§24.7 Integral Representations

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

§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

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

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

24.7.1 $\mathop{B_{{2n}}\/}\nolimits=(-1)^{{n+1}}\frac{4n}{1-2^{{1-2n}}}\int _{0}^{\infty}\frac{t^{{2n-1}}}{e^{{2\pi t}}+1}dt=(-1)^{{n+1}}\frac{2n}{1-2^{{1-2n}}}\int _{0}^{\infty}t^{{2n-1}}e^{{-\pi t}}\mathop{\mathrm{sech}\/}\nolimits\!\left(\pi t\right)dt,$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\int$ : integral, $n$ : integer and $t$ : real or complex
Permalink:: http://dlmf.nist.gov/24.7.E1
Encodings:: TeX, pMML, png

24.7.2 $\mathop{B_{{2n}}\/}\nolimits=(-1)^{{n+1}}4n\int _{0}^{\infty}\frac{t^{{2n-1}}}{e^{{2\pi t}}-1}dt=(-1)^{{n+1}}2n\int _{0}^{\infty}t^{{2n-1}}e^{{-\pi t}}\mathop{\mathrm{csch}\/}\nolimits\!\left(\pi t\right)dt,$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\mathrm{csch}\/}\nolimits z$ : hyperbolic cosecant function, $\int$ : integral, $n$ : integer and $t$ : real or complex
Permalink:: http://dlmf.nist.gov/24.7.E2
Encodings:: TeX, pMML, png

24.7.3 $\mathop{B_{{2n}}\/}\nolimits=(-1)^{{n+1}}\frac{\pi}{1-2^{{1-2n}}}\int _{0}^{\infty}t^{{2n}}{\mathop{\mathrm{sech}\/}\nolimits^{{2}}}\!\left(\pi t\right)dt,$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $dx$ : differential of $x$ , $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\int$ : integral, $n$ : integer and $t$ : real or complex
Permalink:: http://dlmf.nist.gov/24.7.E3
Encodings:: TeX, pMML, png

24.7.4 $\mathop{B_{{2n}}\/}\nolimits=(-1)^{{n+1}}\pi\int _{0}^{\infty}t^{{2n}}{\mathop{\mathrm{csch}\/}\nolimits^{{2}}}\!\left(\pi t\right)dt,$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $dx$ : differential of $x$ , $\mathop{\mathrm{csch}\/}\nolimits z$ : hyperbolic cosecant function, $\int$ : integral, $n$ : integer and $t$ : real or complex
Permalink:: http://dlmf.nist.gov/24.7.E4
Encodings:: TeX, pMML, png

24.7.5 $\mathop{B_{{2n}}\/}\nolimits=(-1)^{n}\frac{2n(2n-1)}{\pi}\*\int _{0}^{\infty}t^{{2n-2}}\mathop{\ln\/}\nolimits\!\left(1-e^{{-2\pi t}}\right)dt.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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

24.7.6 $\mathop{E_{{2n}}\/}\nolimits=(-1)^{n}2^{{2n+1}}\int _{0}^{\infty}t^{{2n}}\mathop{\mathrm{sech}\/}\nolimits\!\left(\pi t\right)dt.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $dx$ : differential of $x$ , $\mathop{\mathrm{sech}\/}\nolimits z$ : 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

§24.7(ii) Bernoulli and Euler Polynomials

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

The following four equations hold for $0<\realpart{x}<1$ .

24.7.7 $\mathop{B_{{2n}}\/}\nolimits\!\left(x\right)=(-1)^{{n+1}}2n\*\int _{0}^{\infty}\frac{\mathop{\cos\/}\nolimits\!\left(2\pi x\right)-e^{{-2\pi t}}}{\mathop{\cosh\/}\nolimits\!\left(2\pi t\right)-\mathop{\cos\/}\nolimits\!\left(2\pi x\right)}t^{{2n-1}}dt,$ $n=1,2,\dots$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : 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

24.7.8 $\mathop{B_{{2n+1}}\/}\nolimits\!\left(x\right)=(-1)^{{n+1}}(2n+1)\*\int _{0}^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(2\pi x\right)}{\mathop{\cosh\/}\nolimits\!\left(2\pi t\right)-\mathop{\cos\/}\nolimits\!\left(2\pi x\right)}t^{{2n}}dt.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

24.7.9 $\mathop{E_{{2n}}\/}\nolimits\!\left(x\right)=(-1)^{n}4\int _{0}^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(\pi x\right)\mathop{\cosh\/}\nolimits\!\left(\pi t\right)}{\mathop{\cosh\/}\nolimits\!\left(2\pi t\right)-\mathop{\cos\/}\nolimits\!\left(2\pi x\right)}t^{{2n}}dt,$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

24.7.10 $\mathop{E_{{2n+1}}\/}\nolimits\!\left(x\right)=(-1)^{{n+1}}4\*\int _{0}^{\infty}\frac{\mathop{\cos\/}\nolimits\!\left(\pi x\right)\mathop{\sinh\/}\nolimits\!\left(\pi t\right)}{\mathop{\cosh\/}\nolimits\!\left(2\pi t\right)-\mathop{\cos\/}\nolimits\!\left(2\pi x\right)}t^{{2n+1}}dt.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : 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

¶ Mellin–Barnes Integral

24.7.11 $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)=\frac{1}{2\pi i}\int _{{-c-i\infty}}^{{-c+i\infty}}(x+t)^{n}\left(\frac{\pi}{\mathop{\sin\/}\nolimits\!\left(\pi t\right)}\right)^{2}dt,$ $0<c<1$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $n$ : integer, $t$ : real or complex and $x$ : real or complex
Referenced by:: §24.7(ii)
Permalink:: http://dlmf.nist.gov/24.7.E11
Encodings:: TeX, pMML, png

§24.7(iii) Compendia

Permalink:: http://dlmf.nist.gov/24.7.iii

For further integral representations see Prudnikov et al. (1986a, §§2.3–2.6) and Gradshteyn and Ryzhik (2000, Chapters 3 and 4).

§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