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1: 16.23 Mathematical Applications

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§16.23(ii) Random Graphs

►A substantial transition occurs in a random graph of $n$ vertices when the number of edges becomes approximately $\frac{1}{2}n$. In Janson et al. (1993) limiting distributions are discussed for the sparse connected components of these graphs, and the asymptotics of three ${}_2{}^{}F_2^{}$ functions are applied to compute the expected value of the excess. …

2: 24.3 Graphs

§24.3 Graphs

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Figure 24.3.1: Bernoulli polynomials $B_n\left(x\right)$, $n=2,3,\mathrm{\dots },6$. Magnify

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Figure 24.3.2: Euler polynomials $E_n\left(x\right)$, $n=2,3,\mathrm{\dots },6$. Magnify

3: 26.19 Mathematical Applications

… ►Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). ►Other areas of combinatorial analysis include graph theory, coding theory, and combinatorial designs. …

4: 16.24 Physical Applications

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§16.24(i) Random Walks

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5: 35.9 Applications

… ►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument ${}_p{}^{}F_q^{}$, with $p\le 2$ and $q\le 1$. … ►In the nascent area of applications of zonal polynomials to the limiting probability distributions of symmetric random matrices, one of the most comprehensive accounts is Rains (1998).

6: 32.14 Combinatorics

… ►The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994). … ►See Forrester and Witte (2001, 2002) for other instances of Painlevé equations in random matrix theory.

7: Preface

… ►Abramowitz and Stegun’s Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables is being completely rewritten with regard to the needs of today. …The DLMF will make full use of advanced communications and computational resources to present downloadable math data, manipulable graphs, tables of numerical values, and math-aware search. …

8: 18.4 Graphics

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§18.4(i) Graphs

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Figure 18.4.1: Jacobi polynomials $P_n^{(1.5,-0.5)}\left(x\right)$, $n=1,2,3,4,5$. Magnify

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Figure 18.4.2: Jacobi polynomials $P_n^{(1.25,0.75)}\left(x\right)$, $n=7,8$. … Magnify

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Figure 18.4.3: Chebyshev polynomials $T_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

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Figure 18.4.4: Legendre polynomials $P_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

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9: 32.3 Graphics

§32.3 Graphics

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Figure 32.3.3: $w_k(x)$ for $-12\le x\le 0.73$ and $k=\mathrm{1.85185\hspace{0.33em}3}$, $\mathrm{1.85185\hspace{0.33em}5}$. The two graphs are indistinguishable when $x$ exceeds $-5.2$, approximately. … Magnify

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Figure 32.3.4: $w_k(x)$ for $-12\le x\le 2.3$ and $k=-\mathrm{0.45142\hspace{0.33em}7}$, $-\mathrm{0.45142\hspace{0.33em}8}$. The two graphs are indistinguishable when $x$ exceeds $-4.8$, approximately. … Magnify

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Figure 32.3.5: $w_k(x)$ and $k\mathrm{Ai}\left(x\right)$ for $-10\le x\le 4$ with $k=0.5$. The two graphs are indistinguishable when $x$ exceeds $-0.4$, approximately. Magnify

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Figure 32.3.10: $u_k(x;\frac{5}{2})$ for $-12\le x\le 4$ with $k=\mathrm{0.24499\hspace{0.33em}2}$, $\mathrm{0.24499\hspace{0.33em}3}$. … Magnify

10: Bonita V. Saunders

… ►She is the Visualization Editor and principal designer of graphs and visualizations for the DLMF. … ►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. …