§25.11 Hurwitz Zeta Function

The function $\zeta (s,a)$ was introduced in Hurwitz (1882) and defined by the series expansion

25.11.1		$$\zeta (s,a)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{(n+a)}^s},$$
$\mathrm{\Re }s>1$, $a\ne 0,-1,-2,\mathrm{\dots }$.
ⓘ Defines: $\zeta (s,a)$: Hurwitz zeta function Symbols: $\mathrm{\Re }$: real part, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Keywords: definition Source: Apostol (1976, p. 251) Referenced by: (25.11.11), (25.11.15), (25.11.17), (25.11.2), (25.11.3), (25.11.4), §25.11(i), (25.14.2), (8.15.2) Permalink: http://dlmf.nist.gov/25.11.E1 Encodings: TeX, pMML, png See also: Annotations for §25.11(i), §25.11 and Ch.25

$\zeta (s,a)$ has a meromorphic continuation in the $s$-plane, its only singularity in $ℂ$ being a simple pole at $s=1$ with residue $1$. As a function of $a$, with $s$ ($\ne 1$) fixed, $\zeta (s,a)$ is analytic in the half-plane $\mathrm{\Re }a>0$. The Riemann zeta function is a special case:

25.11.2		$$\zeta (s,1)=\zeta \left(s\right).$$
ⓘ Symbols: $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function and $s$: complex variable Keywords: specialization Source: Apostol (1976, p. 255) Proof sketch: Derivable from (25.11.1) and (25.2.1). Referenced by: (25.11.10), (25.11.24), (25.12.12) Permalink: http://dlmf.nist.gov/25.11.E2 Encodings: TeX, pMML, png See also: Annotations for §25.11(i), §25.11 and Ch.25

For most purposes it suffices to restrict because of the following straightforward consequences of (25.11.1):

25.11.3		$$\zeta (s,a)=\zeta (s,a+1)+a^{-s},$$
ⓘ Symbols: $\zeta (s,a)$: Hurwitz zeta function, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Keywords: recurrence Proof sketch: Derivable from (25.11.1) by comparing its $n=0$ term with its sum over $n\ge 1$. Permalink: http://dlmf.nist.gov/25.11.E3 Encodings: TeX, pMML, png See also: Annotations for §25.11(i), §25.11 and Ch.25

25.11.4		$$\zeta (s,a)=\zeta (s,a+m)+\sum _{n=0}^{m-1}\frac{1}{{(n+a)}^s},$$
$m=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\zeta (s,a)$: Hurwitz zeta function, $m$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Keywords: recurrence Proof sketch: Derivable from (25.11.1) by comparing its sum over $n=0,\mathrm{\dots },m-1$, $m\ge 1$, with its sum over $n\ge m$. Permalink: http://dlmf.nist.gov/25.11.E4 Encodings: TeX, pMML, png See also: Annotations for §25.11(i), §25.11 and Ch.25

Most references treat real $a$ with .

25.11.5		$$\zeta (s,a)=\sum _{n=0}^N\frac{1}{{(n+a)}^s}+\frac{{(N+a)}^{1-s}}{s-1}-s{\int }_N^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor }{{(x+a)}^{s+1}}dx,$$
$s\ne 1$, $\mathrm{\Re }s>0$, $a>0$, $N=0,1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$, $\lfloor x\rfloor$: floor of $x$, $\int$: integral, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: Euler–Maclaurin formula, improper integral Source: Apostol (1976, (25), p. 269) Referenced by: (25.11.7) Permalink: http://dlmf.nist.gov/25.11.E5 Encodings: TeX, pMML, png See also: Annotations for §25.11(iii), §25.11 and Ch.25

25.11.6		$$\zeta (s,a)=\frac{1}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$
$s\ne 1$, $\mathrm{\Re }s>-1$, $a>0$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$, $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: Euler–Maclaurin formula, improper integral Source: Apostol (1985a, (25), p. 231) Referenced by: Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20) Permalink: http://dlmf.nist.gov/25.11.E6 Encodings: TeX, pMML, png Errata (effective with 1.0.12): Originally the integrand was incorrect because its numerator contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$. Reported 2016-05-08 by Clemens Heuberger See also: Annotations for §25.11(iii), §25.11 and Ch.25

