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1: 1.14 Integral Transforms
§1.14 Integral Transforms
§1.14(i) Fourier Transform
Inversion
Convolution
Fourier Transform
2: Bibliography F
  • S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
  • P. Flajolet and A. Odlyzko (1990) Singularity analysis of generating functions. SIAM J. Discrete Math. 3 (2), pp. 216–240.
  • A. S. Fokas, B. Grammaticos, and A. Ramani (1993) From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.
  • A. S. Fokas, A. R. Its, and A. V. Kitaev (1991) Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142 (2), pp. 313–344.
  • A. S. Fokas, A. R. Its, and X. Zhou (1992) Continuous and Discrete Painlevé Equations. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.
  • 3: Vadim B. Kuznetsov
    Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics. …
    4: 20.11 Generalizations and Analogs
    It is a discrete analog of theta functions. If both m , n are positive, then G ( m , n ) allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): …This is the discrete analog of the Poisson identity (§1.8(iv)). …
    5: Bibliography S
  • K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
  • B. I. Schneider, X. Guan, and K. Bartschat (2016) Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation. Adv. Quantum Chem. 72, pp. 95–127.
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.
  • F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
  • 6: 20 Theta Functions
    Chapter 20 Theta Functions
    7: Bibliography W
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • G. N. Watson (1910) The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.
  • G. Wolf (2008) On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.
  • R. Wong and J. F. Lin (1978) Asymptotic expansions of Fourier transforms of functions with logarithmic singularities. J. Math. Anal. Appl. 64 (1), pp. 173–180.
  • R. Wong and H. Y. Zhang (2009b) On the connection formulas of the third Painlevé transcendent. Discrete Contin. Dyn. Syst. 23 (1-2), pp. 541–560.
  • 8: Bibliography G
  • B. Gabutti and B. Minetti (1981) A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.
  • M. J. Gander and A. H. Karp (2001) Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.
  • GAP (website) The GAP Group, Centre for Interdisciplinary Research in Computational Algebra, University of St. Andrews, United Kingdom.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.
  • 9: Bibliography K
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.
  • T. Koornwinder, A. Kostenko, and G. Teschl (2018) Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.
  • 10: Bibliography B
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • R. Blackmore and B. Shizgal (1985) Discrete ordinate solution of Fokker-Planck equations with non-linear coefficients. Phys. Rev. A 31 (3), pp. 1855–1868.
  • R. Blackmore, U. Weinert, and B. Shizgal (1986) Discrete ordinate solution of a Fokker-Planck equation in laser physics. Transport Theory Statist. Phys. 15 (1-2), pp. 181–210.
  • V. Britanak, P. C. Yip, and K. R. Rao (2007) Discrete Cosine and Sine Transforms. General Properties, Fast Algorithms and Integer Approximations. Elsevier/Academic Press, Amsterdam.