§25.11 Hurwitz Zeta Function

Keywords:: Hurwitz zeta function
Referenced by:: §25.14(i)
Permalink:: http://dlmf.nist.gov/25.11

§25.11(i) Definition
§25.11(ii) Graphics
§25.11(iii) Representations by the Euler–Maclaurin Formula
§25.11(iv) Series Representations
§25.11(v) Special Values
§25.11(vi) Derivatives
§25.11(vii) Integral Representations
§25.11(viii) Further Integral Representations
§25.11(ix) Integrals
§25.11(x) Further Series Representations
§25.11(xi) Sums
§25.11(xii) $a$ -Asymptotic Behavior

§25.11(i) Definition

Notes:: See Apostol (1976, pp. 251–255). Analytic properties of $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ with respect to $a$ follow from (25.11.30).
Keywords:: Hurwitz zeta function, Riemann zeta function
Permalink:: http://dlmf.nist.gov/25.11.i

The function $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ was introduced in Hurwitz (1882) and defined by the series expansion

25.11.1 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\sum _{{n=0}}^{\infty}\frac{1}{(n+a)^{s}},$ $\realpart{s}>1$ , $a\neq 0,-1,-2,\dots$ .

Defines:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function
Symbols:: $\realpart{}$ : real part, $n$ : nonnegative integer, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(i), §25.11(v), §25.11(vi)
Permalink:: http://dlmf.nist.gov/25.11.E1
Encodings:: TeX, pMML, png

$\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ has a meromorphic continuation in the $s$ -plane, its only singularity in $\Complex$ being a simple pole at $s=1$ with residue 1. As a function of $a$ , with $s$ ( $\neq 1$ ) fixed, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ is analytic in the half-plane $\realpart{a}>0$ . The Riemann zeta function is a special case:

25.11.2 $\mathop{\zeta\/}\nolimits\!\left(s,1\right)=\mathop{\zeta\/}\nolimits\!\left(s\right).$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.11.E2
Encodings:: TeX, pMML, png

For most purposes it suffices to restrict $0<\realpart{a}\leq 1$ because of the following straightforward consequences of (25.11.1):

25.11.3 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\mathop{\zeta\/}\nolimits\!\left(s,a+1\right)+a^{{-s}},$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $a$ : real or complex parameter and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.11.E3
Encodings:: TeX, pMML, png

25.11.4 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\mathop{\zeta\/}\nolimits\!\left(s,a+m\right)+\sum _{{n=0}}^{{m-1}}\frac{1}{(n+a)^{s}},$ $m=1,2,3,\dots$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : real or complex parameter and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.11.E4
Encodings:: TeX, pMML, png

Most references treat real $a$ with $0<a\leq 1$ .

§25.11(ii) Graphics

Notes:: These graphics were constructed at NIST.
Keywords:: Hurwitz zeta function
Permalink:: http://dlmf.nist.gov/25.11.ii

Figure 25.11.1: Hurwitz zeta function $\mathop{\zeta\/}\nolimits\!\left(x,a\right)$ , $a$ = 0.3, 0.5, 0.8, 1, $-20\leq x\leq 10$ . The curves are almost indistinguishable for $-14<x<-1$ , approximately.

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $x$ : real variable and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/25.11.F1
Encodings:: pdf, png

Visualization Help

Figure 25.11.2: Hurwitz zeta function $\mathop{\zeta\/}\nolimits\!\left(x,a\right)$ , $-19.5\leq x\leq 10$ , $0.02\leq a\leq 1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $x$ : real variable and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/25.11.F2
Encodings:: VRML, X3D, pdf, png

§25.11(iii) Representations by the Euler–Maclaurin Formula

Notes:: See Apostol (1976, p. 269) and Apostol (1985a). For (25.11.7) take $N=1$ in (25.11.5) and integrate by parts.
Keywords:: Hurwitz zeta function
Permalink:: http://dlmf.nist.gov/25.11.iii