25.11.7		$$\zeta (s,a)=\begin{array}{l}\frac{1}{a^s}+\frac{1}{{(1+a)}^s}\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}\frac{1}{{(1+a)}^{s+2k-1}}\\ -\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{{(x+a)}^{s+2n+1}}dx,\end{array}$$
$s\ne 1$, $a>0$, $n=1,2,3,\mathrm{\dots }$, $\mathrm{\Re }s>-2n$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$, $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: Euler–Maclaurin formula, improper integral Proof sketch: Derivable from (25.11.5) by setting $N=1$ and repeatedly integrating by parts using (1.2.6), (24.2.2), (24.2.4), (24.2.11), (24.2.12), (24.4.34). Permalink: http://dlmf.nist.gov/25.11.E7 Encodings: TeX, pMML, png See also: Annotations for §25.11(iii), §25.11 and Ch.25

For ${\tilde{B}}_n\left(x\right)$ see §24.2(iii).

25.11.8		$$\zeta (s,\frac{1}{2}a)=\zeta (s,\frac{1}{2}a+\frac{1}{2})+2^s\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{{(n+a)}^s},$$
$\mathrm{\Re }s>0$, $s\ne 1$, .
ⓘ Symbols: $\zeta (s,a)$: Hurwitz zeta function, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Keywords: infinite series, series representation Source: Srivastava and Choi (2001, (5), p. 89) Referenced by: (25.11.35), §25.11(x) Permalink: http://dlmf.nist.gov/25.11.E8 Encodings: TeX, pMML, png See also: Annotations for §25.11(iv), §25.11 and Ch.25

25.11.9		$$\zeta (1-s,a)=\frac{2\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}\cos \left(\frac{1}{2}\pi s-2n\pi a\right),$$
$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $a=1$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Keywords: infinite series, series representation Source: Apostol (1976, (9), (10), p. 257) Referenced by: (25.13.3), Erratum (V1.1.4) for Equation (25.11.9) Permalink: http://dlmf.nist.gov/25.11.E9 Encodings: TeX, pMML, png Clarification (effective with 1.1.4): The constraint which originally read “$\mathrm{\Re }s>1$, ” has been extended to be “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $a=1$”. Suggested 2021-08-23 by Gergő Nemes See also: Annotations for §25.11(iv), §25.11 and Ch.25

25.11.10		$$\zeta (s,a)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(s\right)}_n}{n!}\zeta \left(n+s\right){(1-a)}^n,$$
$s\ne 1$, .
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $n$: nonnegative integer, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable Keywords: infinite series, series representation Proof sketch: Derivable using Taylor’s theorem (1.10.1) with $z=a$, $z_0=1$, (25.11.17), (5.2.4), (25.11.2). Referenced by: §25.11(iv), §25.11(iv), Erratum (V1.0.9) for Chapters 7, 25 Permalink: http://dlmf.nist.gov/25.11.E10 Encodings: TeX, pMML, png Notational Change (effective with 1.0.9): We have rewritten the original summation ${\sum }_{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(n+s\right)}{n!\mathrm{\Gamma }\left(s\right)}\zeta \left(n+s\right){(1-a)}^n$ more concisely as ${\sum }_{n=0}^{\mathrm{\infty }}\frac{{\left(s\right)}_n}{n!}\zeta \left(n+s\right){(1-a)}^n$ using the Pochhammer symbol. See also: Annotations for §25.11(iv), §25.11 and Ch.25

When $a=\frac{1}{2}$, (25.11.10) reduces to (25.8.3); compare (25.11.11).

For other series expansions similar to (25.11.10) see Coffey (2008).

Throughout this subsection $\mathrm{\Re }a>0$.

