§25.5 Integral Representations

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Keywords:: Riemann zeta function, along the real line, integral representations
Notes:: See Erdélyi et al. (1953a, Section 1.12), and Magnus et al. (1966, pp. 20–22).
Permalink:: http://dlmf.nist.gov/25.5
See also:: Annotations for Ch.25

§25.5(i) In Terms of Elementary Functions
§25.5(ii) In Terms of Other Functions
§25.5(iii) Contour Integrals

§25.5(i) In Terms of Elementary Functions

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Notes:: See Apostol (1976, Section 12.3) and Erdélyi et al. (1953a, Section 1.12).
Permalink:: http://dlmf.nist.gov/25.5.i
See also:: Annotations for §25.5 and Ch.25

Throughout this subsection $s\neq 1$ .

25.5.1 $\zeta\left(s\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-% 1}}{e^{x}-1}\,\mathrm{d}x,$ $\Re s>1$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, : real variable and : 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 $\zeta\left(s\right)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{e^{% x}x^{s}}{(e^{x}-1)^{2}}\,\mathrm{d}x,$ $\Re s>1$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, : real variable and : 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 $\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma\left(s\right)}\int_{0}^{\infty}% \frac{x^{s-1}}{e^{x}+1}\,\mathrm{d}x,$ $\Re s>0$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, : real variable and : 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 $\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma\left(s+1\right)}\int_{0}^{% \infty}\frac{e^{x}x^{s}}{(e^{x}+1)^{2}}\,\mathrm{d}x,$ $\Re s>0$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, : real variable and : 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 $\zeta\left(s\right)=-s\int_{0}^{\infty}\frac{x-\left\lfloor x\right\rfloor-% \frac{1}{2}}{x^{s+1}}\,\mathrm{d}x,$ $-1<\Re s<0$ .

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\left\lfloor\NVar{x}\right\rfloor$ : floor of , $\int$ : integral, $\Re$ : real part, : real variable and : complex variable
Keywords:: improper integral, integral representation
Source:: Titchmarsh (1986b, (2.1.6), p. 15)
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 $\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\Gamma\left(s\right)}% \int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x% ^{s-1}}{e^{x}}\,\mathrm{d}x,$ $\Re s>-1$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, : real variable and : 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 $\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{B_{2m}}{(2m)% !}{\left(s\right)_{2m-1}}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left% (\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}x% ^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\,\mathrm{d}x,$ $\Re s>-(2n+1)$ , $n=1,2,3,\dots$ .

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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 of , $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $\int$ : integral, $\Re$ : real part, : nonnegative integer, : nonnegative integer, : real variable and : 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 $\zeta\left(s\right)=\frac{1}{2(1-2^{-s})\Gamma\left(s\right)}\int_{0}^{\infty}% \frac{x^{s-1}}{\sinh x}\,\mathrm{d}x,$ $\Re s>1$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\Re$ : real part, : real variable and : complex variable
Keywords:: improper integral, integral representation
Source:: Erdélyi et al. (1953a, (1.12.6), 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 $\zeta\left(s\right)=\frac{2^{s-1}}{\Gamma\left(s+1\right)}\int_{0}^{\infty}% \frac{x^{s}}{(\sinh x)^{2}}\,\mathrm{d}x,$ $\Re s>1$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\Re$ : real part, : real variable and : complex variable
Keywords:: improper integral, integral representation
Source:: Erdélyi et al. (1953a, (1.12.7), 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 $\zeta\left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}\int_{0}^{\infty}\frac{\cos\left(% s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}\cosh\left(\frac{1}{2}\pi x% \right)}\,\mathrm{d}x.$

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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 of , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, : real variable and : complex variable
Keywords:: improper integral, integral representation
Sources:: Lindelöf (1905, (6), p. 98); with ; 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 $\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+2\int_{0}^{\infty}\frac{\sin% \left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^{2\pi x}-1)}\,\mathrm{% d}x.$

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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 of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, : real variable and : complex variable
Keywords:: improper integral, integral representation
Sources:: Lindelöf (1905, (3), p. 97); Erdélyi et al. (1953a, (1.12.12), p. 33)
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 $\zeta\left(s\right)=\frac{2^{s-1}}{s-1}-2^{s}\int_{0}^{\infty}\frac{\sin\left(% s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^{\pi x}+1)}\,\mathrm{d}x.$

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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 of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, : real variable and : complex variable
Keywords:: improper integral, integral representation
Source:: Lindelöf (1905, (4), p. 98); with
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