25.11.5 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\sum _{{n=0}}^{N}\frac{1}{(n+a)^{s}}+\frac{(N+a)^{{1-s}}}{s-1}-s\int _{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{(x+a)^{{s+1}}}dx,$ $s\neq 1$ , $\realpart{s}>0$ , $a>0$ , $N=0,1,2,3,\dots$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $dx$ : differential of $x$ , $\left\lfloor x\right\rfloor$ : floor of $x$ , $\int$ : integral, $\realpart{}$ : real part, $n$ : nonnegative integer, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(iii)
Permalink:: http://dlmf.nist.gov/25.11.E5
Encodings:: TeX, pMML, png

25.11.6 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{1}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-s(s+1)\int _{0}^{\infty}\frac{\mathop{\widetilde{B}_{{2}}\/}\nolimits\!\left(x\right)}{(x+a)^{{s+2}}}dx,$ $s\neq 1$ , $\realpart{s}>-1$ , $a>0$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, $\realpart{}$ : real part, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.11.E6
Encodings:: TeX, pMML, png

25.11.7 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{1}{a^{s}}+\frac{1}{(1+a)^{s}}\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum _{{k=1}}^{n}\binom{s+2k-2}{2k-1}\frac{\mathop{B_{{2k}}\/}\nolimits}{2k}\frac{1}{(1+a)^{{s+2k-1}}}-\binom{s+2n}{2n+1}\int _{1}^{\infty}\frac{\mathop{\widetilde{B}_{{2n+1}}\/}\nolimits\!\left(x\right)}{(x+a)^{{s+2n+1}}}dx,$ $s\neq 1$ , $a>0$ , $n=1,2,3,\dots$ , $\realpart{s}>-2n$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\binom{m}{n}$ : binomial coefficient, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, $\realpart{}$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(iii)
Permalink:: http://dlmf.nist.gov/25.11.E7
Encodings:: TeX, pMML, png

For $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ see §24.2(iii).

§25.11(iv) Series Representations

Notes:: See Srivastava and Choi (2001, p. 89) and Apostol (1976, p. 257). For (25.11.10) use Taylor’s theorem (§§1.4(vi), 1.10(i)) and (25.11.17).
Keywords:: Hurwitz zeta function
Permalink:: http://dlmf.nist.gov/25.11.iv

25.11.8 $\mathop{\zeta\/}\nolimits\!\left(s,\tfrac{1}{2}a\right)=\mathop{\zeta\/}\nolimits\!\left(s,\tfrac{1}{2}a+\tfrac{1}{2}\right)+2^{s}\sum _{{n=0}}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}},$ $\realpart{s}>0$ , $s\neq 1$ , $0<a\leq 1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\realpart{}$ : real part, $n$ : nonnegative integer, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(x)
Permalink:: http://dlmf.nist.gov/25.11.E8
Encodings:: TeX, pMML, png

25.11.9 $\mathop{\zeta\/}\nolimits\!\left(1-s,a\right)=\frac{2\mathop{\Gamma\/}\nolimits\!\left(s\right)}{(2\pi)^{s}}\*\sum _{{n=1}}^{\infty}\frac{1}{n^{s}}\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\pi s-2n\pi a\right),$ $\realpart{s}>1$ , $0<a\leq 1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\realpart{}$ : real part, $n$ : nonnegative integer, $a$ : real or complex parameter and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.11.E9
Encodings:: TeX, pMML, png

25.11.10 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\sum _{{n=0}}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(n+s\right)}{n!\mathop{\Gamma\/}\nolimits\!\left(s\right)}\mathop{\zeta\/}\nolimits\!\left(n+s\right)(1-a)^{n},$ $s\neq 1$ , $|a-1|<1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(iv), §25.11(iv)
Permalink:: http://dlmf.nist.gov/25.11.E10
Encodings:: TeX, pMML, png

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

§25.11(v) Special Values

Notes:: See Apostol (1976, pp. 268 and 264) and Erdélyi et al. (1953a, p. 45). For (25.11.11) apply (25.2.2) and (25.11.1). For (25.11.15) use (25.11.1) and analytic continuation. For (25.11.16) see Apostol (1976, p. 263).
Keywords:: Hurwitz zeta function
Permalink:: http://dlmf.nist.gov/25.11.v

Throughout this subsection $\realpart{a}>0$ .

