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Mehler–Fock 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: 14.31 Other Applications
§14.31(ii) Conical Functions
These functions are also used in the MehlerFock 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 MehlerFock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …
3: 14.20 Conical (or Mehler) Functions
§14.20 Conical (or Mehler) Functions
Solutions are known as conical or Mehler functions. …
§14.20(vi) Generalized MehlerFock Transformation
4: Bibliography O
  • F. Oberhettinger and T. P. Higgins (1961) Tables of Lebedev, Mehler and Generalized Mehler Transforms. Mathematical Note Technical Report 246, Boeing Scientific Research Lab, Seattle.
  • F. Oberhettinger (1990) Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.
  • F. Oberhettinger and L. Badii (1973) Tables of Laplace Transforms. Springer-Verlag, Berlin-New York.
  • F. Oberhettinger (1972) Tables of Bessel Transforms. Springer-Verlag, Berlin-New York.
  • F. Oberhettinger (1974) Tables of Mellin Transforms. Springer-Verlag, Berlin-New York.
  • 5: 9.1 Special Notation
    Other notations that have been used are as follows: Ai ( - x ) and Bi ( - x ) for Ai ( x ) and Bi ( x ) (Jeffreys (1928), later changed to Ai ( x ) and Bi ( x ) ); U ( x ) = π Bi ( x ) , V ( x ) = π Ai ( x ) (Fock (1945)); A ( x ) = 3 - 1 / 3 π Ai ( - 3 - 1 / 3 x ) (Szegő (1967, §1.81)); e 0 ( x ) = π Hi ( - x ) , e ~ 0 ( x ) = - π Gi ( - x ) (Tumarkin (1959)).
    6: 14.1 Special Notation
    The main functions treated in this chapter are the Legendre functions P ν ( x ) , Q ν ( x ) , P ν ( z ) , Q ν ( z ) ; Ferrers functions P ν μ ( x ) , Q ν μ ( x ) (also known as the Legendre functions on the cut); associated Legendre functions P ν μ ( z ) , Q ν μ ( z ) , Q ν μ ( z ) ; conical functions P - 1 2 + i τ μ ( x ) , Q - 1 2 + i τ μ ( x ) , Q ^ - 1 2 + i τ μ ( x ) , P - 1 2 + i τ μ ( x ) , Q - 1 2 + i τ μ ( x ) (also known as Mehler functions). …
    7: 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.
  • F. Feuillebois (1991) Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.
  • V. A. Fock (1965) Electromagnetic Diffraction and Propagation Problems. International Series of Monographs on Electromagnetic Waves, Vol. 1, Pergamon Press, Oxford.
  • V. Fock (1945) Diffraction of radio waves around the earth’s surface. Acad. Sci. USSR. J. Phys. 9, pp. 255–266.
  • 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.
  • 8: Bibliography G
  • B. Gabutti and B. Minetti (1981) A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.
  • F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
  • G. Gasper (1975) Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.
  • A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.
  • 9: 10.9 Integral Representations
    Mehler–Sonine and Related Integrals
    10.9.22 J ν ( x ) = 1 2 π i - i i Γ ( - t ) ( 1 2 x ) ν + 2 t Γ ( ν + t + 1 ) d t , ν > 0 , x > 0 ,
    10.9.23 J ν ( z ) = 1 2 π i - - i c - + i c Γ ( t ) Γ ( ν - t + 1 ) ( 1 2 z ) ν - 2 t d t ,
    10.9.24 H ν ( 1 ) ( z ) = - e - 1 2 ν π i 2 π 2 c - i c + i Γ ( t ) Γ ( t - ν ) ( - 1 2 i z ) ν - 2 t d t , 0 < ph z < π ,
    10.9.25 H ν ( 2 ) ( z ) = e 1 2 ν π i 2 π 2 c - i c + i Γ ( t ) Γ ( t - ν ) ( 1 2 i z ) ν - 2 t d t , - π < ph z < 0 .
    10: 14.34 Software
    §14.34(iv) Conical (Mehler) and/or Toroidal Functions