24 Bernoulli and Euler Polynomials Properties 24.6 Explicit Formulas 24.8 Series Expansions

§24.7 Integral Representations

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Permalink:: http://dlmf.nist.gov/24.7
See also:: Annotations for Ch.24

§24.7(i) Bernoulli and Euler Numbers

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Keywords:: Bernoulli numbers, Euler numbers, integral representation, integral representations
Notes:: See Erdélyi et al. (1953a, pp. 38, 39, and 42). For (24.7.5) see Ramanujan (1927, p. 7).
Permalink:: http://dlmf.nist.gov/24.7.i
See also:: Annotations for §24.7 and Ch.24

The identities in this subsection hold for $n=1,2,\dotsc$ . (24.7.6) also holds for $n=0$ .

24.7.1	$\displaystyle B_{2n}$	$\displaystyle=(-1)^{n+1}\frac{4n}{1-2^{1-2n}}\int_{0}^{\infty}\frac{t^{2n-1}}{% e^{2\pi t}+1}\,\mathrm{d}t=(-1)^{n+1}\frac{2n}{1-2^{1-2n}}\int_{0}^{\infty}t^{% 2n-1}e^{-\pi t}\operatorname{sech}\left(\pi t\right)\,\mathrm{d}t,$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\int$ : integral, $n$ : integer and $t$ : real or complex Permalink: http://dlmf.nist.gov/24.7.E1 Encodings: TeX, pMML, png See also: Annotations for §24.7(i), §24.7 and Ch.24
24.7.2	$\displaystyle B_{2n}$	$\displaystyle=(-1)^{n+1}4n\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t}-1}\,% \mathrm{d}t=(-1)^{n+1}2n\int_{0}^{\infty}t^{2n-1}e^{-\pi t}\operatorname{csch}% \left(\pi t\right)\,\mathrm{d}t,$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\int$ : integral, $n$ : integer and $t$ : real or complex Permalink: http://dlmf.nist.gov/24.7.E2 Encodings: TeX, pMML, png See also: Annotations for §24.7(i), §24.7 and Ch.24
24.7.3	$\displaystyle B_{2n}$	$\displaystyle=(-1)^{n+1}\frac{\pi}{1-2^{1-2n}}\int_{0}^{\infty}t^{2n}{% \operatorname{sech}}^{2}\left(\pi t\right)\,\mathrm{d}t,$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\int$ : integral, $n$ : integer and $t$ : real or complex Permalink: http://dlmf.nist.gov/24.7.E3 Encodings: TeX, pMML, png See also: Annotations for §24.7(i), §24.7 and Ch.24
24.7.4	$\displaystyle B_{2n}$	$\displaystyle=(-1)^{n+1}\pi\int_{0}^{\infty}t^{2n}{\operatorname{csch}}^{2}% \left(\pi t\right)\,\mathrm{d}t,$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\int$ : integral, $n$ : integer and $t$ : real or complex Permalink: http://dlmf.nist.gov/24.7.E4 Encodings: TeX, pMML, png See also: Annotations for §24.7(i), §24.7 and Ch.24
24.7.5	$\displaystyle B_{2n}$	$\displaystyle=(-1)^{n}\frac{2n(2n-1)}{\pi}\*\int_{0}^{\infty}t^{2n-2}\ln\left(% 1-e^{-2\pi t}\right)\,\mathrm{d}t.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : integer and $t$ : real or complex Referenced by: §24.7(i) Permalink: http://dlmf.nist.gov/24.7.E5 Encodings: TeX, pMML, png See also: Annotations for §24.7(i), §24.7 and Ch.24
24.7.6	$\displaystyle E_{2n}$	$\displaystyle=(-1)^{n}2^{2n+1}\int_{0}^{\infty}t^{2n}\operatorname{sech}\left(% \pi t\right)\,\mathrm{d}t.$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\int$ : integral, $n$ : integer and $t$ : real or complex Referenced by: §24.7(i) Permalink: http://dlmf.nist.gov/24.7.E6 Encodings: TeX, pMML, png See also: Annotations for §24.7(i), §24.7 and Ch.24

§24.7(ii) Bernoulli and Euler Polynomials

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Keywords:: Bernoulli polynomials, Euler polynomials, integral representations
Notes:: See Erdélyi et al. (1953a, pp. 38 and 43). For (24.7.11) see Paris and Kaminski (2001, p. 173).
Permalink:: http://dlmf.nist.gov/24.7.ii
See also:: Annotations for §24.7 and Ch.24

The following four equations hold for $0<\Re x<1$ .