25.11.11 $\mathop{\zeta\/}\nolimits\!\left(s,\tfrac{1}{2}\right)=(2^{s}-1)\mathop{\zeta\/}\nolimits\!\left(s\right),$ $s\neq 1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function and $s$ : complex variable
Referenced by:: §25.11(iv), §25.11(v)
Permalink:: http://dlmf.nist.gov/25.11.E11
Encodings:: TeX, pMML, png

25.11.12 $\mathop{\zeta\/}\nolimits\!\left(n+1,a\right)=\frac{(-1)^{{n+1}}{\mathop{\psi\/}\nolimits^{{(n)}}}\!\left(a\right)}{n!},$ $n=1,2,3,\dots$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $n$ : nonnegative integer and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/25.11.E12
Encodings:: TeX, pMML, png

25.11.13 $\mathop{\zeta\/}\nolimits\!\left(0,a\right)=\tfrac{1}{2}-a.$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/25.11.E13
Encodings:: TeX, pMML, png

25.11.14 $\mathop{\zeta\/}\nolimits\!\left(-n,a\right)=-\frac{\mathop{B_{{n+1}}\/}\nolimits\!\left(a\right)}{n+1},$ $n=0,1,2,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $n$ : nonnegative integer and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/25.11.E14
Encodings:: TeX, pMML, png

25.11.15 $\mathop{\zeta\/}\nolimits\!\left(s,ka\right)=k^{{-s}}\*\sum _{{n=0}}^{{k-1}}\mathop{\zeta\/}\nolimits\!\left(s,a+\frac{n}{k}\right),$ $s\neq 1$ , $k=1,2,3,\dots$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $k$ : nonnegative integer, $n$ : nonnegative integer, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(v), §25.11(vi)
Permalink:: http://dlmf.nist.gov/25.11.E15
Encodings:: TeX, pMML, png

25.11.16 $\mathop{\zeta\/}\nolimits\!\left(1-s,\frac{h}{k}\right)=\frac{2\mathop{\Gamma\/}\nolimits\!\left(s\right)}{(2\pi k)^{s}}\*\sum _{{r=1}}^{k}\mathop{\cos\/}\nolimits\!\left(\frac{\pi s}{2}-\frac{2\pi rh}{k}\right)\mathop{\zeta\/}\nolimits\!\left(s,\frac{r}{k}\right),$ $s\neq 0,1$ ; $h,k$ integers, $1\leq h\leq k$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $k$ : nonnegative integer and $s$ : complex variable
Referenced by:: §25.11(v)
Permalink:: http://dlmf.nist.gov/25.11.E16
Encodings:: TeX, pMML, png

§25.11(vi) Derivatives

Notes:: See Apostol (1985a, p. 231) and Miller and Adamchik (1998). For (25.11.17) differentiate (25.11.1). For (25.11.18) see Erdélyi et al. (1953a, p. 26). For (25.11.24) use (25.11.15) with $a=1/k$ , multiply by $k^{s}$ and differentiate.
Keywords:: Hurwitz zeta function, derivatives
Permalink:: http://dlmf.nist.gov/25.11.vi

¶ $a$ -Derivative

25.11.17 $\frac{\partial}{\partial a}\mathop{\zeta\/}\nolimits\!\left(s,a\right)=-s\mathop{\zeta\/}\nolimits\!\left(s+1,a\right),$ $s\neq 0,1$ ; $\realpart{a}>0$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $\realpart{}$ : real part, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(iv), §25.11(vi)
Permalink:: http://dlmf.nist.gov/25.11.E17
Encodings:: TeX, pMML, png

¶ $s$ -Derivatives

In (25.11.18)–(25.11.24) primes on $\mathop{\zeta\/}\nolimits$ denote derivatives with respect to $s$ . Similarly in §§25.11(viii) and 25.11(xii).

