24 Bernoulli and Euler Polynomials Properties 24.12 Zeros 24.14 Sums

§24.13 Integrals

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Permalink:: http://dlmf.nist.gov/24.13
See also:: Annotations for Ch.24

§24.13(i) Bernoulli Polynomials

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Keywords:: Bernoulli polynomials, integrals
Notes:: (24.13.1) and (24.13.2) follow from (24.4.34) and (24.4.1). For (24.13.3)–(24.13.5) see Nörlund (1922, p. 143). For (24.13.6) see Apostol (1976, p. 276) or Nörlund (1924, p. 31).
Referenced by:: Erratum (V1.0.7) for References
Permalink:: http://dlmf.nist.gov/24.13.i
Addition (effective with 1.0.7):: The reference to Agoh and Dilcher (2011) has been added after (24.13.6).
See also:: Annotations for §24.13 and Ch.24

24.13.1	$\displaystyle\int B_{n}\left(t\right)\,\mathrm{d}t$	$\displaystyle=\frac{B_{n+1}\left(t\right)}{n+1}+\text{const.},$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex A&S Ref: 23.1.11 Referenced by: §24.13(i) Permalink: http://dlmf.nist.gov/24.13.E1 Encodings: TeX, pMML, png See also: Annotations for §24.13(i), §24.13 and Ch.24
24.13.2	$\displaystyle\int_{x}^{x+1}B_{n}\left(t\right)\,\mathrm{d}t$	$\displaystyle=x^{n},$
$n=1,2,\dots$ ,
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer, $t$ : real or complex and $x$ : real or complex Referenced by: §24.13(i) Permalink: http://dlmf.nist.gov/24.13.E2 Encodings: TeX, pMML, png See also: Annotations for §24.13(i), §24.13 and Ch.24
24.13.3	$\displaystyle\int_{x}^{x+(1/2)}B_{n}\left(t\right)\,\mathrm{d}t$	$\displaystyle=\frac{E_{n}\left(2x\right)}{2^{n+1}},$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer, $t$ : real or complex and $x$ : real or complex Referenced by: §24.13(i) Permalink: http://dlmf.nist.gov/24.13.E3 Encodings: TeX, pMML, png See also: Annotations for §24.13(i), §24.13 and Ch.24
24.13.4	$\displaystyle\int_{0}^{1/2}B_{n}\left(t\right)\,\mathrm{d}t$	$\displaystyle=\frac{1-2^{n+1}}{2^{n}}\frac{B_{n+1}}{n+1},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex Permalink: http://dlmf.nist.gov/24.13.E4 Encodings: TeX, pMML, png See also: Annotations for §24.13(i), §24.13 and Ch.24
24.13.5	$\displaystyle\int_{1/4}^{3/4}B_{n}\left(t\right)\,\mathrm{d}t$	$\displaystyle=\frac{E_{n}}{2^{2n+1}}.$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $E_{\NVar{n}}$ : Euler numbers, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex Referenced by: §24.13(i) Permalink: http://dlmf.nist.gov/24.13.E5 Encodings: TeX, pMML, png See also: Annotations for §24.13(i), §24.13 and Ch.24

For $m,n=1,2,\dotsc$ ,

24.13.6		$\int_{0}^{1}B_{n}\left(t\right)B_{m}\left(t\right)\,\mathrm{d}t=\frac{(-1)^{n-% 1}m!n!}{(m+n)!}B_{m+n}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $m$ : integer, $n$ : integer and $t$ : real or complex A&S Ref: 23.1.12 Referenced by: §24.13(i) Permalink: http://dlmf.nist.gov/24.13.E6 Encodings: TeX, pMML, png See also: Annotations for §24.13(i), §24.13 and Ch.24

For integrals of the form $\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)\,\mathrm{d}t$ and $\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)B_{k}\left(t\right)\,\mathrm% {d}t$ see Agoh and Dilcher (2011).

§24.13(ii) Euler Polynomials

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Keywords:: Euler polynomials, integrals
Notes:: For (24.13.7) and (24.13.10) see Nörlund (1922, p. 143). For (24.13.11) see Nörlund (1924, p. 36).
Permalink:: http://dlmf.nist.gov/24.13.ii
See also:: Annotations for §24.13 and Ch.24

24.13.7		$\int E_{n}\left(t\right)\,\mathrm{d}t=\frac{E_{n+1}\left(t\right)}{n+1}+\text{% const.},$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex A&S Ref: 23.1.11 Referenced by: §24.13(ii) Permalink: http://dlmf.nist.gov/24.13.E7 Encodings: TeX, pMML, png See also: Annotations for §24.13(ii), §24.13 and Ch.24

24.13.8		$\int_{0}^{1}E_{n}\left(t\right)\,\mathrm{d}t=-2\frac{E_{n+1}\left(0\right)}{n+% 1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}B_{n+2},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex Permalink: http://dlmf.nist.gov/24.13.E8 Encodings: TeX, pMML, png See also: Annotations for §24.13(ii), §24.13 and Ch.24

24.13.9		$\int_{0}^{1/2}E_{2n}\left(t\right)\,\mathrm{d}t=-\frac{E_{2n+1}\left(0\right)}% {2n+1}=\frac{2(2^{2n+2}-1)B_{2n+2}}{(2n+1)(2n+2)},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex Permalink: http://dlmf.nist.gov/24.13.E9 Encodings: TeX, pMML, png See also: Annotations for §24.13(ii), §24.13 and Ch.24

24.13.10		$\int_{0}^{1/2}E_{2n-1}\left(t\right)\,\mathrm{d}t=\frac{E_{2n}}{n2^{2n+1}},$
$n=1,2,\dots$ .
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex Referenced by: §24.13(ii) Permalink: http://dlmf.nist.gov/24.13.E10 Encodings: TeX, pMML, png See also: Annotations for §24.13(ii), §24.13 and Ch.24

For $m,n=1,2,\dotsc$ ,

24.13.11		$\int_{0}^{1}E_{n}\left(t\right)E_{m}\left(t\right)\,\mathrm{d}t=(-1)^{n}4\frac% {(2^{m+n+2}-1)m!n!}{(m+n+2)!}B_{m+n+2}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $m$ : integer, $n$ : integer and $t$ : real or complex A&S Ref: 23.1.12 Referenced by: §24.13(ii) Permalink: http://dlmf.nist.gov/24.13.E11 Encodings: TeX, pMML, png See also: Annotations for §24.13(ii), §24.13 and Ch.24

§24.13(iii) Compendia

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Keywords:: Bernoulli polynomials, Euler polynomials, Laplace transforms, compendia, integrals
Permalink:: http://dlmf.nist.gov/24.13.iii
See also:: Annotations for §24.13 and Ch.24

For Laplace and inverse Laplace transforms see Prudnikov et al. (1992a, §§3.28.1–3.28.2) and Prudnikov et al. (1992b, §§3.26.1–3.26.2). For other integrals see Prudnikov et al. (1990, pp. 55–57).

24.12 Zeros 24.14 Sums

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§24.13 Integrals

Contents

§24.13(i) Bernoulli Polynomials

§24.13(ii) Euler Polynomials

§24.13(iii) Compendia