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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: 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 (1972) Tables of Bessel Transforms. Springer-Verlag, Berlin-New York.
  • J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.
  • C. Osácar, J. Palacián, and M. Palacios (1995) Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.
  • 4: 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
    5: 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). …
    6: 14.34 Software
    §14.34(iv) Conical (Mehler) and/or Toroidal Functions
    7: 18.11 Relations to Other Functions
    §18.11(ii) Formulas of Mehler–Heine Type
    8: Bibliography G
  • 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.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.
  • W. Groenevelt (2007) Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.
  • 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: 14.12 Integral Representations
    §14.12(i) 1 < x < 1
    Mehler–Dirichlet Formula