About the Project
NIST

integral equations and representations

AdvancedHelp

(0.003 seconds)

1—10 of 84 matching pages

1: 31.10 Integral Equations and Representations
§31.10 Integral Equations and Representations
Kernel Functions
For integral equations satisfied by the Heun polynomial Hp n , m ( z ) we have σ = 1 2 - δ - j , j = 0 , 1 , , n . …
Kernel Functions
2: 12.18 Methods of Computation
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …
3: 16.25 Methods of Computation
Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …
4: Bibliography E
  • G. P. Egorychev (1984) Integral Representation and the Computation of Combinatorial Sums. Translations of Mathematical Monographs, Vol. 59, American Mathematical Society, Providence, RI.
  • A. Erdélyi (1942a) Integral equations for Heun functions. Quart. J. Math., Oxford Ser. 13, pp. 107–112.
  • A. Erdélyi (1942b) The Fuchsian equation of second order with four singularities. Duke Math. J. 9 (1), pp. 48–58.
  • T. Estermann (1959) On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.
  • J. A. Ewell (1990) A new series representation for ζ ( 3 ) . Amer. Math. Monthly 97 (3), pp. 219–220.
  • 5: Bibliography S
  • D. Schmidt and G. Wolf (1979) A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.
  • R. Shail (1980) On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.
  • H. Shanker (1940a) On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
  • R. Sips (1965) Représentation asymptotique de la solution générale de l’équation de Mathieu-Hill. Acad. Roy. Belg. Bull. Cl. Sci. (5) 51 (11), pp. 1415–1446.
  • B. D. Sleeman (1969) Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.
  • 6: 28.28 Integrals, Integral Representations, and Integral Equations
    §28.28 Integrals, Integral Representations, and Integral Equations
    §28.28(i) Equations with Elementary Kernels
    7: Bibliography K
  • A. Ya. Kazakov and S. Yu. Slavyanov (1996) Integral equations for special functions of Heun class. Methods Appl. Anal. 3 (4), pp. 447–456.
  • A. I. Kheyfits (2004) Closed-form representations of the Lambert W function. Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.
  • A. V. Kitaev, C. K. Law, and J. B. McLeod (1994) Rational solutions of the fifth Painlevé equation. Differential Integral Equations 7 (3-4), pp. 967–1000.
  • Y. A. Kravtsov (1964) Asymptotic solution of Maxwell’s equations near caustics. Izv. Vuz. Radiofiz. 7, pp. 1049–1056.
  • M. D. Kruskal (1974) The Korteweg-de Vries Equation and Related Evolution Equations. In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), A. C. Newell (Ed.), Lectures in Appl. Math., Vol. 15, pp. 61–83.
  • 8: Bibliography F
  • M. V. Fedoryuk (1989) The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).
  • P. Flajolet and B. Salvy (1998) Euler sums and contour integral representations. Experiment. Math. 7 (1), pp. 15–35.
  • A. S. Fokas and M. J. Ablowitz (1982) On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.
  • T. Fort (1948) Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.
  • Y. Fukui and T. Horiguchi (1992) Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice. Phys. A 190 (3-4), pp. 346–362.
  • 9: Bibliography
  • D. W. Albrecht, E. L. Mansfield, and A. E. Milne (1996) Algorithms for special integrals of ordinary differential equations. J. Phys. A 29 (5), pp. 973–991.
  • A. Apelblat (1991) Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42 (5), pp. 708–714.
  • F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.
  • R. Askey and J. Fitch (1969) Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 26 (2), pp. 411–437.
  • R. Askey (1974) Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.
  • 10: Bibliography L
  • C. G. Lambe and D. R. Ward (1934) Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.
  • Y. T. Li and R. Wong (2008) Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.
  • J. L. López and N. M. Temme (1999b) Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.
  • T. A. Lowdon (1970) Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.
  • N. A. Lukaševič (1971) The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).