# generalized Mehler–Fock transformation

(0.003 seconds)

## 1—10 of 452 matching pages

##### 2: 16.2 Definition and Analytic Properties
###### Polynomials
Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via …
##### 6: 19.2 Definitions
###### §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, $P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ is a real-valued solution of (14.20.1); in terms of $Q^{\mu}_{-\frac{1}{2}\pm\mathrm{i}\tau}\left(x\right)$ (which are complex-valued in general): …
##### 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: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi}\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi}\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-\ifrac{1}{3}}\pi\mathrm{Ai}\left(-3^{-\ifrac{1}{3}}x\right)$ (Szegő (1967, §1.81)); $e_{0}(x)=\pi\mathrm{Hi}(-x)$, $\widetilde{e}_{0}(x)=-\pi\mathrm{Gi}(-x)$ (Tumarkin (1959)).