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

§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 .

24.7.1 $B_{2n}=(-1)^{n+1}\frac{4n}{1-2^{1-2n}}\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t% }+1}\,\mathrm{d}t=(-1)^{n+1}\frac{2n}{1-2^{1-2n}}\int_{0}^{\infty}t^{2n-1}e^{-% \pi t}\operatorname{sech}\left(\pi t\right)\,\mathrm{d}t,$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\int$ : integral, : integer and : 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 $B_{2n}=(-1)^{n+1}4n\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t}-1}\,\mathrm{d}t% =(-1)^{n+1}2n\int_{0}^{\infty}t^{2n-1}e^{-\pi t}\operatorname{csch}\left(\pi t% \right)\,\mathrm{d}t,$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\int$ : integral, : integer and : 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 $B_{2n}=(-1)^{n+1}\frac{\pi}{1-2^{1-2n}}\int_{0}^{\infty}t^{2n}{\operatorname{% sech}}^{2}\left(\pi t\right)\,\mathrm{d}t,$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\int$ : integral, : integer and : 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 $B_{2n}=(-1)^{n+1}\pi\int_{0}^{\infty}t^{2n}{\operatorname{csch}}^{2}\left(\pi t% \right)\,\mathrm{d}t,$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\int$ : integral, : integer and : 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 $B_{2n}=(-1)^{n}\frac{2n(2n-1)}{\pi}\*\int_{0}^{\infty}t^{2n-2}\ln\left(1-e^{-2% \pi t}\right)\,\mathrm{d}t.$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : 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, $\ln\NVar{z}$ : principal branch of logarithm function, : integer and : 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 $E_{2n}=(-1)^{n}2^{2n+1}\int_{0}^{\infty}t^{2n}\operatorname{sech}\left(\pi t% \right)\,\mathrm{d}t.$

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\int$ : integral, : integer and : 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 $B_{2n}\left(x\right)=(-1)^{n+1}2n\*\int_{0}^{\infty}\frac{\cos\left(2\pi x% \right)-e^{-2\pi t}}{\cosh\left(2\pi t\right)-\cos\left(2\pi x\right)}t^{2n-1}% \,\mathrm{d}t,$ $n=1,2,\dots$ ,

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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 , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, : integer, : real or complex and : 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 $B_{2n+1}\left(x\right)=(-1)^{n+1}(2n+1)\*\int_{0}^{\infty}\frac{\sin\left(2\pi x% \right)}{\cosh\left(2\pi t\right)-\cos\left(2\pi x\right)}t^{2n}\,\mathrm{d}t.$

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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 , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, : integer, : real or complex and : 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 $E_{2n}\left(x\right)=(-1)^{n}4\int_{0}^{\infty}\frac{\sin\left(\pi x\right)% \cosh\left(\pi t\right)}{\cosh\left(2\pi t\right)-\cos\left(2\pi x\right)}t^{2% n}\,\mathrm{d}t,$

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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 , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, : integer, : real or complex and : 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 $E_{2n+1}\left(x\right)=(-1)^{n+1}4\*\int_{0}^{\infty}\frac{\cos\left(\pi x% \right)\sinh\left(\pi t\right)}{\cosh\left(2\pi t\right)-\cos\left(2\pi x% \right)}t^{2n+1}\,\mathrm{d}t.$

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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 , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, : integer, : real or complex and : 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 $B_{n}\left(x\right)=\frac{1}{2\pi i}\int_{-c-i\infty}^{-c+i\infty}(x+t)^{n}% \left(\frac{\pi}{\sin\left(\pi t\right)}\right)^{2}\,\mathrm{d}t,$