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

§24.7 Integral Representations

ⓘ

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

§24.7(i) Bernoulli and Euler Numbers
§24.7(ii) Bernoulli and Euler Polynomials
§24.7(iii) Compendia

§24.7(i) Bernoulli and Euler Numbers

ⓘ

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

ⓘ

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

ⓘ

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

ⓘ

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

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§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