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11: 16.23 Mathematical Applications

… ►A variety of problems in classical mechanics and mathematical physics lead to Picard–Fuchs equations. … ►A substantial transition occurs in a random graph of

n

vertices when the number of edges becomes approximately

\frac{1}{2}n

. … ►The Bieberbach conjecture states that if

\sum_{n=0}^{\infty}a_{n}z^{n}

is a conformal map of the unit disk to any complex domain, then

|a_{n}|\leq n|a_{1}|

. In the proof of this conjecture de Branges (1985) uses the inequality …The proof of this inequality is given in Askey and Gasper (1976). …

12: Donald St. P. Richards

… ►He is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis: A Conference in Honor of R. A. Kunze (with T. …

13: 18.15 Asymptotic Approximations

… ►For higher coefficients see Baratella and Gatteschi (1988), and for another estimate of the error term in a related expansion see Wong and Zhao (2003). For large

\beta

, fixed

\alpha

, and

0\leq n/\beta\leq c

, Dunster (1999) gives asymptotic expansions of

P^{(\alpha,\beta)}_{n}\left(z\right)

that are uniform in unbounded complex

z

-domains containing

z=\pm 1

. …The latter expansions are in terms of Bessel functions, and are uniform in complex

z

-domains not containing neighborhoods of 1. … ►For a bound on the error term in (18.15.10) see Szegő (1975, Theorem 8.21.11). … ►

In Terms of Elementary Functions

…

14: 2.5 Mellin Transform Methods

… ►Next from Table 2.5.1 we observe that the integrals for the transform pair

\mathscr{M}\mskip-3.0muf_{j}\mskip 3.0mu\left(1-z\right)

and

\mathscr{M}\mskip-3.0muh_{k}\mskip 3.0mu\left(z\right)

are absolutely convergent in the domain

D_{jk}

specified in Table 2.5.2, and these domains are nonempty as a consequence of (2.5.19) and (2.5.20). …

15: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►This insures the vanishing of the boundary terms in (1.18.26), and also is a choice which indicates that

\mathcal{D}(T)=\mathcal{D}({T}^{*})

, as

f(x)

and

g(x)

satisfy the same boundary conditions and thus define the same domains. … ►Let

T

be a linear operator on

V

with dense domain

\mathcal{D}(T)

and with range

\mathcal{R}(T)=\{Tv\mid v\in\mathcal{D}(T)\}

. …

16: Bibliography B

… ►

L. V. Babushkina, M. K. Kerimov, and A. I. Nikitin (1988b) Algorithms for evaluating spherical Bessel functions in the complex domain. Zh. Vychisl. Mat. i Mat. Fiz. 28 (12), pp. 1779–1788, 1918.

ⓘ

Notes:: English translation in U.S.S.R. Comput. Math. and Math. Phys. 28(1988), no. 6, 122–128
External Links:: ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §10.77(iv).
Encodings:: BibTeX

…

17: 2.8 Differential Equations with a Parameter

… ►in which

\xi

ranges over a bounded or unbounded interval or domain

\mathbf{\Delta}

, and

\psi(\xi)

C^{\infty}

or analytic on

\mathbf{\Delta}

. … ►

2.8.11

W_{n,1}(u,\xi)=e^{u\xi}\left(\sum_{s=0}^{n-1}\frac{A_{s}(\xi)}{u^{s}}+O\left(% \frac{1}{u^{n}}\right)\right),

\xi\in\mathbf{\Delta}_{1}(\alpha_{1})

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $n$ : positive integer, $W_{n,j}(u,\xi)$ : solution, $A_{s}(\xi)$ : coefficients, $\mathbf{\Delta}_{j}(\alpha_{j})$ : domain and $u$ : large real or complex parameter
Referenced by:: §2.8(ii), §2.8(ii)
Permalink:: http://dlmf.nist.gov/2.8.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §2.8(ii), §2.8 and Ch.2

►

2.8.12

W_{n,2}(u,\xi)=e^{-u\xi}\left(\sum_{s=0}^{n-1}(-1)^{s}\frac{A_{s}(\xi)}{u^{s}}% +O\left(\frac{1}{u^{n}}\right)\right),

\xi\in\mathbf{\Delta}_{2}(\alpha_{2})

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $n$ : positive integer, $W_{n,j}(u,\xi)$ : solution, $A_{s}(\xi)$ : coefficients, $\mathbf{\Delta}_{j}(\alpha_{j})$ : domain and $u$ : large real or complex parameter
Referenced by:: §2.8(ii), §2.8(ii)
Permalink:: http://dlmf.nist.gov/2.8.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §2.8(ii), §2.8 and Ch.2

… ►The regions of validity

\mathbf{\Delta}_{j}(\alpha_{j})

comprise those points

\xi

that can be joined to

\alpha_{j}

\mathbf{\Delta}

by a path

\mathscr{Q}_{j}

along which

\Re v

is nondecreasing

(j=1)

or nonincreasing

(j=2)

v

passes from

\alpha_{j}

\xi

. …

18: 1.5 Calculus of Two or More Variables

… ►A more general concept of integrability (both finite and infinite) for functions on domains in

{\mathbb{R}}^{n}

is Lebesgue integrability. …

… ►Bonita V. Saunders, born in Portsmouth, Virginia, is a member of the Applied and Computational Mathematics Division of the Information Technology Laboratory at the National Institute of Standards and Technology. … ►In 1985 she was the first African American and first woman to obtain a Ph. … ►In 2001 she was selected by the National Association of Mathematicians (NAM) to present the Claytor Lecture at the Joint Mathematics Meetings in New Orleans in memory of W. …In 2019 she was named a Fellow of the Washington Academy of Sciences. ►As the principal developer of graphics for the DLMF, she has collaborated with other NIST mathematicians, computer scientists, and student interns to produce informative graphs and dynamic interactive visualizations of elementary and higher mathematical functions over both simply and multiply connected domains. …