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asymptotic solutions of transcendental equations

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1: 2.2 Transcendental Equations
§2.2 Transcendental Equations
where F 0 = f 0 and s F s ( s 1 ) is the coefficient of x 1 in the asymptotic expansion of ( f ( x ) ) s (Lagrange’s formula for the reversion of series). …
2: Bibliography E
  • A. Erdélyi (1944) Certain expansions of solutions of the Heun equation. Quart. J. Math., Oxford Ser. 15, pp. 62–69.
  • A. Erdélyi (1956) Asymptotic Expansions. Dover Publications Inc., New York.
  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1953a) Higher Transcendental Functions. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London.
  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1953b) Higher Transcendental Functions. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London.
  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1955) Higher Transcendental Functions. Vol. III. McGraw-Hill Book Company, Inc., New York-Toronto-London.
  • 3: Bibliography F
  • B. R. Fabijonas, D. W. Lozier, and F. W. J. Olver (2004) Computation of complex Airy functions and their zeros using asymptotics and the differential equation. ACM Trans. Math. Software 30 (4), pp. 471–490.
  • J. L. Fields (1966) A note on the asymptotic expansion of a ratio of gamma functions. Proc. Edinburgh Math. Soc. (2) 15, pp. 43–45.
  • 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.
  • A. S. Fokas and Y. C. Yortsos (1981) The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.
  • F. N. Fritsch, R. E. Shafer, and W. P. Crowley (1973) Solution of the transcendental equation w e w = x . Comm. ACM 16 (2), pp. 123–124.
  • 4: Bibliography G
  • L. Gårding (1947) The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.
  • I. Gargantini and P. Henrici (1967) A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.
  • V. I. Gromak and N. A. Lukaševič (1982) Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).
  • V. I. Gromak (1976) The solutions of Painlevé’s fifth equation. Differ. Uravn. 12 (4), pp. 740–742 (Russian).
  • V. I. Gromak (1978) One-parameter systems of solutions of Painlevé equations. Differ. Uravn. 14 (12), pp. 2131–2135 (Russian).
  • 5: Bibliography C
  • L. D. Carr, C. W. Clark, and W. P. Reinhardt (2000) Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity. Phys. Rev. A 62 (063610), pp. 1–10.
  • P. A. Clarkson (1991) Nonclassical Symmetry Reductions and Exact Solutions for Physically Significant Nonlinear Evolution Equations. In Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids (Edmonton, AB, 1990), W. Rozmus and J. A. Tuszynski (Eds.), pp. 72–79.
  • P. A. Clarkson (2005) Special polynomials associated with rational solutions of the fifth Painlevé equation. J. Comput. Appl. Math. 178 (1-2), pp. 111–129.
  • D. S. Clemm (1969) Algorithm 352: Characteristic values and associated solutions of Mathieu’s differential equation. Comm. ACM 12 (7), pp. 399–407.
  • O. Costin (1999) Correlation between pole location and asymptotic behavior for Painlevé I solutions. Comm. Pure Appl. Math. 52 (4), pp. 461–478.
  • 6: Bibliography M
  • R. S. Maier (2007) The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.
  • T. Masuda (2004) Classical transcendental solutions of the Painlevé equations and their degeneration. Tohoku Math. J. (2) 56 (4), pp. 467–490.
  • M. Mazzocco (2001a) Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.
  • M. Mazzocco (2001b) Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.
  • S. Moch, P. Uwer, and S. Weinzierl (2002) Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. J. Math. Phys. 43 (6), pp. 3363–3386.