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Poisson integral

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11: 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)). …
12: Errata
  • Chapter 1 Additions

    The following additions were made in Chapter 1:

  • 13: Bibliography T
  • G. ’t Hooft and M. Veltman (1979) Scalar one-loop integrals. Nuclear Phys. B 153 (3-4), pp. 365–401.
  • I. C. Tang (1969) Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.
  • J. S. Thompson (1996) High Speed Numerical Integration of Fermi Dirac Integrals. Master’s Thesis, Naval Postgraduate School, Monterey, CA.
  • J. Todd (1954) Evaluation of the exponential integral for large complex arguments. J. Research Nat. Bur. Standards 52, pp. 313–317.
  • C. A. Tracy and H. Widom (1997) On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes. Phys. A 244 (1-4), pp. 402–413.
  • 14: 20.11 Generalizations and Analogs
    This is the discrete analog of the Poisson identity (§1.8(iv)). … As in §20.11(ii), the modulus k of elliptic integrals19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in q -series via (20.9.1). …
    15: 18.2 General Orthogonal Polynomials
    More generally than (18.2.1)–(18.2.3), w ( x ) d x may be replaced in (18.2.1) by d μ ( x ) , where the measure μ is the Lebesgue–Stieltjes measure μ α corresponding to a bounded nondecreasing function α on the closure of ( a , b ) with an infinite number of points of increase, and such that a b | x | n d μ ( x ) < for all n . …
    Poisson kernel
    For OP’s p n with h n 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). …