special function solutions

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11: Peter A. Clarkson

… ►He is a member of the editorial boards of nine international journals and has served as Chair, Vice-Chair, and Secretary of the SIAM Activity Group on Orthogonal Polynomials and Special Functions. ►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. … Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …

12: 18.38 Mathematical Applications

… ►While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. …

13: 28.1 Special Notation

§28.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are the Mathieu functions …and the modified Mathieu functions … ►Alternative notations for the functions are as follows. …

14: 32.1 Special Notation

§32.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►The functions treated in this chapter are the solutions of the Painlevé equations ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$.

15: 2.8 Differential Equations with a Parameter

… ►Many special functions satisfy an equation of the form … ►The transformation is now specialized in such a way that: (a) $\xi$ and $z$ are analytic functions of each other at the transition point (if any); (b) the approximating differential equation obtained by neglecting $\psi (\xi )$ (or part of $\psi (\xi )$) has solutions that are functions of a single variable. … ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). …

16: 10.47 Definitions and Basic Properties

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§10.47(ii) Standard Solutions

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Equation (10.47.2)

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§10.47(iii) Numerically Satisfactory Pairs of Solutions

►For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols $J$, $Y$, $H$, and $\nu$ replaced by $𝗃$, $𝗒$, $𝗁$, and $n$, respectively. …

17: 2.9 Difference Equations

… ►Many special functions that depend on parameters satisfy a three-term linear recurrence relation …

18: 2.11 Remainder Terms; Stokes Phenomenon

… ►Here $\mathrm{erfc}$ is the complementary error function (§7.2(i)), and …Also, … ►

§2.11(v) Exponentially-Improved Expansions (continued)

►Expansions similar to (2.11.15) can be constructed for many other special functions. … …

19: 3.7 Ordinary Differential Equations

… ►For applications to special functions $f$, $g$, and $h$ are often simple rational functions. … ►For general information on solutions of equation (3.7.1) see §1.13. … ►

§3.7(ii) Taylor-Series Method: Initial-Value Problems

… ► … ►The Sturm–Liouville eigenvalue problem is the construction of a nontrivial solution of the system …

20: 27.9 Quadratic Characters

§27.9 Quadratic Characters

… ►If $p$ does not divide $n$, then $(n|p)$ has the value $1$ when the quadratic congruence $x^2\equiv n(modp)$ has a solution, and the value $-1$ when this congruence has no solution. The Legendre symbol $(n|p)$, as a function of $n$, is a Dirichlet character (mod $p$). …Special values include: …