25.11.18 ${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(0,a\right)=\mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(a\right)-\tfrac{1}{2}\mathop{\ln\/}\nolimits\!\left(2\pi\right),$ $a>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $a$ : real or complex parameter
Referenced by:: ¶ ‣ §25.11(vi), §25.11(vi)
Permalink:: http://dlmf.nist.gov/25.11.E18
Encodings:: TeX, pMML, png

25.11.19 ${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(s,a\right)=-\frac{\mathop{\ln\/}\nolimits a}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{{1-s}}}{(s-1)^{2}}+s(s+1)\int _{0}^{\infty}\frac{\mathop{\widetilde{B}_{{2}}\/}\nolimits\!\left(x\right)\mathop{\ln\/}\nolimits\!\left(x+a\right)}{(x+a)^{{s+2}}}dx-(2s+1)\int _{0}^{\infty}\frac{\mathop{\widetilde{B}_{{2}}\/}\nolimits\!\left(x\right)}{(x+a)^{{s+2}}}dx,$ $\realpart{s}>-1$ , $s\neq 1$ , $a>0$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, $\realpart{}$ : real part, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.11.E19
Encodings:: TeX, pMML, png

25.11.20 $(-1)^{k}{\mathop{\zeta\/}\nolimits^{{(k)}}}\!\left(s,a\right)=\frac{(\mathop{\ln\/}\nolimits a)^{k}}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{{1-s}}\sum _{{r=0}}^{{k-1}}\frac{(\mathop{\ln\/}\nolimits a)^{r}}{r!(s-1)^{{k-r+1}}}-s(s+1)\int _{0}^{\infty}\frac{\mathop{\widetilde{B}_{{2}}\/}\nolimits\!\left(x\right)(\mathop{\ln\/}\nolimits\!\left(x+a\right))^{k}}{(x+a)^{{s+2}}}dx+k(2s+1)\int _{0}^{\infty}\frac{\mathop{\widetilde{B}_{{2}}\/}\nolimits\!\left(x\right)(\mathop{\ln\/}\nolimits\!\left(x+a\right))^{{k-1}}}{(x+a)^{{s+2}}}dx-k(k-1)\int _{0}^{\infty}\frac{\mathop{\widetilde{B}_{{2}}\/}\nolimits\!\left(x\right)(\mathop{\ln\/}\nolimits\!\left(x+a\right))^{{k-2}}}{(x+a)^{{s+2}}}dx,$ $\realpart{s}>-1$ , $s\neq 1$ , $a>0$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, $\realpart{}$ : real part, $x$ : real variable, $a$ : real or complex parameter, $s$ : complex variable and $k$ : integer
Permalink:: http://dlmf.nist.gov/25.11.E20
Encodings:: TeX, pMML, png

25.11.21 ${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(1-2n,\frac{h}{k}\right)=\frac{(\mathop{\psi\/}\nolimits\!\left(2n\right)-\mathop{\ln\/}\nolimits\!\left(2\pi k\right))\mathop{B_{{2n}}\/}\nolimits\!\left(h/k\right)}{2n}-\frac{(\mathop{\psi\/}\nolimits\!\left(2n\right)-\mathop{\ln\/}\nolimits\!\left(2\pi\right))\mathop{B_{{2n}}\/}\nolimits}{2nk^{{2n}}}+\frac{(-1)^{{n+1}}\pi}{(2\pi k)^{{2n}}}\sum _{{r=1}}^{{k-1}}\mathop{\sin\/}\nolimits\!\left(\frac{2\pi rh}{k}\right){\mathop{\psi\/}\nolimits^{{(2n-1)}}}\!\left(\frac{r}{k}\right)+\frac{(-1)^{{n+1}}2\cdot(2n-1)!}{(2\pi k)^{{2n}}}\sum _{{r=1}}^{{k-1}}\mathop{\cos\/}\nolimits\!\left(\frac{2\pi rh}{k}\right){\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(2n,\frac{r}{k}\right)+\frac{{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(1-2n\right)}{k^{{2n}}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : sine function, $n$ : nonnegative integer, $h$ : integer and $k$ : integer
Permalink:: http://dlmf.nist.gov/25.11.E21
Encodings:: TeX, pMML, png

where $h,k$ are integers with $1\leq h\leq k$ and $n=1,2,3,\dots$ .

