asymptotic solutions of differential equations
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21: 2.9 Difference Equations
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►For asymptotic approximations to solutions of second-order difference equations analogous to the Liouville–Green (WKBJ) approximation for differential equations (§2.7(iii)) see Spigler and Vianello (1992, 1997) and Spigler et al. (1999).
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22: 36.15 Methods of Computation
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§36.15(ii) Asymptotics
►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. Close to the bifurcation set but far from , the uniform asymptotic approximations of §36.12 can be used. … ►§36.15(v) Differential Equations
►For numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).23: Bibliography K
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Asymptotic behavior of the solutions of the Painlevé equation of the first kind.
Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).
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24: 28.8 Asymptotic Expansions for Large
§28.8 Asymptotic Expansions for Large
… ►Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). …It is stated that corresponding uniform approximations can be obtained for other solutions, including the eigensolutions, of the differential equations by application of the results, but these approximations are not included. … ►Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). … ►With additional restrictions on , uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). …25: 11.9 Lommel Functions
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►For uniform asymptotic expansions, for large and fixed , of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390).
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26: 28.34 Methods of Computation
27: 10.20 Uniform Asymptotic Expansions for Large Order
§10.20 Uniform Asymptotic Expansions for Large Order
… ►Define to be the solution of the differential equation … ► ►§10.20(iii) Double Asymptotic Properties
►For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of see §10.41(v).28: 10.74 Methods of Computation
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►For large positive real values of the uniform asymptotic expansions of §§10.20(i) and 10.20(ii) can be used.
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§10.74(ii) Differential Equations
►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods. As 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. … ►For further information, including parallel methods for solving the differential equations, see Lozier and Olver (1993). …29: Mathematical Introduction
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►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3).
►In addition, there is a comprehensive account of the great variety of analytical methods that are used for deriving and applying the extremely important asymptotic properties of the special functions, including double asymptotic properties (Chapter 2 and §§10.41(iv), 10.41(v)).
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►This is because is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as is an entire function of each of its parameters , , and : this results in fewer restrictions and simpler equations.
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►For all equations and other technical information this Handbook and the DLMF either provide references to the literature for proof or describe steps that can be followed to construct a proof.
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►For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed.
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30: 9.17 Methods of Computation
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►For large the asymptotic expansions of §§9.7 and 9.12(viii) should be used instead.
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