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Keywords:: Riemann zeta function, along the real line, integral representations
Notes:: See Titchmarsh (1986b, Section 2.6) and de Bruijn (1937).
Permalink:: http://dlmf.nist.gov/25.5.ii
See also:: Annotations for §25.5 and Ch.25

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

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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 of , $\int$ : integral, : real variable and : complex variable
Keywords:: improper integral, integral representation
Source:: Titchmarsh (1986b, p. 22)
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 $\omega(x)\equiv\sum_{n=1}^{\infty}e^{-n^{2}\pi x}=\frac{1}{2}\left(\theta_{3}% \left(0\middle|ix\right)-1\right).$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : 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, : nonnegative integer and : 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), $0<\Re s<1$ , $\psi\left(x\right)$ 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 $\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\left(\pi s\right)}{\pi}\*\int_{0}% ^{\infty}(\ln\left(1+x\right)-\psi\left(1+x\right))x^{-s}\,\mathrm{d}x,$

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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 of , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, : real variable and : 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 $\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\left(\pi s\right)}{\pi(s-1)}\*% \int_{0}^{\infty}\left(\frac{1}{1+x}-\psi'\left(1+x\right)\right)x^{1-s}\,% \mathrm{d}x,$

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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 of , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\int$ : integral, $\sin\NVar{z}$ : sine function, : real variable and : 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 $\zeta\left(1+s\right)=\frac{\sin\left(\pi s\right)}{\pi}\int_{0}^{\infty}\left% (\gamma+\psi\left(1+x\right)\right)x^{-s-1}\,\mathrm{d}x,$

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Symbols:: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\int$ : integral, $\sin\NVar{z}$ : sine function, : real variable and : 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 $\zeta\left(1+s\right)=\frac{\sin\left(\pi s\right)}{\pi s}\int_{0}^{\infty}% \psi'\left(1+x\right)x^{-s}\,\mathrm{d}x,$

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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 of , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\int$ : integral, $\sin\NVar{z}$ : sine function, : real variable and : 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 $\zeta\left(m+s\right)=(-1)^{m-1}\frac{\Gamma\left(s\right)\sin\left(\pi s% \right)}{\pi\Gamma\left(m+s\right)}\*\int_{0}^{\infty}{\psi}^{(m)}\left(1+x% \right)x^{-s}\,\mathrm{d}x,$ $m=1,2,3,\dots$ .

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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 of , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\int$ : integral, $\sin\NVar{z}$ : sine function, : nonnegative integer, : real variable and : 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(iii) Contour Integrals

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Keywords:: Riemann zeta function, contour integrals, integral representations
Notes:: See Apostol (1976, Section 12.4) and Erdélyi et al. (1953a, Section 1.12).
Permalink:: http://dlmf.nist.gov/25.5.iii
Clarification (effective with 1.2.1):: Just below (25.5.21), we added “The particular loop contour for (25.5.20), (25.5.21) (see Figure 5.9.1)”.
See also:: Annotations for §25.5 and Ch.25

25.5.20 $\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{2\pi i}\int_{-\infty}^{(0+)}% \frac{z^{s-1}}{e^{-z}-1}\,\mathrm{d}z,$ $s\neq 1,2,\dots$ ,

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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 of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, : real or complex parameter, : complex variable and : complex variable
Keywords:: contour integral, integral representation
Sources:: Apostol (1976, (6), p. 253); with ; Erdélyi et al. (1953a, (1.12.9), p. 32); with
A&S Ref:: 23.2.4 (in slightly different form)
Referenced by:: §25.11(vii), §25.5(iii), §25.5(iii)
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 $\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{2\pi i(1-2^{1-s})}\*\int_{-% \infty}^{(0+)}\frac{z^{s-1}}{e^{-z}+1}\,\mathrm{d}z,$ $s\neq 1,2,\dots$ .

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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 of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, : complex variable and : complex variable
Keywords:: contour integral, integral representation
Sources:: Erdélyi et al. (1953a, (1.12.10), p. 32); with ; Magnus et al. (1966, p. 21); with
Referenced by:: §25.5(iii), §25.5(iii), Erratum (V1.2.1) for Figure 5.9.1
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 $\pm\pi\mathrm{i}$ , $\pm 3\pi\mathrm{i}$ , …. For the contour of integration in (25.5.20) and (25.5.21) see Figure 5.9.1.

25.4 Reflection Formulas 25.6 Integer Arguments

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§25.5 Integral Representations

Contents

§25.5(i) In Terms of Elementary Functions

§25.5(ii) In Terms of Other Functions

§25.5(iii) Contour Integrals