25.11.22 ${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(1-2n,\tfrac{1}{2}\right)=-\frac{\mathop{B_{{2n}}\/}\nolimits\mathop{\ln\/}\nolimits 2}{n\cdot 4^{n}}-\frac{(2^{{2n-1}}-1){\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(1-2n\right)}{2^{{2n-1}}},$ $n=1,2,3,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/25.11.E22
Encodings:: TeX, pMML, png

25.11.23 ${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(1-2n,\tfrac{1}{3}\right)=-\frac{\pi(9^{n}-1)\mathop{B_{{2n}}\/}\nolimits}{8n\sqrt{3}(3^{{2n-1}}-1)}-\frac{\mathop{B_{{2n}}\/}\nolimits\mathop{\ln\/}\nolimits 3}{4n\cdot 3^{{2n-1}}}-\frac{(-1)^{n}{\mathop{\psi\/}\nolimits^{{(2n-1)}}}\!\left(\frac{1}{3}\right)}{2\sqrt{3}(6\pi)^{{2n-1}}}-\frac{\left(3^{{2n-1}}-1\right){\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(1-2n\right)}{2\cdot 3^{{2n-1}}},$ $n=1,2,3,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/25.11.E23
Encodings:: TeX, pMML, png

25.11.24 $\sum _{{r=1}}^{{k-1}}{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(s,\frac{r}{k}\right)=(k^{s}-1){\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(s\right)+k^{s}\mathop{\zeta\/}\nolimits\!\left(s\right)\mathop{\ln\/}\nolimits k,$ $s\neq 1$ , $k=1,2,3,\dots$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $s$ : complex variable and $k$ : integer
Referenced by:: ¶ ‣ §25.11(vi), §25.11(vi)
Permalink:: http://dlmf.nist.gov/25.11.E24
Encodings:: TeX, pMML, png

§25.11(vii) Integral Representations

Notes:: See Apostol (1976, pp. 252–253). For (25.11.25) see Srivastava and Choi (2001, p. 89). For (25.11.26) see Berndt (1972). For (25.11.27) and (25.11.28) argue as indicated above for (25.5.6) and (25.5.7). For (25.11.29) see Lindelöf (1905, p. 106). For (25.11.30) assume $\realpart{s}>1$ , collapse the integration path onto the real axis, apply (25.11.25) and (5.5.3) followed by analytic continuation.
Keywords:: Hurwitz zeta function
Permalink:: http://dlmf.nist.gov/25.11.vii

25.11.25 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int _{0}^{\infty}\frac{x^{{s-1}}e^{{-ax}}}{1-e^{{-x}}}dx,$ $\realpart{s}>1$ , $\realpart{a}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(x), §25.11(vii), §25.11(viii)
Permalink:: http://dlmf.nist.gov/25.11.E25
Encodings:: TeX, pMML, png

25.11.26 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=-s\int _{{-a}}^{\infty}\frac{x-\left\lfloor x\right\rfloor-\frac{1}{2}}{(x+a)^{{s+1}}}dx,$ $-1<\realpart{s}<0$ , $0<a\leq 1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $dx$ : differential of $x$ , $\left\lfloor x\right\rfloor$ : floor of $x$ , $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(vii)
Permalink:: http://dlmf.nist.gov/25.11.E26
Encodings:: TeX, pMML, png

25.11.27 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{1}{2}a^{{-s}}+\frac{a^{{1-s}}}{s-1}+\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int _{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{{s-1}}}{e^{{ax}}}dx,$ $\realpart{s}>-1$ , $s\neq 1$ , $\realpart{a}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(vii)
Permalink:: http://dlmf.nist.gov/25.11.E27
Encodings:: TeX, pMML, png

25.11.28 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{1}{2}a^{{-s}}+\frac{a^{{1-s}}}{s-1}+\sum _{{k=1}}^{n}\frac{\mathop{\Gamma\/}\nolimits\!\left(s+2k-1\right)}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\frac{\mathop{B_{{2k}}\/}\nolimits}{(2k)!}a^{{-2k-s+1}}+\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int _{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum _{{k=1}}^{n}\frac{\mathop{B_{{2k}}\/}\nolimits}{(2k)!}x^{{2k-1}}\right)x^{{s-1}}e^{{-ax}}dx,$ $\realpart{s}>-(2n+1)$ , $s\neq 1$ , $\realpart{a}>0$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\int$ : integral, $\realpart{}$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(vii)
Permalink:: http://dlmf.nist.gov/25.11.E28
Encodings:: TeX, pMML, png

