coalescing transition points

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31—40 of 162 matching pages

31: 36.7 Zeros

… ►The zeros in Table 36.7.1 are points in the

\mathbf{x}=(x,y)

plane, where

\operatorname{ph}\Psi_{2}\left(\mathbf{x}\right)

is undetermined. … ►,

y=0

), the number of rings in the

m

th row, measured from the origin and before the transition to hairpins, is given by …

32: 3.1 Arithmetics and Error Measures

… ►A nonzero normalized binary floating-point machine number

x

is represented as … … ► … ►

IEEE Standard

… ►

Rounding

…

33: Bibliography

… ►

M. Abramowitz and P. Rabinowitz (1954) Evaluation of Coulomb wave functions along the transition line. Physical Rev. (2) 96, pp. 77–79.

ⓘ

External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §33.12(i).
Encodings:: BibTeX

… ►

V. I. Arnol’d (1972) Normal forms of functions near degenerate critical points, the Weyl groups

{A}_{k},{D}_{k},{E}_{k}

and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).

ⓘ

Notes:: In Russian. English translation: Functional Anal. Appl., 6(1973), pp. 254–272.
External Links:: MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

►

V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

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Notes:: Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.
External Links:: Document, MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

►

V. I. Arnol’d (1975) Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).

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Notes:: In Russian. English translation: Russian Math. Surveys, 30(1975), no. 5, pp. 1–75.
External Links:: Document, MathReview (D. Siersma), ZentralBlatt
Referenced by:: §36.2(i), Ch.36.
Encodings:: BibTeX

…

34: 2.10 Sums and Sequences

… ►The singularities of

f(z)

on the unit circle are branch points at

z=e^{\pm i\alpha}

. To match the limiting behavior of

f(z)

at these points we set … ►For uniform expansions when two singularities coalesce on the circle of convergence see Wong and Zhao (2005). …

35: Bibliography S

… ►

J. Segura (2002) The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40 (1), pp. 114–133.

ⓘ

External Links:: ISSN 1095-7170, Document, MathReview, ZentralBlatt
Referenced by:: §3.8(v).
Encodings:: BibTeX

… ►

P. N. Shivakumar and J. Xue (1999) On the double points of a Mathieu equation. J. Comput. Appl. Math. 107 (1), pp. 111–125.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §28.7.
Encodings:: BibTeX

… ►

J. H. Silverman and J. Tate (1992) Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-97825-9, MathReview (Andrew Bremner), ZentralBlatt
Referenced by:: §22.18(iv), §23.1, §23.20(ii), §23.20(ii), §23.20(v).
Encodings:: BibTeX

… ►

D. M. Smith (1991) Algorithm 693: A FORTRAN package for floating-point multiple-precision arithmetic. ACM Trans. Math. Software 17 (2), pp. 273–283.

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Notes:: Multiple-precision Fortran.
External Links:: ISSN 0098-3500, Document, GAMS (module 693; package TOMS), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

I. I. Sobelman (1992) Atomic Spectra and Radiative Transitions. 2nd edition, Springer-Verlag, Berlin.

ⓘ

Notes:: A second printing with corrections was published in 1996.
Referenced by:: §34.12.
Encodings:: BibTeX

…

36: 13.2 Definitions and Basic Properties

… ►In effect, the regular singularities of the hypergeometric differential equation at

b

and

\infty

coalesce into an irregular singularity at

\infty

. … ►In general,

U\left(a,b,z\right)

has a branch point at

z=0

. …

37: 15.11 Riemann’s Differential Equation

… ►The most general form is given by … ►Here

\{a_{1},a_{2}\}

\{b_{1},b_{2}\}

\{c_{1},c_{2}\}

are the exponent pairs at the points

\alpha

\beta

\gamma

, respectively. … ►

15.11.3

w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

ⓘ

Defines:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation
Symbols:: $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

… ►These constants can be chosen to map any two sets of three distinct points

\{\alpha,\beta,\gamma\}

and

\{\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\}

onto each other. … ►

15.11.6

P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}\widetilde{\alpha}&\widetilde{% \beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

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Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $t$ : mapping
Permalink:: http://dlmf.nist.gov/15.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(ii), §15.11 and Ch.15

…

38: 28.7 Analytic Continuation of Eigenvalues

… ►The only singularities are algebraic branch points, with

a_{n}\left(q\right)

and

b_{n}\left(q\right)

finite at these points. The number of branch points is infinite, but countable, and there are no finite limit points. …The branch points are called the exceptional values, and the other points normal values. … ►For a visualization of the first branch point of

a_{0}\left(\mathrm{i}\hat{q}\right)

and

a_{2}\left(\mathrm{i}\hat{q}\right)

see Figure 28.7.1. …

39: 29.3 Definitions and Basic Properties

… ►The eigenvalues coalesce according to …

40: Sidebar 9.SB1: Supernumerary Rainbows

… ►Airy invented his function in 1838 precisely to describe this phenomenon more accurately than Young had done in 1800 when pointing out that supernumerary rainbows require the wave theory of light and are impossible to explain with Newton’s picture of light as a stream of independent corpuscles. …