# SL(2,Z)

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## 11—20 of 798 matching pages

##### 11: 19.19 Taylor and Related Series
For $N=0,1,2,\dots$ define the homogeneous hypergeometric polynomial … If $n=2$, then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)). … and define the $n$-tuple $\mathbf{\tfrac{1}{2}}=(\tfrac{1}{2},\dots,\tfrac{1}{2})$. … The number of terms in $T_{N}$ can be greatly reduced by using variables $\mathbf{Z}=\boldsymbol{{1}}-(\mathbf{z}/A)$ with $A$ chosen to make $E_{1}(\mathbf{Z})=0$. …
19.19.7 $R_{-a}\left(\boldsymbol{{\tfrac{1}{2}}};\mathbf{z}\right)=A^{-a}\sum_{N=0}^{% \infty}\frac{{\left(a\right)_{N}}}{{\left(\tfrac{1}{2}n\right)_{N}}}T_{N}(% \boldsymbol{{\tfrac{1}{2}}},\mathbf{Z}),$
##### 12: 10.44 Sums
10.44.1 $\mathscr{Z}_{\nu}\left(\lambda z\right)=\lambda^{\pm\nu}\sum_{k=0}^{\infty}% \frac{(\lambda^{2}-1)^{k}(\frac{1}{2}z)^{k}}{k!}\mathscr{Z}_{\nu\pm k}\left(z% \right),$ $|\lambda^{2}-1|<1$.
If $\mathscr{Z}=I$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary. …
10.44.3 $\mathscr{Z}_{\nu}\left(u\pm v\right)=\sum_{k=-\infty}^{\infty}(\pm 1)^{k}% \mathscr{Z}_{\nu+k}\left(u\right)I_{k}\left(v\right),$ $|v|<|u|$.
The restriction $|v|<|u|$ is unnecessary when $\mathscr{Z}=I$ and $\nu$ is an integer. …
10.44.6 $K_{n}\left(z\right)=\frac{n!(\tfrac{1}{2}z)^{-n}}{2}\sum_{k=0}^{n-1}(-1)^{k}% \frac{(\tfrac{1}{2}z)^{k}I_{k}\left(z\right)}{k!(n-k)}+(-1)^{n-1}\left(\ln% \left(\tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)I_{n}\left(z\right)+(-1)% ^{n}\sum_{k=1}^{\infty}\frac{(n+2k)I_{n+2k}\left(z\right)}{k(n+k)},$
##### 13: 22.16 Related Functions
With $q$ as in (22.2.1) and $\zeta=\pi x/(2K)$, … In Equations (22.16.24)–(22.16.26), $-2K. … where $\xi=x/{\theta_{3}}^{2}\left(0,q\right)$. … (Sometimes in the literature $\mathrm{Z}\left(x|k\right)$ is denoted by $\mathrm{Z}(\operatorname{am}\left(x,k\right),k^{2})$.) … $\mathrm{Z}\left(x|k\right)$ satisfies the same quasi-addition formula as the function $\mathcal{E}\left(x,k\right)$, given by (22.16.27). …
##### 14: 10.43 Integrals
Let $\mathscr{Z}_{\nu}\left(z\right)$ be defined as in §10.25(ii). …
$\int z^{\nu+1}\mathscr{Z}_{\nu}\left(z\right)\mathrm{d}z=z^{\nu+1}\mathscr{Z}_% {\nu+1}\left(z\right),$
10.43.2 $\int z^{\nu}\mathscr{Z}_{\nu}\left(z\right)\mathrm{d}z=\pi^{\frac{1}{2}}2^{\nu% -1}\Gamma\left(\nu+\tfrac{1}{2}\right)z\*\left(\mathscr{Z}_{\nu}\left(z\right)% \mathbf{L}_{\nu-1}\left(z\right)-\mathscr{Z}_{\nu-1}\left(z\right)\mathbf{L}_{% \nu}\left(z\right)\right),$ $\nu\neq-\tfrac{1}{2}$.
$\int e^{\pm z}z^{\nu}\mathscr{Z}_{\nu}\left(z\right)\mathrm{d}z=\frac{e^{\pm z% }z^{\nu+1}}{2\nu+1}\left(\mathscr{Z}_{\nu}\left(z\right)\mp\mathscr{Z}_{\nu+1}% \left(z\right)\right),$ $\nu\neq-\tfrac{1}{2}$,
$\int e^{\pm z}z^{-\nu}\mathscr{Z}_{\nu}\left(z\right)\mathrm{d}z=\frac{e^{\pm z% }z^{-\nu+1}}{1-2\nu}\left(\mathscr{Z}_{\nu}\left(z\right)\mp\mathscr{Z}_{\nu-1% }\left(z\right)\right),$ $\nu\neq\tfrac{1}{2}$.
##### 15: 10.25 Definitions
In particular, the principal branch of $I_{\nu}\left(z\right)$ is defined in a similar way: it corresponds to the principal value of $(\tfrac{1}{2}z)^{\nu}$, is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. … as $z\to\infty$ in $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$ $(<\tfrac{3}{2}\pi)$. …
###### Symbol $\mathscr{Z}_{\nu}\left(z\right)$
Corresponding to the symbol $\mathscr{C}_{\nu}$ introduced in §10.2(ii), we sometimes use $\mathscr{Z}_{\nu}\left(z\right)$ to denote $I_{\nu}\left(z\right)$, $e^{\nu\pi i}K_{\nu}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. …
##### 16: 22.21 Tables
Curtis (1964b) tabulates $\operatorname{sn}\left(mK/n,k\right)$, $\operatorname{cn}\left(mK/n,k\right)$, $\operatorname{dn}\left(mK/n,k\right)$ for $n=2(1)15$, $m=1(1)n-1$, and $q$ (not $k$) $=0(.005)0.35$ to 20D. Lawden (1989, pp. 280–284 and 293–297) tabulates $\operatorname{sn}\left(x,k\right)$, $\operatorname{cn}\left(x,k\right)$, $\operatorname{dn}\left(x,k\right)$, $\mathcal{E}\left(x,k\right)$, $\mathrm{Z}\left(x|k\right)$ to 5D for $k=0.1(.1)0.9$, $x=0(.1)X$, where $X$ ranges from 1. 5 to 2. 2. … Zhang and Jin (1996, p. 678) tabulates $\operatorname{sn}\left(Kx,k\right)$, $\operatorname{cn}\left(Kx,k\right)$, $\operatorname{dn}\left(Kx,k\right)$ for $k=\frac{1}{4},\frac{1}{2}$ and $x=0(.1)4$ to 7D. …
##### 17: 35.1 Special Notation
 $a,b$ complex variables. … complex symmetric matrix. … zonal polynomials.