25.11.29 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{1}{2}a^{{-s}}+\frac{a^{{1-s}}}{s-1}+2\int _{0}^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(s\mathop{\mathrm{arctan}\/}\nolimits\!\left(x/a\right)\right)}{(a^{2}+x^{2})^{{s/2}}(e^{{2\pi x}}-1)}dx,$ $s\neq 1$ , $\realpart{a}>0$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(vii)
Permalink:: http://dlmf.nist.gov/25.11.E29
Encodings:: TeX, pMML, png

25.11.30 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-s\right)}{2\pi i}\int _{{-\infty}}^{{(0+)}}\frac{e^{{az}}z^{{s-1}}}{1-e^{z}}dz,$ $s\neq 1$ , $\realpart{a}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $a$ : real or complex parameter, $s$ : complex variable and $z$ : complex variable
Referenced by:: §25.11(i), §25.11(vii)
Permalink:: http://dlmf.nist.gov/25.11.E30
Encodings:: TeX, pMML, png

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

§25.11(viii) Further Integral Representations

Notes:: See Adamchik (1998). For (25.11.31) use (25.11.25).
Referenced by:: ¶ ‣ §25.11(vi)
Permalink:: http://dlmf.nist.gov/25.11.viii

25.11.31 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int _{0}^{\infty}\frac{x^{{s-1}}e^{{-ax}}}{2\mathop{\cosh\/}\nolimits x}dx=4^{{-s}}\left(\mathop{\zeta\/}\nolimits\!\left(s,\tfrac{1}{4}+\tfrac{1}{4}a\right)-\mathop{\zeta\/}\nolimits\!\left(s,\tfrac{3}{4}+\tfrac{1}{4}a\right)\right),$ $\realpart{s}>0$ , $\realpart{a}>-1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(viii)
Permalink:: http://dlmf.nist.gov/25.11.E31
Encodings:: TeX, pMML, png

25.11.32 $\int _{0}^{a}x^{n}\mathop{\psi\/}\nolimits\!\left(x\right)dx=(-1)^{{n-1}}{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-n\right)+(-1)^{n}h(n)\frac{\mathop{B_{{n+1}}\/}\nolimits}{n+1}-\sum _{{k=0}}^{n}(-1)^{k}\binom{n}{k}h(k)\frac{\mathop{B_{{k+1}}\/}\nolimits(a)}{k+1}a^{{n-k}}+\sum _{{k=0}}^{n}(-1)^{k}\binom{n}{k}{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-k,a\right)a^{{n-k}},$ $n=1,2,\dots$ , $\realpart{a}>0$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\binom{m}{n}$ : binomial coefficient, $dx$ : differential of $x$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\realpart{}$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $a$ : real or complex parameter and $h(n)$ : sum
Permalink:: http://dlmf.nist.gov/25.11.E32
Encodings:: TeX, pMML, png

where

25.11.33 $h(n)=\sum _{{k=1}}^{n}k^{{-1}}.$

Symbols:: $k$ : nonnegative integer, $n$ : nonnegative integer and $h(n)$ : sum
Referenced by:: §25.16(ii)
Permalink:: http://dlmf.nist.gov/25.11.E33
Encodings:: TeX, pMML, png

25.11.34 $n\int _{0}^{a}{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(1-n,x\right)dx={\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-n,a\right)-{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-n\right)+\frac{\mathop{B_{{n+1}}\/}\nolimits-\mathop{B_{{n+1}}\/}\nolimits\!\left(a\right)}{n(n+1)},$ $n=1,2,\dots$ , $\realpart{a}>0$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $n$ : nonnegative integer, $x$ : real variable and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/25.11.E34
Encodings:: TeX, pMML, png

§25.11(ix) Integrals

Keywords:: Hurwitz zeta function, integrals
Permalink:: http://dlmf.nist.gov/25.11.ix

See Prudnikov et al. (1990, §2.3), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).

