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11: Frank W. J. Olver

… ►g. … ►Most notably, he served as the Editor-in-Chief and Mathematics Editor of the online NIST Digital Library of Mathematical Functions and its 966-page print companion, the NIST Handbook of Mathematical Functions (Cambridge University Press, 2010). … ►

1 Algebraic and Analytic Methods

…

12: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Two elements

u

and

v

V

are orthogonal if

\left\langle u,v\right\rangle=0

. … ►Functions

f,g\in L^{2}\left(X,\,\mathrm{d}\alpha\right)

for which

\left\langle f-g,f-g\right\rangle=0

are identified with each other. … ►The adjoint

{T}^{*}

T

does satisfy

\left\langle Tf,g\right\rangle=\left\langle f,{T}^{*}g\right\rangle

where

\left\langle f,g\right\rangle=\int_{a}^{b}f(x)g(x)\,\mathrm{d}x

. … ►where

\widehat{f}(\lambda_{n})

\frac{1}{\sqrt{\pi}}\int_{0}^{\pi}f(y){\mathrm{e}}^{-2\mathrm{i}ny}\,\mathrm{d% }y=\left\langle f,\phi_{\mathrm{exp}}(n)\right\rangle

, being that of (1.8.3) and (1.8.4). … ►This dilatation transformation, which does require analyticity of

q(x)

in (1.18.28), or an analytic approximation thereto, leaves the poles, corresponding to the discrete spectrum, invariant, as they are, as is the branch point, actual singularities of

\left\langle\left(z-T\right)^{-1}f,f\right\rangle

. …

13: DLMF Project News

error generating summary

14: 1.2 Elementary Algebra

… ►where

\ell

n

n-1

according as

n

is even or odd. … ►where

\ell

= last term of the series =

a+(n-1)d

. … ►The geometric mean

G

and harmonic mean

H

n

positive numbers

a_{1},a_{2},\dots,a_{n}

are given by … ►A column vector of length

n

is an

n\times 1

matrix … ►the

l^{1}

norm …

15: Viewing DLMF Interactive 3D Graphics

… ►Below we provide some notes and links to online material which might be helpful in viewing our visualizations, but please see our Disclaimer. … ►1, some advanced features of X3DOM are currently not fully supported (see x3dom.org). …

16: 1.1 Special Notation

… ► ►►►►

$x, y$	real variables.
…
$\left\langle f,g\right\rangle$	inner, or scalar, product for real or complex vectors or functions.
…
${\mathbf{A}}^{-1}$	inverse of the square matrix $\mathbf{A}$
…

…

17: 27.1 Special Notation

… ► ►►►►►►►

$d, k, m, n$	positive integers (unless otherwise indicated).
…
$\left(d_{1},\dots,d_{n}\right)$	greatest common divisor of $d_{1},\dots,d_{n}$ .
$\sum_{d\mathbin{\|}n}$ , $\prod_{d\mathbin{\|}n}$	sum, product taken over divisors of $n$ .
$\sum_{\left(m,n\right)=1}$	sum taken over $m$ , $1\leq m\leq n$ and $m$ relatively prime to $n$ .
$p,p_{1},p_{2},\dots$	prime numbers (or primes): integers ( $>1$ ) with only two positive integer divisors, $1$ and the number itself.
$\sum_{p}$ , $\prod_{p}$	sum, product extended over all primes.
…

18: Notices

… ►

Index of Selected Software Within the DLMF Chapters

Within each of the DLMF chapters themselves we will provide a list of research software for the functions discussed in that chapter. The purpose of these listings is to provide references to the research literature on the engineering of software for special functions. To qualify for listing, the development of the software must have been the subject of a research paper published in the peer-reviewed literature. If such software is available online for free download we will provide a link to the software.

In general, we will not index other software within DLMF chapters unless the software is unique in some way, such as being the only known software for computing a particular function.

…

19: Bibliography L

… ►

C. G. Lambe (1952) Lamé-Wangerin functions. Quart. J. Math., Oxford Ser. (2) 3, pp. 107–114.

ⓘ

External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §29.17(ii).
Encodings:: BibTeX

… ►

E. R. Love (1972a) Addendum to: “Changing the order of integration”. J. Austral. Math. Soc. 14, pp. 383–384.

ⓘ

External Links:: Document, MathReview (G. W. Johnson), ZentralBlatt
Referenced by:: §1.5(v).
Encodings:: BibTeX

… ►

D. W. Lozier and F. W. J. Olver (1994) Numerical Evaluation of Special Functions. In Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics (Vancouver, BC, 1993), Proc. Sympos. Appl. Math., Vol. 48, pp. 79–125.

ⓘ

Notes:: An updated version exists online.
External Links:: FullText, MathReview (John P. Coleman), ZentralBlatt
Referenced by:: Ch.3.
Encodings:: BibTeX

… ►

Y. L. Luke and J. Wimp (1963) Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray. Math. Comp. 17 (84), pp. 395–404.

ⓘ

External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

Y. L. Luke (1969a) The Special Functions and their Approximations, Vol. 1. Academic Press, New York.

ⓘ

External Links:: MathReview (R. G. Langebartel), ZentralBlatt
Referenced by:: Ch.13, §15.10(ii), §15.12(ii), §15.12(iii), Ch.15, §16.11(iii), §16.11(iii), §16.17, §16.18, §16.18, §16.19, §16.20, §16.21, §16.22, §16.5, §16.5, §16.5, §16.7, §16.8(ii), §16.8(iii), Ch.16, Ch.3.
Encodings:: BibTeX

…

20: Bibliography N

… ►

G. Nemes (2013b) Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.

ⓘ

External Links:: ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt
Referenced by:: §5.11(ii), §5.11(ii).
Encodings:: BibTeX

… ►

G. Nemes (2014b) The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

ⓘ

External Links:: ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt
Referenced by:: (11.11.10), (11.11.8), §11.11(iii).
Encodings:: BibTeX

… ►

G. Nemes (2015c) The resurgence properties of the incomplete gamma function II. Stud. Appl. Math. 135 (1), pp. 86–116.

ⓘ

External Links:: ISSN 0022-2526, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.11(iii).
Encodings:: BibTeX

… ►

G. Nikolov and V. Pillwein (2015) An extension of Turán’s inequality. Math. Inequal. Appl. 18 (1), pp. 321–335.

ⓘ

External Links:: ISSN 1331-4343, Document, Link, MathReview (Gradimir V. Milovanovic), ZentralBlatt
Referenced by:: §18.14(ii).
Encodings:: BibTeX

… ►

H. M. Nussenzveig (1992) Diffraction Effects in Semiclassical Scattering. Montroll Memorial Lecture Series in Mathematical Physics, Cambridge University Press.

ⓘ

Notes:: Also available electronically through Cambridge Books Online, ISBN 9780511599903.
External Links:: ISBN 9780521383189, Document
Referenced by:: §9.16, §9.16.
Encodings:: BibTeX

…