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SL(2,Z) bilinear transformation

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1: 1.14 Integral Transforms
§1.14 Integral Transforms
§1.14(i) Fourier Transform
§1.14(iii) Laplace Transform
Fourier Transform
Laplace Transform
2: 23.15 Definitions
Also 𝒜 denotes a bilinear transformation on τ , given by …The set of all bilinear transformations of this form is denoted by SL ( 2 , ) (Serre (1973, p. 77)). A modular function f ( τ ) is a function of τ that is meromorphic in the half-plane τ > 0 , and has the property that for all 𝒜 SL ( 2 , ) , or for all 𝒜 belonging to a subgroup of SL ( 2 , ) , …(Some references refer to 2 as the level). …
3: 23.18 Modular Transformations
§23.18 Modular Transformations
and λ ( τ ) is a cusp form of level zero for the corresponding subgroup of SL ( 2 , ) . … J ( τ ) is a modular form of level zero for SL ( 2 , ) . …
23.18.5 η ( 𝒜 τ ) = ε ( 𝒜 ) ( i ( c τ + d ) ) 1 / 2 η ( τ ) ,
Note that η ( τ ) is of level 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 , ) , 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 𝐙 , … Then (35.2.1) converges absolutely on the region ( 𝐙 ) > 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 + 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 f 2 , where …
6: 10.29 Recurrence Relations and Derivatives
With 𝒵 ν ( z ) defined as in §10.25(ii),
𝒵 ν 1 ( z ) 𝒵 ν + 1 ( z ) = ( 2 ν / z ) 𝒵 ν ( z ) ,
𝒵 ν 1 ( z ) + 𝒵 ν + 1 ( z ) = 2 𝒵 ν ( z ) .
For results on modified quotients of the form z 𝒵 ν ± 1 ( z ) / 𝒵 ν ( z ) see Onoe (1955) and Onoe (1956). … For k = 0 , 1 , 2 , , …
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 ; 2 Z s , 2 Z s ) and the eigenvalues …are determined by the N zeros, x i N of the Pollaczek polynomial P N ( l + 1 ) ( x ; 2 Z s , 2 Z s ) . … The polynomials P N ( l + 1 ) ( x ; 2 Z s , 2 Z 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 + ( 2 Z / 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 P III , if w ( z ) = ζ 1 / 2 u ( ζ ) with ζ = z 2 , then … In P IV , if w ( z ) = 2 2 ( u ( ζ ) ) 2 with ζ = 2 z and α = 2 ν + 1 , then … where μ 1 , μ 2 , μ 3 are constants, f 1 , f 2 , f 3 are functions of z , with … where μ 1 , μ 2 , μ 3 , μ 4 are constants, f 1 , f 2 , f 3 , f 4 are functions of z , with …
9: 10.36 Other Differential Equations
The quantity λ 2 in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by λ 2 if at the same time the symbol 𝒞 in the given solutions is replaced by 𝒵 . …
10.36.1 z 2 ( z 2 + ν 2 ) w ′′ + z ( z 2 + 3 ν 2 ) w ( ( z 2 + ν 2 ) 2 + z 2 ν 2 ) w = 0 , w = 𝒵 ν ( z ) ,
10.36.2 z 2 w ′′ + z ( 1 ± 2 z ) w + ( ± z ν 2 ) w = 0 , w = e z 𝒵 ν ( z ) .
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 π ε 0 s ) = Z 1 Z 2 α c / s , where Z j are atomic numbers, ε 0 is the electric constant, α is the fine structure constant, and 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 e , the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, a 0 = / ( m e c α ) , and to a multiple of the Rydberg constant, …