# .新宝2世界杯结束『世界杯佣金分红55%，咨询专员：@ky975』.tbx-k2q1w9-2022年11月29日4时59分17秒

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## 1—10 of 816 matching pages

##### 1: 4.17 Special Values and Limits
4.17.3 $\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}.$
##### 2: 28.6 Expansions for Small $q$
28.6.2 $a_{1}\left(q\right)=1+q-\tfrac{1}{8}q^{2}-\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^% {4}+\tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}+\tfrac{55}{94\;37184}q^{7% }-\tfrac{83}{353\;89440}q^{8}+\cdots,$
28.6.3 $b_{1}\left(q\right)=1-q-\tfrac{1}{8}q^{2}+\tfrac{1}{64}q^{3}-\tfrac{1}{1536}q^% {4}-\tfrac{11}{36864}q^{5}+\tfrac{49}{5\;89824}q^{6}-\tfrac{55}{94\;37184}q^{7% }-\tfrac{83}{353\;89440}q^{8}+\cdots,$
The coefficients of the power series of $a_{2n}\left(q\right)$, $b_{2n}\left(q\right)$ and also $a_{2n+1}\left(q\right)$, $b_{2n+1}\left(q\right)$ are the same until the terms in $q^{2n-2}$ and $q^{2n}$, respectively. … Here $j=1$ for $a_{2n}\left(q\right)$, $j=2$ for $b_{2n+2}\left(q\right)$, and $j=3$ for $a_{2n+1}\left(q\right)$ and $b_{2n+1}\left(q\right)$. … where $k$ is the unique root of the equation $2E\left(k\right)=K\left(k\right)$ in the interval $(0,1)$, and $k^{\prime}=\sqrt{1-k^{2}}$. …
##### 3: 25.20 Approximations
• Cody et al. (1971) gives rational approximations for $\zeta\left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\leq s\leq 5$, $5\leq s\leq 11$, $11\leq s\leq 25$, $25\leq s\leq 55$. Precision is varied, with a maximum of 20S.

• Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta\left(s+1\right)$ and $\zeta\left(s+k\right)$, $k=2,3,4,5,8$, for $0\leq s\leq 1$ (23D).

• Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover $\zeta\left(s\right)$ for $0\leq s\leq 1$ (15D), $\zeta\left(s+1\right)$ for $0\leq s\leq 1$ (20D), and $\ln\xi\left(\tfrac{1}{2}+ix\right)$25.4) for $-1\leq x\leq 1$ (20D). For errata see Piessens and Branders (1972).

• Morris (1979) gives rational approximations for $\operatorname{Li}_{2}\left(x\right)$25.12(i)) for $0.5\leq x\leq 1$. Precision is varied with a maximum of 24S.

• Antia (1993) gives minimax rational approximations for $\Gamma\left(s+1\right)F_{s}(x)$, where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals $-\infty and $2\leq x<\infty$, with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$. For each $s$ there are three sets of approximations, with relative maximum errors $10^{-4},10^{-8},10^{-12}$.

