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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: 31.2 Differential Equations
F -Homotopic Transformations
By composing these three steps, there result 2 3 = 8 possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1).
Homographic Transformations
Composite Transformations
There are 8 24 = 192 automorphisms of equation (31.2.1) by compositions of F -homotopic and homographic transformations. …
3: 1.9 Calculus of a Complex Variable
Conformal Transformation
Bilinear Transformation
or its limiting form, and is invariant under bilinear transformations. Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …
4: 15.14 Integrals
§15.14 Integrals
The Mellin transform of the hypergeometric function of negative argument is given by … Fourier transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §§1.14 and 2.14). Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). …Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …
5: 12.16 Mathematical Applications
PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …Integral transforms and sampling expansions are considered in Jerri (1982).
6: 2.5 Mellin Transform Methods
§2.5 Mellin Transform Methods
The Mellin transform of f ( t ) is defined by …The inversion formula is given by …
§2.5(iii) Laplace Transforms with Small Parameters
7: 35.2 Laplace Transform
§35.2 Laplace Transform
Inversion Formula
Convolution Theorem
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 …
8: 15.17 Mathematical Applications
The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … 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. …
9: 19.15 Advantages of Symmetry
Symmetry in x , y , z of R F ( x , y , z ) , R G ( x , y , z ) , and R J ( x , y , z , p ) replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). (19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. …
10: 14.31 Other Applications
§14.31(ii) Conical Functions
These functions are also used in the Mehler–Fock integral transform14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …