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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 k = 0 , 1 , 2 , , …
10.29.5 𝒵 ν ( k ) ( z ) = 1 2 k ( 𝒵 ν - k ( z ) + ( k 1 ) 𝒵 ν - k + 2 ( z ) + ( k 2 ) 𝒵 ν - k + 4 ( z ) + + 𝒵 ν + k ( z ) ) .
7: 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 ) .
8: 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, …
9: 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 …
10: 10.43 Integrals
Let 𝒵 ν ( z ) be defined as in §10.25(ii). …
§10.43(v) Kontorovich–Lebedev Transform
The Kontorovich–Lebedev transform of a function g ( x ) is defined as … For asymptotic expansions of the direct transform (10.43.30) see Wong (1981), and for asymptotic expansions of the inverse transform (10.43.31) see Naylor (1990, 1996). For collections of the Kontorovich–Lebedev transform, see Erdélyi et al. (1954b, Chapter 12), Prudnikov et al. (1986b, pp. 404–412), and Oberhettinger (1972, Chapter 5). …