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11: 35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7(iii) Partial Differential Equations
Subject to the conditions (a)–(c), the function f ( T ) = F 1 2 ( a , b ; c ; T ) is the unique solution of each partial differential equationSystems of partial differential equations for the F 1 0 (defined in §35.8) and F 1 1 functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). … Butler and Wood (2002) applies Laplace’s method (§2.3(iii)) to (35.7.5) to derive uniform asymptotic approximations for the functions …
12: Bibliography K
  • A. A. Kapaev (1988) Asymptotic behavior of the solutions of the Painlevé equation of the first kind. Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).
  • 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.
  • V. I. Krylov and N. S. Skoblya (1985) A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.
  • 13: Bibliography N
  • A. Nakamura (1996) Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.
  • J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.
  • G. Nemes (2013a) An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.
  • G. Nemes (2020) An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.
  • M. Newman (1967) Solving equations exactly. J. Res. Nat. Bur. Standards Sect. B 71B, pp. 171–179.
  • 14: 3.11 Approximation Techniques
    Also, in cases where f ( x ) satisfies a linear ordinary differential equation with polynomial coefficients, the expansion (3.11.11) can be substituted in the differential equation to yield a recurrence relation satisfied by the c n . …
    Laplace Transform Inversion
    Numerical inversion of the Laplace transform (§1.14(iii)) …requires f = - 1 F to be obtained from numerical values of F . A general procedure is to approximate F by a rational function R (vanishing at infinity) and then approximate f by r = - 1 R . …
    15: Bibliography S
  • H. E. Salzer (1955) Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.
  • J. L. Schiff (1999) The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer-Verlag, New York.
  • N. T. Shawagfeh (1992) The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.
  • T. Shiota (1986) Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (2), pp. 333–382.
  • R. Spigler (1984) The linear differential equation whose solutions are the products of solutions of two given differential equations. J. Math. Anal. Appl. 98 (1), pp. 130–147.
  • 16: 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.
  • E. W. Leaver (1986) Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.
  • X. Li and R. Wong (1994) Error bounds for asymptotic expansions of Laplace convolutions. SIAM J. Math. Anal. 25 (6), pp. 1537–1553.
  • Y. A. Li and P. J. Olver (2000) Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differential Equations 162 (1), pp. 27–63.
  • N. A. Lukaševič (1971) The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).
  • 17: Frank W. J. Olver
    He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i. …, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. … In a review of that volume, Jet Wimp of Drexel University said that the papers “exemplify a redoubtable mathematical talent, the work of a man who has done more than almost anyone else in the 20th century to bestow on the discipline of applied mathematics the elegance and rigor that its earliest practitioners, such as Gauss and Laplace, would have wished for it. …
    18: 11.11 Asymptotic Expansions of Anger–Weber Functions
    11.11.9_5 a k + 1 ( λ ) = λ 1 - λ 2 λ a k ′′ ( λ ) + a k ( λ ) ( 2 k + 1 ) ( 2 k + 2 ) , k = 0 , 1 , 2 , .
    11.11.10 A - ν ( λ ν ) - 1 π k = 0 ( 2 k ) ! a k ( - λ ) ν 2 k + 1 , ν , | ph ν | π - δ .
    11.11.11 A - ν ( λ ν ) ( 2 π ν ) 1 / 2 e - ν μ k = 0 ( 1 2 ) k b k ( λ ) ν k , ν , | ph ν | π 2 - δ ,
    11.11.13_5 ( 1 2 ) k b k ( λ ) = ( - 1 ) k ( 1 - λ 2 ) 1 / 4 U k ( 1 1 - λ 2 ) , k = 0 , 1 , 2 , ,
    19: 1.14 Integral Transforms
    §1.14(iii) Laplace Transform
    Inversion
    Translation
    Derivatives
    Convolution
    20: 3.5 Quadrature
    Example. Laplace Transform Inversion
    Equation (3.5.36), without the error term, becomes …
    Table 3.5.19: Laplace transform inversion.
    t J 0 ( t ) g ( t )
    In fact from (7.14.4) and the inversion formula for the Laplace transform (§1.14(iii)) we have … A special case is the rule for Hilbert transforms (§1.14(v)): …