expansions of solutions in series of
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21: 2.8 Differential Equations with a Parameter
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βΊThe form of the asymptotic expansion depends on the nature of the transition points in
, that is, points at which has a zero or singularity.
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βΊIn Cases I and II the asymptotic solutions are in terms of the functions that satisfy (2.8.8) with .
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βΊFor another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004).
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§2.8(iv) Case III: Simple Pole
… βΊFor other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). …22: 4.13 Lambert -Function
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βΊThe Lambert -function is the solution of the equation
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βΊOn the -interval there is one real solution, and it is nonnegative and increasing.
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βΊOther solutions of (4.13.1) are other branches of .
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βΊIn the case of and real the series converges for .
…For these results and other asymptotic expansions see Corless et al. (1997).
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23: 3.6 Linear Difference Equations
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βΊThis is of little consequence if the wanted solution is growing in magnitude at least as fast as any other solution of (3.6.3), and the recursion process is stable.
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βΊBecause the recessive solution of a homogeneous equation is the fastest growing solution in the backward direction, it occurred to J.
…A “trial solution” is then computed by backward recursion, in the course of which the original components of the unwanted solution
die away.
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βΊIf, as , the wanted solution
grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable.
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βΊHere , and its actual value depends on the asymptotic behavior of the wanted solution in relation to those of the other solutions.
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24: Bibliography K
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Series expansions for the third incomplete elliptic integral via partial fraction decompositions.
J. Comput. Appl. Math. 207 (2), pp. 331–337.
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The second Painlevé equation in electric probe theory. Some numerical solutions.
Zh. Vychisl. Mat. Mat. Fiz. 38 (6), pp. 992–1000 (Russian).
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The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface.
Technical Physics 49 (1), pp. 1–7.
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Asymptotic expansions of certain -series and a formula of Ramanujan for specific values of the Riemann zeta function.
Acta Arith. 107 (3), pp. 269–298.
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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively -binomial sums and basic hypergeometric series.
Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
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25: 9.17 Methods of Computation
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§9.17(i) Maclaurin Expansions
βΊAlthough the Maclaurin-series expansions of §§9.4 and 9.12(vi) converge for all finite values of , they are cumbersome to use when is large owing to slowness of convergence and cancellation. For large the asymptotic expansions of §§9.7 and 9.12(viii) should be used instead. …However, in the case of and this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v). … βΊAs described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation. …26: Bibliography S
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FGH, a code for the calculation of Coulomb radial wave functions from series expansions.
Comput. Phys. Comm. 146 (2), pp. 250–253.
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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
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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.
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The expansion of Lamé functions into series of associated Legendre functions of the second kind.
Proc. Cambridge Philos. Soc. 62, pp. 441–452.
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An Introduction to Basic Fourier Series.
Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.
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27: 12.4 Power-Series Expansions
§12.4 Power-Series Expansions
… βΊwhere the initial values are given by (12.2.6)–(12.2.9), and and are the even and odd solutions of (12.2.2) given by βΊ
12.4.3
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12.4.4
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βΊThese series converge for all values of .
28: 11.13 Methods of Computation
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βΊ
§11.13(ii) Series Expansions
βΊAlthough the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of , they are cumbersome to use when is large owing to slowness of convergence and cancellation. For large and/or the asymptotic expansions given in §11.6 should be used instead. … βΊTo insure stability the integration path must be chosen so that as we proceed along it the wanted solution grows in magnitude at least as rapidly as the complementary solutions. … βΊThe solution needs to be integrated backwards for small , and either forwards or backwards for large depending whether or not exceeds . …29: 13.29 Methods of Computation
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§13.29(i) Series Expansions
βΊAlthough the Maclaurin series expansion (13.2.2) converges for all finite values of , it is cumbersome to use when is large owing to slowness of convergence and cancellation. …However, this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied by the combination of (13.7.10) and (13.7.11), or by use of the hyperasymptotic expansions given in Olde Daalhuis and Olver (1995a). … βΊAs described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. … βΊwith recessive solution …30: 12.14 The Function
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βΊIn this section solutions of equation (12.2.3) are considered.
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