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1: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

2: 27.17 Other Applications

… ►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. …Apostol and Zuckerman (1951) uses congruences to construct magic squares. …

3: Joyce E. Conlon

… ►In 1999 she joined the NIST Mathematical and Computational Sciences Division, where she contributed to the DLMF project, especially in the construction of the bibliography. …

4: Tom H. Koornwinder

… ►Currently he is on the editorial board for Constructive Approximation, and is editor for the volume on Multivariable Special Functions in the ongoing Askey–Bateman book project. …

5: Bruce R. Miller

… ►In particular, he developed the LaTeXML system used for converting the LaTeX source documents into XML and MathML from which the DLMF website is constructed. …

6: Peter L. Walker

… ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. …

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. … ►In regions in which (10.72.1) has a simple turning point

z_{0}

, that is,

f(z)

and

g(z)

are analytic (or with weaker conditions if

z=x

is a real variable) and

z_{0}

is a simple zero of

f(z)

, asymptotic expansions of the solutions

w

for large

u

can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order

\tfrac{1}{3}

(§9.6(i)). … ►If

f(z)

has a double zero

z_{0}

, or more generally

z_{0}

is a zero of order

m

m=2,3,4,\dotsc

, then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order

1/(m+2)

. … ►In regions in which the function

f(z)

has a simple pole at

z=z_{0}

and

(z-z_{0})^{2}g(z)

is analytic at

z=z_{0}

(the case

\lambda=-1

in §10.72(i)), asymptotic expansions of the solutions

w

of (10.72.1) for large

u

can be constructed in terms of Bessel functions and modified Bessel functions of order

\pm\sqrt{1+4\rho}

, where

\rho

is the limiting value of

(z-z_{0})^{2}g(z)

z\to z_{0}

. … ►Then for large

u

asymptotic approximations of the solutions

w

can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on

u

and

\alpha

). …

8: 13.27 Mathematical Applications

… ►Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. …

9: Mourad E. H. Ismail

… ►Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive Approximation; Journal of Approximation Theory; and Integral Transforms and Special Functions. …

10: Daniel W. Lozier

… ►His research interests have centered on numerical analysis, special functions, computer arithmetic, and mathematical software construction and testing. …