24 Bernoulli and Euler Polynomials Properties 24.12 Zeros 24.14 Sums

§24.13 Integrals

Permalink:: http://dlmf.nist.gov/24.13

§24.13(i) Bernoulli Polynomials
§24.13(ii) Euler Polynomials
§24.13(iii) Compendia

§24.13(i) Bernoulli Polynomials

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).
Keywords:: Bernoulli polynomials, integrals
Permalink:: http://dlmf.nist.gov/24.13.i

24.13.1 $\int\mathop{B_{{n}}\/}\nolimits\!\left(t\right)dt=\frac{\mathop{B_{{n+1}}\/}% \nolimits\!\left(t\right)}{n+1}+\text{const.},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, : differential of , $\int$ : integral, : integer and : 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

24.13.2 $\int_{x}^{{x+1}}\mathop{B_{{n}}\/}\nolimits\!\left(t\right)dt=x^{n},$ $n=1,2,\dots$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, : differential of , $\int$ : integral, : integer, : real or complex and : real or complex
Referenced by:: §24.13(i)
Permalink:: http://dlmf.nist.gov/24.13.E2
Encodings:: TeX, pMML, png

24.13.3 $\int_{x}^{{x+(1/2)}}\mathop{B_{{n}}\/}\nolimits\!\left(t\right)dt=\frac{% \mathop{E_{{n}}\/}\nolimits\!\left(2x\right)}{2^{{n+1}}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, : differential of , $\int$ : integral, : integer, : real or complex and : real or complex
Referenced by:: §24.13(i)
Permalink:: http://dlmf.nist.gov/24.13.E3
Encodings:: TeX, pMML, png

24.13.4 $\int_{0}^{{1/2}}\mathop{B_{{n}}\/}\nolimits\!\left(t\right)dt=\frac{1-2^{{n+1}% }}{2^{n}}\frac{\mathop{B_{{n+1}}\/}\nolimits}{n+1},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, : differential of , $\int$ : integral, : integer and : real or complex
Permalink:: http://dlmf.nist.gov/24.13.E4
Encodings:: TeX, pMML, png

24.13.5 $\int_{{1/4}}^{{3/4}}\mathop{B_{{n}}\/}\nolimits\!\left(t\right)dt=\frac{% \mathop{E_{{n}}\/}\nolimits}{2^{{2n+1}}}.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, : differential of , $\int$ : integral, : integer and : real or complex
Referenced by:: §24.13(i)
Permalink:: http://dlmf.nist.gov/24.13.E5
Encodings:: TeX, pMML, png

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

24.13.6 $\int_{0}^{1}\mathop{B_{{n}}\/}\nolimits\!\left(t\right)\mathop{B_{{m}}\/}% \nolimits\!\left(t\right)dt=\frac{(-1)^{{n-1}}m!n!}{(m+n)!}\mathop{B_{{m+n}}\/% }\nolimits.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, : differential of , : : factorial, $\int$ : integral, : integer, : integer and : 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

§24.13(ii) Euler Polynomials

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

24.13.7 $\int\mathop{E_{{n}}\/}\nolimits\!\left(t\right)dt=\frac{\mathop{E_{{n+1}}\/}% \nolimits\!\left(t\right)}{n+1}+\text{const.},$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, : differential of , $\int$ : integral, : integer and : 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

24.13.8 $\int_{0}^{1}\mathop{E_{{n}}\/}\nolimits\!\left(t\right)dt=-2\frac{\mathop{E_{{% n+1}}\/}\nolimits\!\left(0\right)}{n+1}=\frac{4(2^{{n+2}}-1)}{(n+1)(n+2)}% \mathop{B_{{n+2}}\/}\nolimits,$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, : differential of , $\int$ : integral, : integer and : real or complex
Permalink:: http://dlmf.nist.gov/24.13.E8
Encodings:: TeX, pMML, png

24.13.9 $\int_{0}^{{1/2}}\mathop{E_{{2n}}\/}\nolimits\!\left(t\right)dt=-\frac{\mathop{% E_{{2n+1}}\/}\nolimits\!\left(0\right)}{2n+1}=\frac{2(2^{{2n+2}}-1)\mathop{B_{% {2n+2}}\/}\nolimits}{(2n+1)(2n+2)},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, : differential of , $\int$ : integral, : integer and : real or complex
Permalink:: http://dlmf.nist.gov/24.13.E9
Encodings:: TeX, pMML, png

24.13.10 $\int_{0}^{{1/2}}\mathop{E_{{2n-1}}\/}\nolimits\!\left(t\right)dt=\frac{\mathop% {E_{{2n}}\/}\nolimits}{n2^{{2n+1}}},$ $n=1,2,\dots$ .

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, : differential of , $\int$ : integral, : integer and : real or complex
Referenced by:: §24.13(ii)
Permalink:: http://dlmf.nist.gov/24.13.E10
Encodings:: TeX, pMML, png

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

24.13.11 $\int_{0}^{1}\mathop{E_{{n}}\/}\nolimits\!\left(t\right)\mathop{E_{{m}}\/}% \nolimits\!\left(t\right)dt=(-1)^{n}4\frac{(2^{{m+n+2}}-1)m!n!}{(m+n+2)!}% \mathop{B_{{m+n+2}}\/}\nolimits.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, : differential of , : : factorial, $\int$ : integral, : integer, : integer and : 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

§24.13(iii) Compendia

Keywords:: Bernoulli polynomials, Euler polynomials, integrals
Permalink:: http://dlmf.nist.gov/24.13.iii

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).

§24.13 Integrals

Contents

§24.13(i) Bernoulli Polynomials

§24.13(ii) Euler Polynomials

§24.13(iii) Compendia