24.7.7	$\displaystyle B_{2n}\left(x\right)$	$\displaystyle=(-1)^{n+1}2n\*\int_{0}^{\infty}\frac{\cos\left(2\pi x\right)-e^{% -2\pi t}}{\cosh\left(2\pi t\right)-\cos\left(2\pi x\right)}t^{2n-1}\,\mathrm{d% }t,$
$n=1,2,\dots$ ,
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $n$ : integer, $t$ : real or complex and $x$ : real or complex Permalink: http://dlmf.nist.gov/24.7.E7 Encodings: TeX, pMML, png See also: Annotations for §24.7(ii), §24.7 and Ch.24
24.7.8	$\displaystyle B_{2n+1}\left(x\right)$	$\displaystyle=(-1)^{n+1}(2n+1)\*\int_{0}^{\infty}\frac{\sin\left(2\pi x\right)% }{\cosh\left(2\pi t\right)-\cos\left(2\pi x\right)}t^{2n}\,\mathrm{d}t.$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : integer, $t$ : real or complex and $x$ : real or complex Permalink: http://dlmf.nist.gov/24.7.E8 Encodings: TeX, pMML, png See also: Annotations for §24.7(ii), §24.7 and Ch.24
24.7.9	$\displaystyle E_{2n}\left(x\right)$	$\displaystyle=(-1)^{n}4\int_{0}^{\infty}\frac{\sin\left(\pi x\right)\cosh\left% (\pi t\right)}{\cosh\left(2\pi t\right)-\cos\left(2\pi x\right)}t^{2n}\,% \mathrm{d}t,$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : integer, $t$ : real or complex and $x$ : real or complex Permalink: http://dlmf.nist.gov/24.7.E9 Encodings: TeX, pMML, png See also: Annotations for §24.7(ii), §24.7 and Ch.24
24.7.10	$\displaystyle E_{2n+1}\left(x\right)$	$\displaystyle=(-1)^{n+1}4\*\int_{0}^{\infty}\frac{\cos\left(\pi x\right)\sinh% \left(\pi t\right)}{\cosh\left(2\pi t\right)-\cos\left(2\pi x\right)}t^{2n+1}% \,\mathrm{d}t.$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $n$ : integer, $t$ : real or complex and $x$ : real or complex Permalink: http://dlmf.nist.gov/24.7.E10 Encodings: TeX, pMML, png See also: Annotations for §24.7(ii), §24.7 and Ch.24

Mellin–Barnes Integral

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See also:: Annotations for §24.7(ii), §24.7 and Ch.24

24.7.11		$B_{n}\left(x\right)=\frac{1}{2\pi i}\int_{-c-i\infty}^{-c+i\infty}(x+t)^{n}% \left(\frac{\pi}{\sin\left(\pi t\right)}\right)^{2}\,\mathrm{d}t,$
$0<c<1$ .
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : integer, $t$ : real or complex and $x$ : real or complex Keywords: Mellin transform Referenced by: §24.7(ii) Permalink: http://dlmf.nist.gov/24.7.E11 Encodings: TeX, pMML, png See also: Annotations for §24.7(ii), §24.7(ii), §24.7 and Ch.24

§24.7(iii) Compendia

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Permalink:: http://dlmf.nist.gov/24.7.iii
See also:: Annotations for §24.7 and Ch.24

For further integral representations see Prudnikov et al. (1986a, §§2.3–2.6) and Gradshteyn and Ryzhik (2015, Chapters 3 and 4).

24.6 Explicit Formulas 24.8 Series Expansions

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§24.7 Integral Representations

Contents

§24.7(i) Bernoulli and Euler Numbers

§24.7(ii) Bernoulli and Euler Polynomials

Mellin–Barnes Integral

§24.7(iii) Compendia

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