# uniform

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## 1—10 of 93 matching pages

##### 1: 14.26 Uniform Asymptotic Expansions

###### §14.26 Uniform Asymptotic Expansions

►The uniform asymptotic approximations given in §14.15 for ${P}_{\nu}^{-\mu}\left(x\right)$ and ${\mathit{Q}}_{\nu}^{\mu}\left(x\right)$ for $$ are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). … ►See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.##### 2: 9.15 Mathematical Applications

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►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point.
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##### 3: 13.27 Mathematical Applications

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►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).

##### 4: 18.40 Methods of Computation

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►Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree.
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##### 5: 20.12 Mathematical Applications

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###### §20.12(ii) Uniformization and Embedding of Complex Tori

… ►Thus theta functions “uniformize” the complex torus. This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)). …##### 6: T. Mark Dunster

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►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.
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##### 7: 10.57 Uniform Asymptotic Expansions for Large Order

###### §10.57 Uniform Asymptotic Expansions for Large Order

►Asymptotic expansions for ${\mathsf{j}}_{n}\left((n+\frac{1}{2})z\right)$, ${\mathsf{y}}_{n}\left((n+\frac{1}{2})z\right)$, ${\mathsf{h}}_{n}^{(1)}\left((n+\frac{1}{2})z\right)$, ${\mathsf{h}}_{n}^{(2)}\left((n+\frac{1}{2})z\right)$, ${\mathsf{i}}_{n}^{(1)}\left((n+\frac{1}{2})z\right)$, and ${\mathsf{k}}_{n}\left((n+\frac{1}{2})z\right)$ as $n\to \mathrm{\infty}$ that are uniform with respect to $z$ can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). …##### 8: 10.69 Uniform Asymptotic Expansions for Large Order

##### 9: 10.72 Mathematical Applications

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►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.
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►These expansions are uniform with respect to $z$, including the turning point ${z}_{0}$ and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities.
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►If $f(z)$ has a double zero ${z}_{0}$, or more generally ${z}_{0}$ is a zero of order $m$, $m=2,3,4,\mathrm{\dots}$, then uniform asymptotic approximations (but

*not*expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. … ►These asymptotic expansions are uniform with respect to $z$, including cut neighborhoods of ${z}_{0}$, and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. … ►These approximations are uniform with respect to both $z$ and $\alpha $, including $z={z}_{0}(a)$, the cut neighborhood of $z=0$, and $\alpha =a$. …##### 10: Bibliography Q

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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points.
Proc. Roy. Soc. London Ser. A 456, pp. 407–431.
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