25.11.11		$$\zeta (s,\frac{1}{2})=(2^s-1)\zeta \left(s\right),$$
$s\ne 1$.
ⓘ Symbols: $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function and $s$: complex variable Keywords: special value Proof sketch: Derivable using (25.11.1), (25.2.2). Referenced by: §25.11(iv) Permalink: http://dlmf.nist.gov/25.11.E11 Encodings: TeX, pMML, png See also: Annotations for §25.11(v), §25.11 and Ch.25

25.11.12		$$\zeta (n+1,a)=\frac{{(-1)}^{n+1}{\psi }^{(n)}\left(a\right)}{n!},$$
$n=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\zeta (s,a)$: Hurwitz zeta function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $n$: nonnegative integer and $a$: real or complex parameter Keywords: special value Source: Erdélyi et al. (1953a, (1.11.8), p. 29) Permalink: http://dlmf.nist.gov/25.11.E12 Encodings: TeX, pMML, png See also: Annotations for §25.11(v), §25.11 and Ch.25

25.11.13		$$\zeta (0,a)=\frac{1}{2}-a.$$
ⓘ Symbols: $\zeta (s,a)$: Hurwitz zeta function and $a$: real or complex parameter Keywords: special value Source: Apostol (1976, p. 268) Permalink: http://dlmf.nist.gov/25.11.E13 Encodings: TeX, pMML, png See also: Annotations for §25.11(v), §25.11 and Ch.25

25.11.14		$$\zeta (-n,a)=-\frac{B_{n+1}\left(a\right)}{n+1},$$
$n=0,1,2,\mathrm{\dots }$.
ⓘ Symbols: $B_n\left(x\right)$: Bernoulli polynomials, $\zeta (s,a)$: Hurwitz zeta function, $n$: nonnegative integer and $a$: real or complex parameter Keywords: special value Source: Apostol (1976, (17), p. 264) Permalink: http://dlmf.nist.gov/25.11.E14 Encodings: TeX, pMML, png See also: Annotations for §25.11(v), §25.11 and Ch.25

25.11.15		$$\zeta (s,ka)=k^{-s}\sum _{n=0}^{k-1}\zeta (s,a+\frac{n}{k}),$$
$s\ne 1$, $k=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Keywords: special value Proof sketch: Derivable by starting with (25.11.1), replacing $a\mapsto ka$, pulling out a factor of $k^{-s}$ and performing a change of sum index $n=km+l$ with $m\ge 0$ and $0\le l\le k-1$, which through Euclidean division converts the single sum into a double sum of the correct form. Referenced by: (25.11.24) Permalink: http://dlmf.nist.gov/25.11.E15 Encodings: TeX, pMML, png See also: Annotations for §25.11(v), §25.11 and Ch.25

25.11.16		$$\zeta (1-s,\frac{h}{k})=\frac{2\mathrm{\Gamma }\left(s\right)}{{(2\pi k)}^s}\sum _{r=1}^k\cos \left(\frac{\pi s}{2}-\frac{2\pi rh}{k}\right)\zeta (s,\frac{r}{k}),$$
$s\ne 0,1$; $h,k$ integers, $1\le h\le k$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $k$: nonnegative integer and $s$: complex variable Keywords: special value Source: Apostol (1976, (14), p. 261) Referenced by: (25.11.21) Permalink: http://dlmf.nist.gov/25.11.E16 Encodings: TeX, pMML, png See also: Annotations for §25.11(v), §25.11 and Ch.25

25.11.25		$\zeta (s,a)$	$={\displaystyle \frac{1}{\mathrm{\Gamma }\left(s\right)}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{x^{s-1}{\mathrm{e}}^{-ax}}{1-{\mathrm{e}}^{-x}}}dx,$
$\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: improper integral, integral representation Sources: Srivastava and Choi (2001, (2), p. 89); Apostol (1976, (5), p. 251) Referenced by: (25.11.27), (25.11.30), (25.11.35), §25.11(x) Permalink: http://dlmf.nist.gov/25.11.E25 Encodings: TeX, pMML, png See also: Annotations for §25.11(vii), §25.11 and Ch.25
25.11.26		$\zeta (s,a)$	$=-s{\displaystyle {\int }_{-a}^{\mathrm{\infty }}}{\displaystyle \frac{x-\lfloor x\rfloor -\frac{1}{2}}{{(x+a)}^{s+1}}}dx,$
, .
ⓘ Symbols: $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$, $\lfloor x\rfloor$: floor of $x$, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: improper integral, integral representation Source: Berndt (1972, (5.3), p. 156) Permalink: http://dlmf.nist.gov/25.11.E26 Encodings: TeX, pMML, png See also: Annotations for §25.11(vii), §25.11 and Ch.25

