# SL(2,Z)

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##### 1: 23.15 Definitions
$k=\frac{{\theta_{2}}^{2}\left(0,q\right)}{{\theta_{3}}^{2}\left(0,q\right)},$
The set of all bilinear transformations of this form is denoted by SL $(2,\mathbb{Z})$ (Serre (1973, p. 77)). A modular function $f(\tau)$ is a function of $\tau$ that is meromorphic in the half-plane $\Im\tau>0$, and has the property that for all $\mathcal{A}\in\mbox{SL}(2,\mathbb{Z})$, or for all $\mathcal{A}$ belonging to a subgroup of SL $(2,\mathbb{Z})$, …(Some references refer to $2\ell$ as the level). …
##### 2: 10.29 Recurrence Relations and Derivatives
With $\mathscr{Z}_{\nu}\left(z\right)$ defined as in §10.25(ii),
$\mathscr{Z}_{\nu-1}\left(z\right)-\mathscr{Z}_{\nu+1}\left(z\right)=(2\nu/z)% \mathscr{Z}_{\nu}\left(z\right),$
$\mathscr{Z}_{\nu-1}\left(z\right)+\mathscr{Z}_{\nu+1}\left(z\right)=2\mathscr{% Z}_{\nu}'\left(z\right).$
For results on modified quotients of the form $\ifrac{z\mathscr{Z}_{\nu\pm 1}\left(z\right)}{\mathscr{Z}_{\nu}\left(z\right)}$ see Onoe (1955) and Onoe (1956). … For $k=0,1,2,\dotsc$, …
##### 3: 15.17 Mathematical Applications
First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL $(2,\mathbb{R})$, and spherical functions on certain nonsymmetric Gelfand pairs. …
##### 4: 23.18 Modular Transformations
and $\lambda\left(\tau\right)$ is a cusp form of level zero for the corresponding subgroup of SL $(2,\mathbb{Z})$. … $J\left(\tau\right)$ is a modular form of level zero for SL $(2,\mathbb{Z})$. …
23.18.5 $\eta\left(\mathcal{A}\tau\right)=\varepsilon(\mathcal{A})\left(-i(c\tau+d)% \right)^{1/2}\eta\left(\tau\right),$
23.18.7 ${s(d,c)=\sum_{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\left\lfloor\frac{dr}{c}% \right\rfloor-\frac{1}{2}\right),}$ $c>0$.
Note that $\eta\left(\tau\right)$ is of level $\tfrac{1}{2}$. …
##### 5: 10.36 Other Differential Equations
The quantity $\lambda^{2}$ in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by $-\lambda^{2}$ if at the same time the symbol $\mathscr{C}$ in the given solutions is replaced by $\mathscr{Z}$. …
10.36.1 $z^{2}(z^{2}+\nu^{2})w^{\prime\prime}+z(z^{2}+3\nu^{2})w^{\prime}-\left((z^{2}+% \nu^{2})^{2}+z^{2}-\nu^{2}\right)w=0,$ $w=\mathscr{Z}_{\nu}'\left(z\right)$,
10.36.2 ${z^{2}w^{\prime\prime}+z(1\pm 2z)w^{\prime}+(\pm z-\nu^{2})w=0},$ $w=e^{\mp z}\mathscr{Z}_{\nu}\left(z\right)$.
##### 6: 33.22 Particle Scattering and Atomic and Molecular Spectra
With $e$ denoting here the elementary charge, the Coulomb potential between two point particles with charges $Z_{1}e,Z_{2}e$ and masses $m_{1},m_{2}$ separated by a distance $s$ is $V(s)=Z_{1}Z_{2}e^{2}/(4\pi\varepsilon_{0}s)=Z_{1}Z_{2}\alpha\hbar c/s$, where $Z_{j}$ are atomic numbers, $\varepsilon_{0}$ is the electric constant, $\alpha$ is the fine structure constant, and $\hbar$ is the reduced Planck’s constant. … In these applications, the $Z$-scaled variables $r$ and $\epsilon$ are more convenient.
###### $Z$ Scaling
The $Z$-scaled variables $r$ and $\epsilon$ of §33.14 are given by … For $Z_{1}Z_{2}=-1$ and $m=m_{e}$, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, $a_{0}=\hbar/(m_{e}c\alpha)$, and to a multiple of the Rydberg constant, …
##### 8: 35.2 Laplace Transform
For any complex symmetric matrix $\mathbf{Z}$, … Then (35.2.1) converges absolutely on the region $\Re\left(\mathbf{Z}\right)>\mathbf{X}_{0}$, and $g(\mathbf{Z})$ is a complex analytic function of all elements $z_{j,k}$ of $\mathbf{Z}$. …
35.2.2 $f(\mathbf{X})=\dfrac{1}{(2\pi\mathrm{i})^{m(m+1)/2}}\int\operatorname{etr}% \left(\mathbf{Z}\mathbf{X}\right)g(\mathbf{Z})\,\mathrm{d}{\mathbf{Z}},$
where the integral is taken over all $\mathbf{Z}=\mathbf{U}+\mathrm{i}\mathbf{V}$ such that $\mathbf{U}>\mathbf{X}_{0}$ and $\mathbf{V}$ ranges over $\boldsymbol{\mathcal{S}}$. … If $g_{j}$ is the Laplace transform of $f_{j}$, $j=1,2$, then $g_{1}g_{2}$ is the Laplace transform of the convolution $f_{1}*f_{2}$, where …
##### 9: 25.10 Zeros
Calculations relating to the zeros on the critical line make use of the real-valued function …is chosen to make $Z(t)$ real, and $\operatorname{ph}\Gamma\left(\frac{1}{4}+\frac{1}{2}it\right)$ assumes its principal value. Because $|Z(t)|=|\zeta\left(\frac{1}{2}+it\right)|$, $Z(t)$ vanishes at the zeros of $\zeta\left(\frac{1}{2}+it\right)$, which can be separated by observing sign changes of $Z(t)$. Because $Z(t)$ changes sign infinitely often, $\zeta\left(\frac{1}{2}+it\right)$ has infinitely many zeros with $t$ real. … Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\exp\left(i\vartheta(t)\right)$ to obtain the Riemann–Siegel formula: …
##### 10: 35.4 Partitions and Zonal Polynomials
For any partition $\kappa$, the zonal polynomial $Z_{\kappa}:\boldsymbol{\mathcal{S}}\to\mathbb{R}$ is defined by the properties … Therefore $Z_{\kappa}\left(\mathbf{T}\right)$ is a symmetric polynomial in the eigenvalues of $\mathbf{T}$. … For $k=0,1,2,\dots$, … For $\mathbf{T}\in{\boldsymbol{\Omega}}$ and $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$,
35.4.8 $\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)% \,\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}Z_{\kappa}\left(\mathbf{X}\right% )\,\mathrm{d}{\mathbf{X}}=\Gamma_{m}\left(a+\kappa\right)\,\left|\mathbf{T}% \right|^{-a}Z_{\kappa}\left(\mathbf{T}^{-1}\right),$