The main functions treated in this chapter are the multivariate gamma and beta functions, respectively $\Gamma_{m}\left(a\right)$ and $\mathrm{B}_{m}\left(a,b\right)$, and the special functions of matrix argument: Bessel (of the first kind) $A_{\nu}\left(\mathbf{T}\right)$ and (of the second kind) $B_{\nu}\left(\mathbf{T}\right)$; confluent hypergeometric (of the first kind) ${{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)$ or $\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)$ and (of the second kind) $\Psi\left(a;b;\mathbf{T}\right)$; Gaussian hypergeometric ${{}_{2}F_{1}}\left(a_{1},a_{2};b;\mathbf{T}\right)$ or $\displaystyle{{}_{2}F_{1}}\left({a_{1},a_{2}\atop b};\mathbf{T}\right)$; generalized hypergeometric ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ or $\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)$. An alternative notation for the multivariate gamma function is $\Pi_{m}(a)=\Gamma_{m}\left(a+\tfrac{1}{2}(m+1)\right)$ (Herz (1955, p. 480)). Related notations for the Bessel functions are $\mathcal{J}_{\nu+\frac{1}{2}(m+1)}(\mathbf{T})=A_{\nu}\left(\mathbf{T}\right)/% A_{\nu}\left(\boldsymbol{{0}}\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_{m}(0,\dots,0,\nu\mathpunct{|}\mathbf{S},\mathbf{T})=\left|\mathbf{T}\right|% ^{\nu}B_{\nu}\left(\mathbf{S}\mathbf{T}\right)$ (Terras (1988, pp. 49–64)), and $\mathcal{K}_{\nu}(\mathbf{T})=\left|\mathbf{T}\right|^{\nu}B_{\nu}\left(% \mathbf{S}\mathbf{T}\right)$ (Faraut and Korányi (1994, pp. 357–358)).
##### 18: 10.66 Expansions in Series of Bessel Functions
10.66.1 $\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu}x=\sum_{k=0}^{\infty}\frac{% e^{(3\nu+k)\pi i/4}x^{k}J_{\nu+k}\left(x\right)}{2^{k/2}k!}=\sum_{k=0}^{\infty% }\frac{e^{(3\nu+3k)\pi i/4}x^{k}I_{\nu+k}\left(x\right)}{2^{k/2}k!}.$
$\operatorname{ber}_{n}\left(x\sqrt{2}\right)=\sum_{k=-\infty}^{\infty}(-1)^{n+% k}J_{n+2k}\left(x\right)I_{2k}\left(x\right),$
$\operatorname{bei}_{n}\left(x\sqrt{2}\right)=\sum_{k=-\infty}^{\infty}(-1)^{n+% k}J_{n+2k+1}\left(x\right)I_{2k+1}\left(x\right).$
##### 19: 19.36 Methods of Computation
where, in the notation of (19.19.7) with $a=-\frac{1}{2}$ and $n=3$, … where $n=0,1,2,\dots$, and … This method loses significant figures in $\rho$ if $\alpha^{2}$ and $k^{2}$ are nearly equal unless they are given exact values—as they can be for tables. If $\alpha^{2}=k^{2}$, then the method fails, but the function can be expressed by (19.6.13) in terms of $E\left(\phi,k\right)$, for which Neuman (1969b) uses ascending Landen transformations. … The cases $k_{c}^{2}/2\leq p<\infty$ and $-\infty require different treatment for numerical purposes, and again precautions are needed to avoid cancellations. …
##### 20: 22.1 Special Notation
 $x,y$ real variables. … complementary modulus, $k^{2}+{k^{\prime}}^{2}=1$. If $k\in[0,1]$, then $k^{\prime}\in[0,1]$. …
The functions treated in this chapter are the three principal Jacobian elliptic functions $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, $\operatorname{dn}\left(z,k\right)$; the nine subsidiary Jacobian elliptic functions $\operatorname{cd}\left(z,k\right)$, $\operatorname{sd}\left(z,k\right)$, $\operatorname{nd}\left(z,k\right)$, $\operatorname{dc}\left(z,k\right)$, $\operatorname{nc}\left(z,k\right)$, $\operatorname{sc}\left(z,k\right)$, $\operatorname{ns}\left(z,k\right)$, $\operatorname{ds}\left(z,k\right)$, $\operatorname{cs}\left(z,k\right)$; the amplitude function $\operatorname{am}\left(x,k\right)$; Jacobi’s epsilon and zeta functions $\mathcal{E}\left(x,k\right)$ and $\mathrm{Z}\left(x|k\right)$. … Other notations for $\operatorname{sn}\left(z,k\right)$ are $\mathrm{sn}(z\mathpunct{|}m)$ and $\mathrm{sn}(z,m)$ with $m=k^{2}$; see Abramowitz and Stegun (1964) and Walker (1996). …