25.11.27		$$\zeta (s,a)=\frac{1}{2}{a}^{-s}+\frac{a^{1-s}}{s-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}\right)x^{s-1}{\mathrm{e}}^{-ax}dx,$$
$\mathrm{\Re }s>-1$, $s\ne 1$, $\mathrm{\Re }a>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: improper integral, integral representation Proof sketch: Derivable from (25.11.25) by first rewriting $(1-{\mathrm{e}}^{-x})={\mathrm{e}}^{-x}({\mathrm{e}}^x-1)$, then 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: (25.11.28) Permalink: http://dlmf.nist.gov/25.11.E27 Encodings: TeX, pMML, png Update (effective with 1.1.9): The fraction in the integrand $\frac{x^{s-1}}{{\mathrm{e}}^{ax}}$ was rewritten to read $x^{s-1}{\mathrm{e}}^{-ax}$. See also: Annotations for §25.11(vii), §25.11 and Ch.25

25.11.28		$$\zeta (s,a)=\begin{array}{l}\frac{1}{2}{a}^{-s}+\frac{a^{1-s}}{s-1}+\sum _{k=1}^n\frac{B_{2k}}{(2k)!}{\left(s\right)}_{2k-1}{a}^{1-s-2k}\\ +\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}-\sum _{k=1}^n\frac{B_{2k}}{(2k)!}{x}^{2k-1}\right)x^{s-1}{\mathrm{e}}^{-ax}dx,\end{array}$$
$\mathrm{\Re }s>-(2n+1)$, $s\ne 1$, $\mathrm{\Re }a>0$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: improper integral, integral representation Proof sketch: Derivable from (25.11.27) by adding and subtracting ${\sum }_{k=1}^n{B}_{2k}{x}^{2k-1}/(2k)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that ${({\mathrm{e}}^x-1)}^{-1}-x^{-1}+1/2-{\sum }_{k=1}^n{B}_{2k}{x}^{2k-1}/(2k)!=O\left(x^{2n+1}\right)$ as $x\to 0$, demonstrating the region of convergence. Referenced by: Erratum (V1.0.9) for Chapters 7, 25 Permalink: http://dlmf.nist.gov/25.11.E28 Encodings: TeX, pMML, png Notational Change (effective with 1.0.9): We have rewritten the original summation ${\sum }_{k=1}^n\frac{\mathrm{\Gamma }\left(s+2k-1\right)}{\mathrm{\Gamma }\left(s\right)}\frac{B_{2k}}{(2k)!}{a}^{-2k-s+1}$ more concisely as ${\sum }_{k=1}^n\frac{B_{2k}}{(2k)!}{\left(s\right)}_{2k-1}{a}^{1-s-2k}$ using the Pochhammer symbol. See also: Annotations for §25.11(vii), §25.11 and Ch.25

25.11.29		$$\zeta (s,a)=\frac{1}{2}{a}^{-s}+\frac{a^{1-s}}{s-1}+2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(s\arctan \left(x/a\right)\right)}{{(a^2+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx,$$
$s\ne 1$, $\mathrm{\Re }a>0$.
ⓘ Symbols: $\zeta (s,a)$: Hurwitz 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, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: improper integral, integral representation Source: Erdélyi et al. (1953a, (1.10.7), p. 26) Permalink: http://dlmf.nist.gov/25.11.E29 Encodings: TeX, pMML, png See also: Annotations for §25.11(vii), §25.11 and Ch.25

