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21: 32.8 Rational Solutions

§32.8 Rational Solutions

… ►Special rational solutions of ${\text{P}}_{\text{III}}$ are … ►These solutions have the form … ►These rational solutions have the form … ►

22: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3). Such a solution is given in terms of a Riemann theta function with two phases. …The agreement of these solutions with two-dimensional surface water waves in shallow water was considered in Hammack et al. (1989, 1995).

23: 24.19 Methods of Computation

… ►For number-theoretic applications it is important to compute $B_{2n}(modp)$ for $2n\le p-3$; in particular to find the irregular pairs $(2n,p)$ for which $B_{2n}\equiv 0(modp)$. … ►

A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs $(2n,p)$. Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

24: Bibliography O

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Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.

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External Links:

ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §13.29(i), §13.7(iii), §2.11(v), §9.7(v).

Encodings:

BibTeX

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On the asymptotic and numerical solution of linear ordinary differential equations. SIAM Rev. 40 (3), pp. 463–495.

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External Links:

ISSN 1095-7200, Document, MathReview (W. Balser), ZentralBlatt

Referenced by:

§16.25, §2.7(ii), §3.7(iii).

Encodings:

BibTeX

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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Numerical solution of Riemann-Hilbert problems: Painlevé II. Found. Comput. Math. 11 (2), pp. 153–179.

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External Links:

ISSN 1615-3375, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§32.17, §32.17.

Encodings:

BibTeX

… ►

Solution of Equations in Euclidean and Banach Spaces. Pure and Applied Mathematics, Vol. 9, Academic Press, New York-London.

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Notes:

Third edition of Solution of equations and systems of equations

External Links:

MathReview (J. Burlak), ZentralBlatt

Referenced by:

§3.3(v), §3.8(iii).

Encodings:

BibTeX

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25: 23.20 Mathematical Applications

… ►Values of $x$ are then found as integer solutions of $x^3+ax+b-y^2=0$ (in particular $x$ must be a divisor of $b-y^2$). … ►For further information, including the application of (23.20.7) to the solution of the general quintic equation, see Borwein and Borwein (1987, Chapter 4). …

26: 19.15 Advantages of Symmetry

… ►Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s $F_D$ (Carlson (1961b)). …

27: Errata

… ►In particular, these are: §§18.2(vii)–18.2(xii), §18.14(iv), §18.16(vii), §§18.28(ix)–18.28(xi), §§18.30(iii)–18.30(viii) (Section 18.30), §18.33(vi), §18.36(v), §18.36(vi), §§18.39(iii)–18.39(v), §18.40(i), §18.40(ii) (Section 18.40), as well as many new equations, new figures, namely Figures: 18.39.1, 18.39.2, 18.40.1, 18.40.2, and updates to the main text. … ►In particular, these are: §1.2(v), §1.2(vi), §1.3(iv), §1.10(xi), §1.13(viii), §§1.18(i)–1.18(x) (Section 1.18), as well as many new equations and updates to the main text. … ►

Previously the notation $h(n)$ was used for the harmonic number $H_n$ (defined in (25.11.33)). The more widely used notation $H_n$ will now be used throughout the DLMF. In particular, this change was made in (25.11.32), (25.11.33), (25.16.5) and (25.16.13) (suggested by Gergő Nemes on 2021-08-23).

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Some equations were moved between §19.16(i) and §19.23. In particular, (19.16.2_5), which was previously (19.23.7), now serves as the definition of $R_G(x,y,z)$. Furthermore, (19.23.6_5) was previously (19.16.3).

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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28: Mathematical Introduction

… ►Particular care is taken with topics that are not dealt with sufficiently thoroughly from the standpoint of this Handbook in the available literature. These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). …

29: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►In particular, … ►For asymptotic expansions for large values of $a$ and/or $b$ of the $x$-solution of the equation …

30: 14.19 Toroidal (or Ring) Functions

… ►When $\nu =n-\frac{1}{2}$, $n=0,1,2,\mathrm{\dots }$, $\mu \in ℝ$, and $x\in (1,\mathrm{\infty })$ solutions of (14.2.2) are known as toroidal or ring functions. …In particular, for $\mu =0$ and $\nu =\pm \frac{1}{2}$ see §14.5(v). …