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generalized Mehler–Fock transformation

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1: 8.19 Generalized Exponential Integral
§8.19 Generalized Exponential Integral
§8.19(ii) Graphics
§8.19(ix) Inequalities
§8.19(x) Integrals
§8.19(xi) Further Generalizations
2: 16.2 Definition and Analytic Properties
§16.2(i) Generalized Hypergeometric Series
Polynomials
Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via …
§16.2(v) Behavior with Respect to Parameters
3: 8.21 Generalized Sine and Cosine Integrals
§8.21 Generalized Sine and Cosine Integrals
§8.21(i) Definitions: General Values
§8.21(iv) Interrelations
§8.21(v) Special Values
4: 1.14 Integral Transforms
§1.14 Integral Transforms
§1.14(i) Fourier Transform
§1.14(iii) Laplace Transform
Fourier Transform
Laplace Transform
5: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
Kummer Transformation
Thomae Transformation
§35.8(iv) General Properties
Laplace Transform
6: 19.2 Definitions
§19.2(i) General Elliptic Integrals
§19.2(iii) Bulirsch’s Integrals
Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). …
7: 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. …
8: 14.20 Conical (or Mehler) Functions
§14.20 Conical (or Mehler) Functions
Solutions are known as conical or Mehler functions. … Lastly, for the range 1 < x < , P - 1 2 + i τ - μ ( x ) is a real-valued solution of (14.20.1); in terms of Q - 1 2 ± i τ μ ( x ) (which are complex-valued in general): …
§14.20(vi) Generalized MehlerFock Transformation
14.20.19 α = μ / τ ,
9: 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. W. J. Olver (1978) General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.
  • F. W. J. Olver (1991a) Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral. SIAM J. Math. Anal. 22 (5), pp. 1460–1474.
  • F. W. J. Olver (1994b) The Generalized Exponential Integral. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 497–510.
  • 10: 9.1 Special Notation
    k nonnegative integer, except in §9.9(iii).
    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)).