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
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§14.31(ii) Conical Functions
… ►These functions are also used in the Mehler–Fock integral transform (§14.20(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. …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 Mehler–Fock Transformation
…4: Bibliography O
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Tables of Lebedev, Mehler and Generalized Mehler Transforms.
Mathematical Note
Technical Report 246, Boeing Scientific Research Lab, Seattle.
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Tables of Fourier Transforms and Fourier Transforms of Distributions.
Springer-Verlag, Berlin.
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Tables of Laplace Transforms.
Springer-Verlag, Berlin-New York.
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Tables of Bessel Transforms.
Springer-Verlag, Berlin-New York.
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Tables of Mellin Transforms.
Springer-Verlag, Berlin-New York.
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5: 9.1 Special Notation
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►Other notations that have been used are as follows: and for and (Jeffreys (1928), later changed to and ); , (Fock (1945)); (Szegő (1967, §1.81)); , (Tumarkin (1959)).
6: Bibliography F
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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un.
J. Funct. Anal. 49 (2), pp. 230–268.
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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation.
Math. Comp. 19 (89), pp. 97–104.
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Electromagnetic Diffraction and Propagation Problems.
International Series of Monographs on Electromagnetic Waves,
Vol. 1, Pergamon Press, Oxford.
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Diffraction of radio waves around the earth’s surface.
Acad. Sci. USSR. J. Phys. 9, pp. 255–266.
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On the asymptotic expansion of Mellin transforms.
SIAM J. Math. Anal. 18 (1), pp. 273–282.
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7: 14.1 Special Notation
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►The main functions treated in this chapter are the Legendre functions , , , ; Ferrers functions , (also known as the Legendre functions on the cut); associated Legendre functions , , ; conical functions , , , , (also known as Mehler functions).
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8: Bibliography G
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Contiguous relations and summation and transformation formulae for basic hypergeometric series.
J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
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Formulas of the Dirichlet-Mehler Type.
In Fractional Calculus and its Applications, B. Ross (Ed.),
Lecture Notes in Math., Vol. 457, pp. 207–215.
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Problem 72-21, Laplace transforms of Airy functions.
SIAM Rev. 15 (4), pp. 796–798.
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Exceptional orthogonal polynomials and the Darboux transformation.
J. Phys. A 43 (43), pp. 43016, 16 pp..
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Fourier transforms related to a root system of rank 1.
Transform. Groups 12 (1), pp. 77–116.
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