Poisson%20kernels
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1: 1.15 Summability Methods
2: 20.11 Generalizations and Analogs
3: 18.18 Sums
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§18.18(vii) Poisson Kernels
►See (18.2.41) for the Poisson kernel in case of general OP’s. ►Laguerre
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… ►For the Poisson kernel of Jacobi polynomials (the Bailey formula) see Bailey (1938). …4: 14.31 Other Applications
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§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)). …5: 18.2 General Orthogonal Polynomials
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Kernel property
… ►Kernel Polynomials
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►For OP’s with and orthogonality relation as in (18.2.5) and (18.2.5_5), the Poisson kernel is defined by …Instances where the Poisson kernel is nonnegative are of special interest, see Ismail (2009, Theorem 4.7.12). …6: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications.
J. Number Theory 7 (4), pp. 413–445.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Integral transforms with generalized Legendre functions as kernels.
Compositio Math. 18, pp. 235–287.
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7: Bibliography T
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Level-spacing distributions and the Airy kernel.
Comm. Math. Phys. 159 (1), pp. 151–174.
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On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes.
Phys. A 244 (1-4), pp. 402–413.
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8: Bibliography K
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Bernstein, Pick, Poisson and related integral expressions for Lambert
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Integral Transforms Spec. Funct. 23 (11), pp. 817–829.
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution.
Justus von Liebig Verlag, Darmstadt (German, English).
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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9: 10.75 Tables
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Achenbach (1986) tabulates , , , , , 20D or 18–20S.
Bickley et al. (1952) tabulates or , or , , (.01 or .1) 10(.1) 20, 8S; , , , or , 10S.
Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of and , for , 9S.
Zhang and Jin (1996, p. 322) tabulates , , , , , , , , , 7S.
Zhang and Jin (1996, p. 323) tabulates the first real zeros of , , , , , , , , 8D.