# asymptotic solutions

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##### 1: 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).
##### 2: T. Mark Dunster
He has received a number of National Science Foundation grants, and has published numerous papers in the areas of uniform asymptotic solutions of differential equations, convergent WKB methods, special functions, quantum mechanics, and scattering theory. …
##### 4: 2.2 Transcendental Equations
###### §2.2 Transcendental Equations
where $F_{0}=f_{0}$ and $sF_{s}$ ($s\geq 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of $(f(x))^{s}$ (Lagrange’s formula for the reversion of series). …
##### 5: 10.72 Mathematical Applications
###### §10.72(i) Differential Equations with Turning Points
Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … In regions in which (10.72.1) has a simple turning point $z_{0}$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_{0}$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\tfrac{1}{3}$9.6(i)). … Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha$). …
##### 6: 2.9 Difference Equations
###### §2.9(ii) Coincident Characteristic Values
For error bounds see Zhang et al. (1996). …
##### 7: Bibliography O
• A. B. Olde Daalhuis and F. W. J. Olver (1994) Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one. Proc. Roy. Soc. London Ser. A 445, pp. 39–56.
• F. W. J. Olver and F. Stenger (1965) Error bounds for asymptotic solutions of second-order differential equations having an irregular singularity of arbitrary rank. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 244–249.
• F. W. J. Olver (1993a) Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.
• F. W. J. Olver (1997a) Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity. Methods Appl. Anal. 4 (4), pp. 375–403.
• F. W. J. Olver (1999) On the uniqueness of asymptotic solutions of linear differential equations. Methods Appl. Anal. 6 (2), pp. 165–174.
• ##### 8: 2.7 Differential Equations
###### §2.7(ii) Irregular Singularities of Rank 1
Note that the coefficients in the expansions (2.7.12), (2.7.13) for the “late” coefficients, that is, $a_{s,1}$, $a_{s,2}$ with $s$ large, are the “early” coefficients $a_{j,2}$, $a_{j,1}$ with $j$ small. …See §2.11(v) for other examples. …
##### 9: 2.8 Differential Equations with a Parameter
###### §2.8(i) Classification of Cases
Zeros of $f(z)$ are also called turning points. …
##### 10: Frank W. J. Olver
He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i. …