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1: 32.3 Graphics

§32.3 Graphics

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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

2: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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§28.5(ii) Graphics: Line Graphs of Second Solutions of Mathieu’s Equation

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3: 9.13 Generalized Airy Functions

… ►Properties and graphs of $U_m(t)$, $V_m(t)$, ${\overline{V}}_m(t)$ are included in Olver (1977a) together with properties and graphs of real solutions of the equation …

4: 22.19 Physical Applications

… ►Figure 22.19.1 shows the nature of the solutions $\theta (t)$ of (22.19.3) by graphing $\mathrm{am}(x,k)$ for both $0\le k\le 1$, as in Figure 22.16.1, and $k\ge 1$, where it is periodic. …

5: 4.13 Lambert $W$-Function

… ►The Lambert $W$-function $W\left(z\right)$ is the solution of the equation … ►On the $z$-interval $[0,\mathrm{\infty })$ there is one real solution, and it is nonnegative and increasing. On the $z$-interval $(-{\mathrm{e}}^{-1},0)$ there are two real solutions, one increasing and the other decreasing. … ►The decreasing solution can be identified as $W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$. Other solutions of (4.13.1) are other branches of $W\left(z\right)$. …

6: 16.23 Mathematical Applications

… ►These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. … ►

§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. …

7: Bonita V. Saunders

… ►She is the Visualization Editor and principal designer of graphs and visualizations for the DLMF. … ►Her research interests include numerical grid generation, numerical solution of partial differential equations, and visualization of special functions. … ►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. …

8: 28.3 Graphics

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§28.3(i) Line Graphs: Mathieu Functions with Fixed $q$ and Variable $x$

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Even $\pi$-Periodic Solutions

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Even $\pi$-Antiperiodic Solutions

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Odd $\pi$-Antiperiodic Solutions

… ►For further graphs see Jahnke et al. (1966, pp. 264–265 and 268–275). …

9: 27.21 Tables

… ►Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare $\pi \left(x\right),x/\ln x$, and $\mathrm{li}\left(x\right)$. …Table II lists all solutions $n$ of the equation $f\left(n\right)=m$ for all $m\le 2500$, where $f\left(n\right)$ is defined by (27.14.2). Table III lists all solutions $n\le {10}^4$ of the equation $d\left(n\right)=m$, and Table IV lists all solutions $n$ of the equation $\sigma (n)=m$ for all $m\le {10}^4$. …

10: Frank W. J. Olver

… ►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. … ►He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i. …, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. …