§25.11(x) Further Series Representations

Notes:: See Apostol (1976, p. 249). For (25.11.35) use (25.11.25) and (25.11.8).
Keywords:: Hurwitz zeta function
Permalink:: http://dlmf.nist.gov/25.11.x

25.11.35 $\sum _{{n=0}}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int _{0}^{\infty}\frac{x^{{s-1}}e^{{-ax}}}{1+e^{{-x}}}dx=2^{{-s}}\left(\mathop{\zeta\/}\nolimits\!\left(s,\tfrac{1}{2}a\right)-\mathop{\zeta\/}\nolimits\!\left(s,\tfrac{1}{2}(1+a)\right)\right),$ $\realpart{a}>0$ , $\realpart{s}>0$ ; or $\realpart{a}=0$ , $\imagpart{a}\neq 0$ , $0<\realpart{s}<1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\imagpart{}$ : imaginary part, $\int$ : integral, $\realpart{}$ : real part, $n$ : nonnegative integer, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(x), §25.11(x)
Permalink:: http://dlmf.nist.gov/25.11.E35
Encodings:: TeX, pMML, png

When $a=1$ , (25.11.35) reduces to (25.2.3).

25.11.36 $\sum _{{n=1}}^{\infty}\frac{\chi(n)}{n^{s}}=k^{{-s}}\sum _{{r=1}}^{k}\chi(r)\mathop{\zeta\/}\nolimits\!\left(s,\frac{r}{k}\right),$ $\realpart{s}>1$ ,

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\realpart{}$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $s$ : complex variable and $\chi(n)$ : Dirichlet character
Permalink:: http://dlmf.nist.gov/25.11.E36
Encodings:: TeX, pMML, png

where $\chi(n)$ is a Dirichlet character $\;\;(\mathop{{\rm mod}}k)$ (§27.8).

§25.11(xi) Sums

Notes:: See Adamchik and Srivastava (1998).
Keywords:: Catalan’s constant, Hurwitz zeta function, Riemann zeta function
Permalink:: http://dlmf.nist.gov/25.11.xi

25.11.37 $\sum _{{k=1}}^{\infty}\frac{(-1)^{k}}{k}\mathop{\zeta\/}\nolimits\!\left(nk,a\right)=-n\mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(a\right)+\mathop{\ln\/}\nolimits\!\left(\prod _{{j=0}}^{{n-1}}\mathop{\Gamma\/}\nolimits\!\left(a-e^{{(2j+1)\pi i/n}}\right)\right),$ $n=2,3,4,\dots$ , $\realpart{a}\geq 1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $e$ : base of exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\realpart{}$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/25.11.E37
Encodings:: TeX, pMML, png

25.11.38 $\sum _{{k=1}}^{\infty}\binom{n+k}{k}\mathop{\zeta\/}\nolimits\!\left(n+k+1,a\right)z^{k}=\frac{(-1)^{n}}{n!}\left({\mathop{\psi\/}\nolimits^{{(n)}}}\!\left(a\right)-{\mathop{\psi\/}\nolimits^{{(n)}}}\!\left(a-z\right)\right),$ $n=1,2,3,\dots$ , $\realpart{a}>0$ , $|z|<|a|$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\binom{m}{n}$ : binomial coefficient, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\realpart{}$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $a$ : real or complex parameter and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/25.11.E38
Encodings:: TeX, pMML, png

25.11.39 $\sum _{{k=2}}^{\infty}\frac{k}{2^{k}}\mathop{\zeta\/}\nolimits\!\left(k+1,\tfrac{3}{4}\right)=8G,$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $k$ : nonnegative integer and $G$ : Catalan’s constant
Permalink:: http://dlmf.nist.gov/25.11.E39
Encodings:: TeX, pMML, png

where $G$ is Catalan’s constant:

25.11.40 $G=\sum _{{n=0}}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}=0.91596\; 55941\; 772\dots.$

Notes:: For more digits see OEIS Sequence A006752; see also Sloane (2003).
Symbols:: $n$ : nonnegative integer and $G$ : Catalan’s constant
Permalink:: http://dlmf.nist.gov/25.11.E40
Encodings:: TeX, pMML, png

For further sums see Prudnikov et al. (1990, pp. 396–397) and Hansen (1975, pp. 358–360).

