac{1}{(1-2^{1-s})\Gamma\left(s\right)}\int_{0}^{\infty}\frac{% x^{s-1}}{e^{x}+1}\mathrm{d}x,$ $\Re s>0$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\Re$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Erdélyi et al. (1953a, (1.12.5), p. 32) A&S Ref: 23.2.8 Referenced by: (25.5.4) 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 $\displaystyle\zeta\left(s\right)$ $\displaystyle=\frac{1}{(1-2^{1-s})\Gamma\left(s+1\right)}\int_{0}^{\infty}% \frac{e^{x}x^{s}}{(e^{x}+1)^{2}}\mathrm{d}x,$ $\Re s>0$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\Re$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Proof sketch: Derivable from (25.5.3) by integration by parts. 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 $\zeta\left(s\right)=-s\int_{0}^{\infty}\frac{x-\left\lfloor x\right\rfloor-% \frac{1}{2}}{x^{s+1}}\mathrm{d}x,$ $-1<\Re s<0$.
 25.5.6 $\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\Gamma\left(s\right)}% \int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x% ^{s-1}}{e^{x}}\mathrm{d}x,$ $\Re s>-1$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\Re$: real part, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Proof sketch: 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: (25.5.7) 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 $\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{B_{2m}}{(2m)% !}{\left(s\right)_{2m-1}}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left% (\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}x% ^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\mathrm{d}x,$ $\Re s>-(2n+1)$, $n=1,2,3,\dots$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $O\left(\NVar{x}\right)$: order not exceeding, $\Gamma\left(\NVar{z}\right)$: gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\Re$: real part, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Proof sketch: 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: 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 $\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}\frac{\Gamma\left(s+2m-1\right)}{\Gamma\left% (s\right)}$ 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 $\displaystyle\zeta\left(s\right)$ $\displaystyle=\frac{1}{2(1-2^{-s})\Gamma\left(s\right)}\int_{0}^{\infty}\frac{% x^{s-1}}{\sinh x}\mathrm{d}x,$ $\Re s>1$. 25.5.9 $\displaystyle\zeta\left(s\right)$ $\displaystyle=\frac{2^{s-1}}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{x^{% s}}{(\sinh x)^{2}}\mathrm{d}x,$ $\Re s>1$.
 25.5.10 $\zeta\left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}\int_{0}^{\infty}\frac{\cos\left(% s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}\cosh\left(\frac{1}{2}\pi x% \right)}\mathrm{d}x.$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential, $\cosh\NVar{z}$: hyperbolic cosine function, $\int$: integral, $\operatorname{arctan}\NVar{z}$: arctangent function, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Sources: Lindelöf (1905, (6), p. 98); with $x=2t$; Erdélyi et al. (1953a, (1.12.15), p. 33) 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 $\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+2\int_{0}^{\infty}\frac{\sin% \left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^{2\pi x}-1)}\mathrm{d}x.$
 25.5.12 $\zeta\left(s\right)=\frac{2^{s-1}}{s-1}-2^{s}\int_{0}^{\infty}\frac{\sin\left(% s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^{\pi x}+1)}\mathrm{d}x.$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\operatorname{arctan}\NVar{z}$: arctangent function, $\sin\NVar{z}$: sine function, $x$: real variable and $s$: complex variable Keywords: improper integral, integral representation Source: Lindelöf (1905, (4), p. 98); with $x=2t$ 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(ii) In Terms of Other Functions

 25.5.13 $\zeta\left(s\right)=\frac{\pi^{s/2}}{s(s-1)\Gamma\left(\frac{1}{2}s\right)}+% \frac{\pi^{s/2}}{\Gamma\left(\frac{1}{2}s\right)}\*\int_{1}^{\infty}\left(x^{s% /2}+x^{(1-s)/2}\right)\frac{\omega(x)}{x}\mathrm{d}x,$ $s\neq 1$,

where

 25.5.14 $\omega(x)\equiv\sum_{n=1}^{\infty}e^{-n^{2}\pi x}=\frac{1}{2}\left(\theta_{3}% \left(0\middle|ix\right)-1\right).$

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), $0<\Re s<1$, $\psi\left(x\right)$ is the digamma function, and $\gamma$ is Euler’s constant (§5.2). (25.5.16) is also valid for $0<\Re s<2$, $s\neq 1$.

 25.5.15 $\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\left(\pi s\right)}{\pi}\*\int_{0}% ^{\infty}(\ln\left(1+x\right)-\psi\left(1+x\right))x^{-s}\mathrm{d}x,$
 25.5.16 $\displaystyle\zeta\left(s\right)$ $\displaystyle=\frac{1}{s-1}+\frac{\sin\left(\pi s\right)}{\pi(s-1)}\*\int_{0}^% {\infty}\left(\frac{1}{1+x}-\psi'\left(1+x\right)\right)x^{1-s}\mathrm{d}x,$ 25.5.17 $\displaystyle\zeta\left(1+s\right)$ $\displaystyle=\frac{\sin\left(\pi s\right)}{\pi}\int_{0}^{\infty}\left(\gamma+% \psi\left(1+x\right)\right)x^{-s-1}\mathrm{d}x,$
 25.5.18 $\displaystyle\zeta\left(1+s\right)$ $\displaystyle=\frac{\sin\left(\pi s\right)}{\pi s}\int_{0}^{\infty}\psi'\left(% 1+x\right)x^{-s}\mathrm{d}x,$ 25.5.19 $\displaystyle\zeta\left(m+s\right)$ $\displaystyle=(-1)^{m-1}\frac{\Gamma\left(s\right)\sin\left(\pi s\right)}{\pi% \Gamma\left(m+s\right)}\*\int_{0}^{\infty}{\psi}^{(m)}\left(1+x\right)x^{-s}% \mathrm{d}x,$ $m=1,2,3,\dots$.

## §25.5(iii) Contour Integrals

 25.5.20 $\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{2\pi i}\int_{-\infty}^{(0+)}% \frac{z^{s-1}}{e^{-z}-1}\mathrm{d}z,$ $s\neq 1,2,\dots$, ⓘ 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, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable Keywords: contour integral, integral representation Sources: Apostol (1976, (6), p. 253); with $a=1$; Erdélyi et al. (1953a, (1.12.9), p. 32); with $z=-t$ 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 $-\infty$, 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 $-\infty$. Equivalently,

 25.5.21 $\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{2\pi i(1-2^{1-s})}\*\int_{-% \infty}^{(0+)}\frac{z^{s-1}}{e^{-z}+1}\mathrm{d}z,$ $s\neq 1,2,\dots$. ⓘ 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, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: 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, (1.12.10), p. 32); with $z=-t$; Magnus et al. (1966, p. 21); with $z=-t$ 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 $\pm\pi\mathrm{i}$, $\pm 3\pi\mathrm{i}$, ….