SL(2,Z) bilinear transformation

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1: 1.14 Integral Transforms

§1.14 Integral Transforms

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§1.14(i) Fourier Transform

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§1.14(iii) Laplace Transform

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Fourier Transform

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Laplace Transform

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2: 23.15 Definitions

… ►Also $𝒜$ 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 $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{<span class="ltx\_text hilite">SL</span>}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL $(2,\mathbb{Z})$, …(Some references refer to $2\mathrm{\ell }$ 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 $$\eta \left(𝒜\tau \right)=\epsilon (𝒜){\left(-\mathrm{i}(c\tau +d)\right)}^{1/2}\eta \left(\tau \right),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{i}$: imaginary unit, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer and $\epsilon (𝒜)$: function

Permalink:

http://dlmf.nist.gov/23.18.E5

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Annotations for §23.18, §23.18 and Ch.23

… ►Note that $\eta \left(\tau \right)$ is of level $\frac{1}{2}$. …

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 $(2,ℝ)$, 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 transform (§1.14(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 $𝐙$, … ►Then (35.2.1) converges absolutely on the region $\mathrm{\Re }\left(𝐙\right)>{𝐗}_0$, and $g(𝐙)$ is a complex analytic function of all elements $z_{j,k}$ of $𝐙$. … ►where the integral is taken over all $𝐙=𝐔+\mathrm{i}𝐕$ such that $𝐔>{𝐗}_0$ and $𝐕$ ranges over $𝓢$. … ►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 $f_1\ast f_2$, where …

6: 10.29 Recurrence Relations and Derivatives

… ►With ${𝒵}_{\nu }\left(z\right)$ defined as in §10.25(ii), ►

${𝒵}_{\nu -1}\left(z\right)-{𝒵}_{\nu +1}\left(z\right)=(2\nu /z){𝒵}_{\nu }\left(z\right),$

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${𝒵}_{\nu -1}\left(z\right)+{𝒵}_{\nu +1}\left(z\right)=2{𝒵}_{\nu }^{\prime }\left(z\right).$

… ►For results on modified quotients of the form $z{𝒵}_{\nu \pm 1}\left(z\right)/{𝒵}_{\nu }\left(z\right)$ see Onoe (1955) and Onoe (1956). … ►For $k=0,1,2,\mathrm{\dots }$, …

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 $c_N(x)=P_N^{(l+1)}(x;\frac{2Z}{s},-\frac{2Z}{s})$ and the eigenvalues …are determined by the $N$ zeros, $x_i^N$ of the Pollaczek polynomial $P_N^{(l+1)}(x;\frac{2Z}{s},-\frac{2Z}{s})$. … ►The polynomials $P_N^{(l+1)}(x;\frac{2Z}{s},-\frac{2Z}{s})$, 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 , results in a zero or negative weight function. …

8: 32.2 Differential Equations

… ►They are distinct modulo Möbius (bilinear) transformations … ►In ${\text{P}}_{\text{III}}$, if $w(z)={\zeta }^{-1/2}u(\zeta )$ with $\zeta =z^2$, then … ►In ${\text{P}}_{\text{IV}}$, 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 $𝒞$ in the given solutions is replaced by $𝒵$. … ►

10.36.1 $$z^2(z^2+{\nu }^2)w^{\prime \prime }+z(z^2+3{\nu }^2)w^{\prime }-\left({(z^2+{\nu }^2)}^2+z^2-{\nu }^2\right)w=0,$$ $w={𝒵}_{\nu }^{\prime }\left(z\right)$,

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Symbols:

${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.36.E1

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Annotations for §10.36 and Ch.10

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10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

Encodings:

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See also:

Annotations for §10.36 and Ch.10

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10: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►With $\mathrm{e}$ denoting here the elementary charge, the Coulomb potential between two point particles with charges $Z_1\mathrm{e},Z_2\mathrm{e}$ and masses $m_1,m_2$ separated by a distance $s$ is $V(s)=Z_1{Z}_2{\mathrm{e}}^2/(4\pi {\epsilon }_{0}s)=Z_1{Z}_2\alpha \mathrm{\hslash }c/s$, where $Z_j$ are atomic numbers, ${\epsilon }_0$ is the electric constant, $\alpha$ is the fine structure constant, and $\mathrm{\hslash }$ is the reduced Planck’s constant. … ►In these applications, the $Z$-scaled variables $r$ and $ϵ$ are more convenient. ►

$Z$ Scaling

►The $Z$-scaled variables $r$ and $ϵ$ of §33.14 are given by … ►For $Z_1{Z}_2=-1$ and $m=m_{\mathrm{e}}$, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, $a_0=\mathrm{\hslash }/(m_{\mathrm{e}}c\alpha )$, and to a multiple of the Rydberg constant, …