§25.11(xii) $a$ -Asymptotic Behavior

Notes:: For (25.11.41) and (25.11.42) see Apostol (1952). For (25.11.43) see Paris (2005b). For (25.11.44) and (25.11.45) see Elizalde (1986).
Keywords:: Hurwitz zeta function, derivatives
Referenced by:: ¶ ‣ §25.11(vi)
Permalink:: http://dlmf.nist.gov/25.11.xii

As $a\to 0$ with $s$ $(\neq 1)$ fixed,

25.11.41 $\mathop{\zeta\/}\nolimits\!\left(s,a+1\right)=\mathop{\zeta\/}\nolimits\!\left(s\right)-s\mathop{\zeta\/}\nolimits\!\left(s+1\right)a+\mathop{O\/}\nolimits\!\left(a^{2}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(xii)
Permalink:: http://dlmf.nist.gov/25.11.E41
Encodings:: TeX, pMML, png

As $\beta\to\pm\infty$ with $s$ fixed, $\realpart{s}>1$ ,

25.11.42 $\mathop{\zeta\/}\nolimits\!\left(s,\alpha+i\beta\right)\to 0,$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function and $s$ : complex variable
Referenced by:: §25.11(xii)
Permalink:: http://dlmf.nist.gov/25.11.E42
Encodings:: TeX, pMML, png

uniformly with respect to bounded nonnegative values of $\alpha$ .

As $a\to\infty$ in the sector $|\mathop{\mathrm{ph}\/}\nolimits a|\leq\pi-\delta(<\pi)$ , with $s(\neq 1)$ and $\delta$ fixed, we have the asymptotic expansion

25.11.43 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)-\frac{a^{{1-s}}}{s-1}-\frac{1}{2}a^{{-s}}\sim\sum _{{k=1}}^{{\infty}}\frac{\mathop{B_{{2k}}\/}\nolimits}{(2k)!}\frac{\mathop{\Gamma\/}\nolimits\!\left(s+2k-1\right)}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}a^{{1-s-2k}}.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $!$ : $n!$ : factorial, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $a$ : real or complex parameter and $s$ : complex variable
Referenced by:: §25.11(xii), §25.11(xii)
Permalink:: http://dlmf.nist.gov/25.11.E43
Encodings:: TeX, pMML, png

Similarly, as $a\to\infty$ in the sector $|\mathop{\mathrm{ph}\/}\nolimits a|\leq\frac{1}{2}\pi-\delta(<\frac{1}{2}\pi)$ ,

25.11.44 ${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-1,a\right)-\frac{1}{12}+\frac{1}{4}a^{2}-\left(\frac{1}{12}-\frac{1}{2}a+\frac{1}{2}a^{2}\right)\mathop{\ln\/}\nolimits a\sim-\sum _{{k=1}}^{\infty}\frac{\mathop{B_{{2k+2}}\/}\nolimits}{(2k+2)(2k+1)2k}a^{{-2k}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality, $k$ : nonnegative integer and $a$ : real or complex parameter
Referenced by:: §25.11(xii)
Permalink:: http://dlmf.nist.gov/25.11.E44
Encodings:: TeX, pMML, png

and

25.11.45 ${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-2,a\right)-\frac{1}{12}a+\frac{1}{9}a^{3}-\left(\frac{1}{6}a-\frac{1}{2}a^{2}+\frac{1}{3}a^{3}\right)\mathop{\ln\/}\nolimits a\sim\sum _{{k=1}}^{\infty}\frac{2\!\mathop{B_{{2k+2}}\/}\nolimits}{(2k+2)(2k+1)2k(2k-1)}a^{{-(2k-1)}}.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality, $k$ : nonnegative integer and $a$ : real or complex parameter
Referenced by:: §25.11(xii)
Permalink:: http://dlmf.nist.gov/25.11.E45
Encodings:: TeX, pMML, png

For the more general case ${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(-m,a\right)$ , $m=1,2,\dots$ , see Elizalde (1986).

For an exponentially-improved form of (25.11.43) see Paris (2005b).