§24.11 Asymptotic Approximations

As $n\to\infty$

 24.11.1 $\displaystyle(-1)^{n+1}B_{2n}$ $\displaystyle\sim\frac{2(2n)!}{(2\pi)^{2n}},$ 24.11.2 $\displaystyle(-1)^{n+1}B_{2n}$ $\displaystyle\sim 4\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n},$ 24.11.3 $\displaystyle(-1)^{n}E_{2n}$ $\displaystyle\sim\frac{2^{2n+2}(2n)!}{\pi^{2n+1}},$ 24.11.4 $\displaystyle(-1)^{n}E_{2n}$ $\displaystyle\sim 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.$ ⓘ 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 $n$: integer Referenced by: §24.11 Permalink: http://dlmf.nist.gov/24.11.E4 Encodings: TeX, pMML, png See also: Annotations for 24.11 and 24

Also,

 24.11.5 $\displaystyle(-1)^{\left\lfloor n/2\right\rfloor-1}\frac{(2\pi)^{n}}{2(n!)}B_{% n}\left(x\right)$ $\displaystyle\to\begin{cases}\cos\left(2\pi x\right),&n\text{ even},\\ \sin\left(2\pi x\right),&n\text{ odd},\end{cases}$ 24.11.6 $\displaystyle(-1)^{\left\lfloor(n+1)/2\right\rfloor}\frac{\pi^{n+1}}{4(n!)}E_{% n}\left(x\right)$ $\displaystyle\to\begin{cases}\sin\left(\pi x\right),&n\text{ even},\\ \cos\left(\pi x\right),&n\text{ odd},\end{cases}$

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

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