uniformization problem

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1: 31.16 Mathematical Applications

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§31.16(i) Uniformization Problem for Heun’s Equation

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2: 9.16 Physical Applications

… ►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. … ►An application of the Scorer functions is to the problem of the uniform loading of infinite plates (Rothman (1954b, a)).

3: Bibliography T

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Uniform asymptotic expansions of integrals: A selection of problems. J. Comput. Appl. Math. 65 (1-3), pp. 395–417.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

Ch.2.

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4: 10.73 Physical Applications

… ►Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain. …

5: 28.33 Physical Applications

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§28.33(ii) Boundary-Value Problems

►Physical problems involving Mathieu functions include vibrational problems in elliptical coordinates; see (28.32.1). …The general solution of the problem is a superposition of the separated solutions. … ►

§28.33(iii) Stability and Initial-Value Problems

… ►References for other initial-value problems include: …

6: 2.4 Contour Integrals

… ►The problem of obtaining an asymptotic approximation to $I(\alpha ,z)$ that is uniform with respect to $\alpha$ in a region containing $\widehat{\alpha }$ is similar to the problem of a coalescing endpoint and saddle point outlined in §2.3(v). …

7: 10.72 Mathematical Applications

… ►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. The canonical form of differential equation for these problems is given by … ►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. … ►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 $\alpha =a$. …

8: 2.8 Differential Equations with a Parameter

… ►The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to $\xi$. … ►The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to $\xi$. … ►The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to $\xi$. … ►The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to $\xi$. … ►Corresponding to the problems for integrals outlined in §§2.3(v), 2.4(v), and 2.4(vi), there are analogous problems for differential equations. …

9: Bibliography Q

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A note on an open problem about the first Painlevé equation. Acta Math. Appl. Sin. Engl. Ser. 24 (2), pp. 203–210.

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External Links:

ISSN 0168-9673, Document, MathReview (Dmitrii Petrovich Novikov), ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

BibTeX

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On two problems concerning means. J. Hangzhou Inst. Elec. Engrg. 17, pp. 1–7 (Chinese).

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Notes:

In Chinese

Referenced by:

§19.9(i).

Encodings:

BibTeX

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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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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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10: 20.12 Mathematical Applications

… ►For applications of ${\theta }_3(0,q)$ to problems involving sums of squares of integers see §27.13(iv), and for extensions see Estermann (1959), Serre (1973, pp. 106–109), Koblitz (1993, pp. 176–177), and McKean and Moll (1999, pp. 142–143). … ►

§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)). …