# SL(2,Z) bilinear transformation

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##### 2: 23.15 Definitions
Also $\mathcal{A}$ denotes a bilinear transformation on $\tau$, given by …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). …
##### 3: 23.18 Modular Transformations
###### §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 $\eta\left(\mathcal{A}\tau\right)=\varepsilon(\mathcal{A})\left(-i(c\tau+d)% \right)^{1/2}\eta\left(\tau\right),$
Note that $\eta\left(\tau\right)$ is of level $\tfrac{1}{2}$. …
##### 4: 15.17 Mathematical Applications
The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … 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. Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform1.14(ii)) or as a specialization of a group Fourier transform. … Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …
##### 5: 35.2 Laplace Transform
###### §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 $\mathbf{Z}$. … 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 …
##### 6: 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 $k=0,1,2,\dotsc$, …
##### 7: 18.39 Applications in the Physical Sciences
This is also the normalization and notation of Chapter 33 for $Z=1$ , and the notation of Weinberg (2013, Chapter 2). … Thus the $c_{N}(x)=P^{(l+1)}_{N}\left(x;\frac{2Z}{s},-\frac{2Z}{s}\right)$ and the eigenvalues …are determined by the $N$ zeros, $x^{N}_{i}$ of the Pollaczek polynomial $P^{(l+1)}_{N}\left(x;\frac{2Z}{s},-\frac{2Z}{s}\right)$. … The polynomials $P^{(l+1)}_{N}\left(x;\frac{2Z}{s},-\frac{2Z}{s}\right)$, for both positive and negative $Z$, define the Coulomb–Pollaczek polynomials (CP OP’s in what follows), see Yamani and Reinhardt (1975, Appendix B, and §IV). … Note that violation of the Favard inequality, $l+1+(2Z/s)>0,$ possible when $Z<0$, results in a zero or negative weight function. …
##### 8: 32.2 Differential Equations
They are distinct modulo Möbius (bilinear) transformationsIn $\mbox{P}_{\mbox{\scriptsize III}}$, if $w(z)=\zeta^{-1/2}u(\zeta)$ with $\zeta=z^{2}$, then … In $\mbox{P}_{\mbox{\scriptsize IV}}$, if $w(z)=2\sqrt{2}(u(\zeta))^{2}$ with $\zeta=\sqrt{2}z$ and $\alpha=2\nu+1$, then … where $\mu_{1}$, $\mu_{2}$, $\mu_{3}$ are constants, $f_{1}$, $f_{2}$, $f_{3}$ are functions of $z$, with … where $\mu_{1}$, $\mu_{2}$, $\mu_{3}$, $\mu_{4}$ are constants, $f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$ are functions of $z$, with …
##### 9: 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 ${z^{2}w^{\prime\prime}+z(1\pm 2z)w^{\prime}+(\pm z-\nu^{2})w=0},$ $w=e^{\mp z}\mathscr{Z}_{\nu}\left(z\right)$.
##### 10: 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, …