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21: 20.12 Mathematical Applications

… ►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)). …

22: 28.22 Connection Formulas

§28.22 Connection Formulas

… ►

23: 1.13 Differential Equations

… ►

§1.13(i) Existence of Solutions

►A domain in the complex plane is simply-connected if it has no “holes”; more precisely, if its complement in the extended plane

\mathbb{C}\cup\{\infty\}

is connected. … ►where

z\in D

, a simply-connected domain, and

f(z)

g(z)

are analytic in

D

, has an infinite number of analytic solutions in

D

. … ►

1.13.6

Aw_{1}(z)+Bw_{2}(z)=0,

\forall z\in D

ⓘ

Symbols:: $\in$ : element of, $\forall$ : for every, $z$ : variable, $D$ : simply-connected domain, $w_{1}(z)$ : solution, $w_{2}(z)$ : solution, $A$ : constant and $B$ : constant
Permalink:: http://dlmf.nist.gov/1.13.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §1.13(i), §1.13(i), §1.13 and Ch.1

…

24: 19.21 Connection Formulas

§19.21 Connection Formulas

… ►The complete cases of

R_{F}

and

R_{G}

have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … ►Connection formulas for

R_{-a}\left(\mathbf{b};\mathbf{z}\right)

are given in Carlson (1977b, pp. 99, 101, and 123–124). …

25: Bibliography O

… ►

F. W. J. Olver (1977a) Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

ⓘ

External Links:: Document, MathReview (Anthony Leung), ZentralBlatt
Referenced by:: §2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).
Encodings:: BibTeX

►

F. W. J. Olver (1977b) Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

ⓘ

External Links:: Document, MathReview (W. Wasow), ZentralBlatt
Referenced by:: §2.8(v).
Encodings:: BibTeX

… ►

F. W. J. Olver (1978) General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.

ⓘ

External Links:: ISSN 0080-4614, FullText, MathReview (W. Wasow), ZentralBlatt
Referenced by:: §2.8(v), (9.13.18), §9.13(i), §9.13(i).
Encodings:: BibTeX

…

26: 9.2 Differential Equation

… ►

§9.2(v) Connection Formulas

…

27: 9.16 Physical Applications

… ►These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). …

28: 10.27 Connection Formulas

§10.27 Connection Formulas

…

29: 10.57 Uniform Asymptotic Expansions for Large Order

… ►Subsequently, for

{\mathsf{i}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)

the connection formula (10.47.11) is available. …

30: 11.13 Methods of Computation

… ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that

\mathbf{H}_{\nu}\left(x\right)

and

\mathbf{L}_{\nu}\left(x\right)

can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. …