24 Bernoulli and Euler PolynomialsProperties24.6 Explicit Formulas24.8 Series Expansions

§24.7 Integral Representations

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http://dlmf.nist.gov/24.7

See also:

Annotations for Ch.24

Contents

1. §24.7(i) Bernoulli and Euler Numbers
2. §24.7(ii) Bernoulli and Euler Polynomials
3. §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).

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http://dlmf.nist.gov/24.7.i

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Annotations for §24.7 and Ch.24

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

24.7.1		$B_{2n}$	$\begin{array}{ll}& ={(-1)}^{n+1}{\displaystyle \frac{4n}{1-2^{1-2n}}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{t^{2n-1}}{{\mathrm{e}}^{2\pi t}+1}}dt\\ & ={(-1)}^{n+1}{\displaystyle \frac{2n}{1-2^{1-2n}}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{t}^{2n-1}{\mathrm{e}}^{-\pi t}\mathrm{sech}\left(\pi t\right)dt,\end{array}$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: 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		$B_{2n}$	$={(-1)}^{n+1}4n{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{t^{2n-1}}{{\mathrm{e}}^{2\pi t}-1}}dt={(-1)}^{n+1}2n{\displaystyle {\int }_0^{\mathrm{\infty }}}{t}^{2n-1}{\mathrm{e}}^{-\pi t}\mathrm{csch}\left(\pi t\right)dt,$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: 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		$B_{2n}$	$={(-1)}^{n+1}{\displaystyle \frac{\pi }{1-2^{1-2n}}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{t}^{2n}{\mathrm{sech}}^2\left(\pi t\right)dt,$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{sech}z$: 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		$B_{2n}$	$={(-1)}^{n+1}\pi {\displaystyle {\int }_0^{\mathrm{\infty }}}{t}^{2n}{\mathrm{csch}}^2\left(\pi t\right)dt,$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{csch}z$: 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		$B_{2n}$	$={(-1)}^n{\displaystyle \frac{2n(2n-1)}{\pi }}{\displaystyle {\int }_0^{\mathrm{\infty }}}{t}^{2n-2}\ln \left(1-{\mathrm{e}}^{-2\pi t}\right)dt.$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: 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		$E_{2n}$	$={(-1)}^n{2}^{2n+1}{\displaystyle {\int }_0^{\mathrm{\infty }}}{t}^{2n}\mathrm{sech}\left(\pi t\right)dt.$
ⓘ Symbols: $E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{sech}z$: 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 .

24.7.7		$B_{2n}\left(x\right)$	$={(-1)}^{n+1}2n{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{\cos \left(2\pi x\right)-{\mathrm{e}}^{-2\pi t}}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}}{t}^{2n-1}dt,$
$n=1,2,\mathrm{\dots }$,
ⓘ Symbols: $B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh z$: 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		$B_{2n+1}\left(x\right)$	$={(-1)}^{n+1}(2n+1){\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{\sin \left(2\pi x\right)}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}}{t}^{2n}dt.$
ⓘ Symbols: $B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: 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		$E_{2n}\left(x\right)$	$={(-1)}^{n}4{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{\sin \left(\pi x\right)\cosh \left(\pi t\right)}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}}{t}^{2n}dt,$
ⓘ Symbols: $E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: 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		$E_{2n+1}\left(x\right)$	$={(-1)}^{n+1}4{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{\cos \left(\pi x\right)\sinh \left(\pi t\right)}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}}{t}^{2n+1}dt.$
ⓘ Symbols: $E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\sinh z$: 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		$$B_n\left(x\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-c-\mathrm{i}\mathrm{\infty }}^{-c+\mathrm{i}\mathrm{\infty }}{(x+t)}^n{\left(\frac{\pi }{\sin \left(\pi t\right)}\right)}^{2}dt,$$
.
ⓘ Symbols: $B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: 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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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 Formulas24.8 Series Expansions

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