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Poisson integral

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1: 1.15 Summability Methods
Poisson Kernel
is the Poisson integral of f ( t ) . … is the conjugate Poisson integral of f ( x ) . … …
2: 1.9 Calculus of a Complex Variable
Poisson Integral
3: 10.9 Integral Representations
Poisson’s and Related Integrals
4: 19.18 Derivatives and Differential Equations
The next four differential equations apply to the complete case of R F and R G in the form R - a ( 1 2 , 1 2 ; z 1 , z 2 ) (see (19.16.20) and (19.16.23)). The function w = R - a ( 1 2 , 1 2 ; x + y , x - y ) satisfies an Euler–Poisson–Darboux equation: …
5: Bibliography K
  • S. H. Khamis (1965) Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution. Justus von Liebig Verlag, Darmstadt (German, English).
  • 6: Bibliography H
  • L. Habsieger (1988) Une q -intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.
  • P. I. Hadži (1975a) Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).
  • H. Hancock (1958) Elliptic Integrals. Dover Publications Inc., New York.
  • I. D. Hill (1973) Algorithm AS66: The normal integral. Appl. Statist. 22 (3), pp. 424–427.
  • M. Hoyles, S. Kuyucak, and S. Chung (1998) Solutions of Poisson’s equation in channel-like geometries. Comput. Phys. Comm. 115 (1), pp. 45–68.
  • 7: 1.8 Fourier Series
    1.8.4 c n = 1 2 π - π π f ( x ) e - i n x d x .
    1.8.5 1 2 a 0 2 + n = 1 ( a n 2 + b n 2 ) 1 π - π π ( f ( x ) ) 2 d x .
    (1.8.10) continues to apply if either a or b or both are infinite and/or f ( x ) has finitely many singularities in ( a , b ) , provided that the integral converges uniformly (§1.5(iv)) at a , b , and the singularities for all sufficiently large λ . …
    Poisson’s Summation Formula
    1.8.16 n = - e - ( n + x ) 2 ω = π ω ( 1 + 2 n = 1 e - n 2 π 2 / ω cos ( 2 n π x ) ) , ω > 0 .
    8: Bibliography B
  • G. E. Barr (1968) A note on integrals involving parabolic cylinder functions. SIAM J. Appl. Math. 16 (1), pp. 71–74.
  • W. Bartky (1938) Numerical calculation of a generalized complete elliptic integral. Rev. Mod. Phys. 10, pp. 264–269.
  • P. A. Becker (1997) Normalization integrals of orthogonal Heun functions. J. Math. Phys. 38 (7), pp. 3692–3699.
  • B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
  • P. J. Bushell (1987) On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.
  • 9: 3.5 Quadrature
    See also Poisson’s summation formula (§1.8(iv)). …
    §3.5(vii) Oscillatory Integrals
    Integrals of the form … For the Bromwich integral
    10: 14.31 Other Applications
    §14.31(i) Toroidal Functions
    Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). … 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)). …