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24 Bernoulli and Euler Polynomials Properties 24.10 Arithmetic Properties 24.12 Zeros

§24.11 Asymptotic Approximations

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Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials, asymptotic approximations
Notes:: For (24.11.1)–(24.11.4) see Leeming (1977). For (24.11.5) and (24.11.6) see Dilcher (1987a).
Referenced by:: Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/24.11
Replacement (effective with 1.0.11):: In the sentence at the end of this section the original reference to López and Temme (1999b) was replaced with a reference to López and Temme (1999c). (The reference to López and Temme (1999b) was moved to §24.16(i).)
See also:: Annotations for Ch.24

As $n\to\infty$

24.11.1 $(-1)^{n+1}B_{2n}\sim\frac{2(2n)!}{(2\pi)^{2n}},$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ) and : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §24.11 and Ch.24

24.11.2 $(-1)^{n+1}B_{2n}\sim 4\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n},$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and : integer
Permalink:: http://dlmf.nist.gov/24.11.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §24.11 and Ch.24

24.11.3 $(-1)^{n}E_{2n}\sim\frac{2^{2n+2}(2n)!}{\pi^{2n+1}},$

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ) and : integer
Permalink:: http://dlmf.nist.gov/24.11.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §24.11 and Ch.24

24.11.4 $(-1)^{n}E_{2n}\sim 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.$

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §24.11 and Ch.24

Also,

24.11.5 $(-1)^{\left\lfloor n/2\right\rfloor-1}\frac{(2\pi)^{n}}{2(n!)}B_{n}\left(x% \right)\to\begin{cases}\cos\left(2\pi x\right),&n\text{ even},\\ \sin\left(2\pi x\right),&n\text{ odd},\end{cases}$

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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, : factorial (as in ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of , $\sin\NVar{z}$ : sine function, : integer and : real or complex
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §24.11 and Ch.24

24.11.6 $(-1)^{\left\lfloor(n+1)/2\right\rfloor}\frac{\pi^{n+1}}{4(n!)}E_{n}\left(x% \right)\to\begin{cases}\sin\left(\pi x\right),&n\text{ even},\\ \cos\left(\pi x\right),&n\text{ odd},\end{cases}$

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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, : factorial (as in ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of , $\sin\NVar{z}$ : sine function, : integer and : real or complex
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §24.11 and Ch.24

uniformly for on compact subsets of $\mathbb{C}$ .

For further results see Temme (1995b) and López and Temme (1999c).

24.10 Arithmetic Properties 24.12 Zeros

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