25.11.30		$$\zeta (s,a)=\frac{\mathrm{\Gamma }\left(1-s\right)}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}\frac{{\mathrm{e}}^{az}{z}^{s-1}}{1-{\mathrm{e}}^z}dz,$$
$s\ne 1$, $\mathrm{\Re }a>0$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz 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, $\mathrm{\Re }$: real part, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable Keywords: contour integral, integral representation Source: Erdélyi et al. (1953a, (1.10.5), p. 25); with $z=-t$ Proof sketch: This contour integral uses Hankel’s contour Figure 5.9.1. Assume $\mathrm{\Re }s>1$, collapse the integration path onto the negative real axis, apply (25.11.25), followed by analytic continuation for all $s\ne 1$, $\mathrm{\Re }a>0$, since the integrand is analytic in $z$, the convergence at the ends of the path is exponential for all $s\ne -1$ and the Hankel contour can be kept well clear of the singularity at the origin in the $z$-plane. Referenced by: §25.11(i), §25.11(vii), Erratum (V1.2.1) for Figure 5.9.1 Permalink: http://dlmf.nist.gov/25.11.E30 Encodings: TeX, pMML, png See also: Annotations for §25.11(vii), §25.11 and Ch.25

where the integration contour (see Figure 5.9.1) is a loop around the negative real axis as described for (25.5.20).

25.11.31		$$\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{2\cosh x}dx=4^{-s}\left(\zeta (s,\frac{1}{4}+\frac{1}{4}a)-\zeta (s,\frac{3}{4}+\frac{1}{4}a)\right),$$
$\mathrm{\Re }s>0$, $\mathrm{\Re }a>-1$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: improper integral, integral representation Proof sketch: The improper integral can be evaluated by using $2\cosh x={\mathrm{e}}^x(1+{\mathrm{e}}^{-2x})$, changing variables $2x=t$, and then applying (25.11.35). Permalink: http://dlmf.nist.gov/25.11.E31 Encodings: TeX, pMML, png See also: Annotations for §25.11(viii), §25.11 and Ch.25

25.11.32		$${\int }_0^a{x}^n\psi \left(x\right)dx=\begin{array}{l}{(-1)}^{n-1}{\zeta }^{\prime }\left(-n\right)+{(-1)}^n{H}_n\frac{B_{n+1}}{n+1}-\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)H_k\frac{B_{k+1}(a)}{k+1}{a}^{n-k}\\ +\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right){\zeta }^{\prime }(-k,a)a^{n-k},\end{array}$$
$n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>0$,
ⓘ Symbols: $B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$, $\psi \left(z\right)$: psi (or digamma) function, $H_n$: harmonic number, $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $h(n)$: harmonic number Keywords: definite integral, integral representation Source: Adamchik (1998, (17), p. 197) Referenced by: Erratum (V1.1.4) for Notation Permalink: http://dlmf.nist.gov/25.11.E32 Encodings: TeX, pMML, png Notation (effective with 1.1.4): The notation previously used for the harmonic number $h(n)$ (resp. $h(k)$) has been replaced to be $H_n$ (resp. $H_k$). Suggested 2021-08-23 by Gergő Nemes See also: Annotations for §25.11(viii), §25.11 and Ch.25

where $H_n$ are the harmonic numbers:

25.11.33		$$H_n=\sum _{k=1}^n{k}^{-1}.$$
ⓘ Defines: $H_n$: harmonic number Symbols: $k$: nonnegative integer, $n$: nonnegative integer and $h(n)$: harmonic number Keywords: harmonic number Source: Knuth (1968, Section 1.2.7, p. 73) Referenced by: §25.11(viii), §25.16(ii), Erratum (V1.1.4) for Notation Permalink: http://dlmf.nist.gov/25.11.E33 Encodings: TeX, pMML, png Notation (effective with 1.1.4): The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_n$. Suggested 2021-08-23 by Gergő Nemes See also: Annotations for §25.11(viii), §25.11 and Ch.25

25.11.34		$$n{\int }_0^a{\zeta }^{\prime }(1-n,x)dx={\zeta }^{\prime }(-n,a)-{\zeta }^{\prime }\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a\right)}{n(n+1)},$$
$n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>0$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$, $\int$: integral, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $a$: real or complex parameter Keywords: definite integral, integral representation Source: Adamchik (1998, (15), p. 196) Permalink: http://dlmf.nist.gov/25.11.E34 Encodings: TeX, pMML, png See also: Annotations for §25.11(viii), §25.11 and Ch.25