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reciprocal-modulus transformation

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1: 1.14 Integral Transforms
§1.14 Integral Transforms
1.14.1 ( f ) ( x ) = f ( x ) = 1 2 π - f ( t ) e i x t d t .
1.14.17 ( f ) ( s ) = f ( s ) = 0 e - s t f ( t ) d t .
1.14.32 ( f ) ( s ) = f ( s ) = 0 x s - 1 f ( x ) d x .
The Hilbert transform of a real-valued function f ( t ) is defined in the following equivalent ways: …
2: 19.7 Connection Formulas
Reciprocal-Modulus Transformation
Imaginary-Modulus Transformation
Imaginary-Argument Transformation
For two further transformations of this type see Erdélyi et al. (1953b, p. 316). …
3: Bibliography F
  • H. E. Fettis (1965) Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.
  • H. E. Fettis (1970) On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.
  • F. Feuillebois (1991) Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.
  • A. S. Fokas and M. J. Ablowitz (1982) On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.
  • A. S. Fokas and Y. C. Yortsos (1981) The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.
  • 4: 1.16 Distributions
    §1.16(vii) Fourier Transforms of Tempered Distributions
    Then its Fourier transform is …
    1.16.35 ( u ) , ϕ = u , ( ϕ ) , ϕ 𝒯 n .
    The Fourier transform ( u ) of a tempered distribution is again a tempered distribution, and …
    §1.16(viii) Fourier Transforms of Special Distributions
    5: 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). …
    6: 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).
    7: 35.2 Laplace Transform
    §35.2 Laplace Transform
    Definition
    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: 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
    9: 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. …
    10: 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. …