diffusion%20equations
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21: 28.1 Special Notation
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►The eigenvalues of Mathieu’s equation are denoted by
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integers. | |
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real or complex parameters of Mathieu’s equation with . | |
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, | , | , | , | , |
Abramowitz and Stegun (1964, Chapter 20)
…22: 32.8 Rational Solutions
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§32.8(ii) Second Painlevé Equation
… ►§32.8(iii) Third Painlevé Equation
… ►§32.8(iv) Fourth Painlevé Equation
… ►§32.8(v) Fifth Painlevé Equation
… ►23: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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Finite Differences and Difference Equations in the Real Domain.
Clarendon Press, Oxford.
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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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24: 9.18 Tables
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Miller (1946) tabulates , for , for ; , for ; , for ; , , , (respectively , , , ) for . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.
Zhang and Jin (1996, p. 337) tabulates , , , for to 8S and for to 9D.
Sherry (1959) tabulates , , , , ; 20S.
Zhang and Jin (1996, p. 339) tabulates , , , , , , , , ; 8D.
25: 31.13 Asymptotic Approximations
§31.13 Asymptotic Approximations
… ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).26: 25.12 Polylogarithms
27: 36.5 Stokes Sets
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►where are the two smallest positive roots of the equation
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►where satisfies the equation
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►Here is the root of the equation
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►where is the root of the equation
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►where is the positive root of the equation
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28: 8 Incomplete Gamma and Related
Functions
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29: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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Formes canoniques des équations confluentes de l’équation de Heun.
Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.
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Sur les équations confluentes de l’équation de Heun.
Ann. Soc. Sci. Bruxelles Sér. I 92 (3), pp. 151–189.
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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30: Bibliography
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Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation.
Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates.
Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.
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