• ##### 4: 24.2 Definitions and Generating Functions
$B_{2n+1}=0$ ,
$(-1)^{n+1}B_{2n}>0$ , $n=1,2,\dots$.
$E_{2n+1}=0$ ,
$(-1)^{n}E_{2n}>0$ .
24.2.9 $E_{n}=2^{n}E_{n}\left(\tfrac{1}{2}\right)=\text{integer},$
##### 5: 28.16 Asymptotic Expansions for Large $q$
Let $s=2m+1$, $m=0,1,2,\dots$, and $\nu$ be fixed with $m<\nu. …
28.16.1 $\lambda_{\nu}\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{1}{8}(s^{2}+1)-\dfrac{1}% {2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\dfrac{1}{2^{17}h^% {3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}% +486)-\dfrac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.$
##### 6: 28.15 Expansions for Small $q$
28.15.1 $\lambda_{\nu}\left(q\right)=\nu^{2}+\frac{1}{2(\nu^{2}-1)}q^{2}+\frac{5\nu^{2}% +7}{32(\nu^{2}-1)^{3}(\nu^{2}-4)}q^{4}+\frac{9\nu^{4}+58\nu^{2}+29}{64(\nu^{2}% -1)^{5}(\nu^{2}-4)(\nu^{2}-9)}q^{6}+\cdots.$
28.15.2 $a-\nu^{2}-\cfrac{q^{2}}{a-(\nu+2)^{2}-\cfrac{q^{2}}{a-(\nu+4)^{2}-\cdots}}=% \cfrac{q^{2}}{a-(\nu-2)^{2}-\cfrac{q^{2}}{a-(\nu-4)^{2}-\cdots}}.$
28.15.3 $\operatorname{me}_{\nu}\left(z,q\right)=e^{\mathrm{i}\nu z}-\frac{q}{4}\left(% \frac{1}{\nu+1}e^{\mathrm{i}(\nu+2)z}-\frac{1}{\nu-1}e^{\mathrm{i}(\nu-2)z}% \right)+\frac{q^{2}}{32}\left(\frac{1}{(\nu+1)(\nu+2)}e^{\mathrm{i}(\nu+4)z}+% \frac{1}{(\nu-1)(\nu-2)}e^{\mathrm{i}(\nu-4)z}-\frac{2(\nu^{2}+1)}{(\nu^{2}-1)% ^{2}}e^{\mathrm{i}\nu z}\right)+\cdots;$
##### 7: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
For $k=2$ $M_{2}$ is the number of permutations of $\{1,2,\ldots,n\}$ with $a_{1}$ cycles of length 1, $a_{2}$ cycles of length 2, $\dotsc$, and $a_{n}$ cycles of length $n$: …$M_{3}$ is the number of set partitions of $\{1,2,\ldots,n\}$ with $a_{1}$ subsets of size 1, $a_{2}$ subsets of size 2, $\dotsc$, and $a_{n}$ subsets of size $n$: …For each $n$ all possible values of $a_{1},a_{2},\ldots,a_{n}$ are covered. … where the summation is over all nonnegative integers $n_{1},n_{2},\ldots,n_{k}$ such that $n_{1}+n_{2}+\cdots+n_{k}=n$. …
##### 8: 10.12 Generating Function and Associated Series
$\cos\left(z\sin\theta\right)=J_{0}\left(z\right)+2\sum_{k=1}^{\infty}J_{2k}% \left(z\right)\cos\left(2k\theta\right),$
$\sin\left(z\sin\theta\right)=2\sum_{k=0}^{\infty}J_{2k+1}\left(z\right)\sin% \left((2k+1)\theta\right),$
$\cos\left(z\cos\theta\right)=J_{0}\left(z\right)+2\sum_{k=1}^{\infty}(-1)^{k}J% _{2k}\left(z\right)\cos\left(2k\theta\right),$
$\cos z=J_{0}\left(z\right)-2J_{2}\left(z\right)+2J_{4}\left(z\right)-2J_{6}% \left(z\right)+\dotsb,$
$\sin z=2J_{1}\left(z\right)-2J_{3}\left(z\right)+2J_{5}\left(z\right)-\dotsb,$
##### 9: 24.19 Methods of Computation
If $\widetilde{N}_{2n}$ denotes the right-hand side of (24.19.1) but with the second product taken only for $p\leq\left\lfloor(\pi e)^{-1}2n\right\rfloor+1$, then $N_{2n}=\left\lceil\widetilde{N}_{2n}\right\rceil$ for $n\geq 2$. … For other information see Chellali (1988) and Zhang and Jin (1996, pp. 1–11). … For number-theoretic applications it is important to compute $B_{2n}\pmod{p}$ for $2n\leq p-3$; in particular to find the irregular pairs $(2n,p)$ for which $B_{2n}\equiv 0\pmod{p}$. …
• Buhler et al. (1992) uses the expansion

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

• A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs $(2n,p)$. Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

• ##### 10: 4.19 Maclaurin Series and Laurent Series
4.19.3 $\tan z=z+\frac{z^{3}}{3}+\frac{2}{15}z^{5}+\frac{17}{315}z^{7}+\cdots+\frac{(-% 1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$ $|z|<\frac{1}{2}\pi$,
4.19.4 $\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}z^{3}+\frac{31}{15120}z^{5}+\cdots% +\frac{(-1)^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$ $0<|z|<\pi$,
4.19.8 $\ln\left(\cos z\right)=\sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}(2^{2n}-1)B_{2% n}}{n(2n)!}z^{2n},$ $|z|<\frac{1}{2}\pi$,
4.19.9 $\ln\left(\frac{\tan z}{z}\right)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{2n}(2^{% 2n-1}-1)B_{2n}}{n(2n)!}z^{2n},$ $|z|<\frac{1}{2}\pi$.