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31: Errata

… ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

The following additions were made in Chapter 1:

• New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

• The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).

• In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).

• In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

• New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

• New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

• Another form of Parseval’s formula, (1.14.7_5).

• We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

• Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

• An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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32: Bibliography W

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Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.

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External Links:

ISSN 0029-599X, Document, MathReview (Sergey M. Zagorodnyuk), ZentralBlatt

Referenced by:

§2.9(iii).

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Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.

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External Links:

ISSN 0025-5718, Document, MathReview (B. G. Pachpatte), ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

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Linear Turning Point Theory. Applied Mathematical Sciences No. 54, Springer-Verlag, New York.

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External Links:

ISBN 0-387-96046-5, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

Ch.9.

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Linear and Nonlinear Waves. John Wiley & Sons, New York.

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Notes:

Reprinted in 1999.

External Links:

ISBN 0471359424, MathReview, ZentralBlatt

Referenced by:

§9.16.

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Asymptotics of linear recurrences. Anal. Appl. (Singap.) 12 (4), pp. 463–484.

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External Links:

ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

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33: 31.14 General Fuchsian Equation

… ►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). The algorithm returns a list of solutions if they exist. …

34: 15.5 Derivatives and Contiguous Functions

… ►By repeated applications of (15.5.11)–(15.5.18) any function $F(a+k,b+\mathrm{\ell };c+m;z)$, in which $k,\mathrm{\ell },m$ are integers, can be expressed as a linear combination of $F(a,b;c;z)$ and any one of its contiguous functions, with coefficients that are rational functions of $a,b,c$, and $z$. …

35: 3.8 Nonlinear Equations

… ►If $p=1$ and , then the convergence is said to be linear or geometric. … … ►

Regula Falsi

… ►Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: … ►This is useful when $f(z)$ satisfies a second-order linear differential equation because of the ease of computing $f^{\prime \prime }(z_n)$. …

36: 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).

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On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.7(ii).

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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).

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Hyperasymptotic solutions of second-order linear differential equations. II. Methods Appl. Anal. 2 (2), pp. 198–211.

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External Links:

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

Referenced by:

§10.17(v), §10.40(iv), §2.11(v), §9.7(v).

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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.

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External Links:

ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§2.11(v), §2.7(ii).

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37: About Color Map

… ►Surface visualizations in the DLMF represent functions of the form $z=f(x,y)$ by the height $z$ or the magnitude, $|z|$, for complex functions, over the $x\times y$ plane. We use color to augment these vizualizations, either to reinforce the recognition of the height, or to convey an extra dimension to represent the phase of complex valued functions. … ►The following figure illustrates the piece-wise linear mapping of the height to each of the color components red, green and blue, written as $⟨R,G,B⟩$. … ►By painting the surfaces with a color that encodes the phase, $\mathrm{ph}f$, both the magnitude and phase of complex valued functions can be displayed. … ►We therefore use a piecewise linear mapping as illustrated below, that takes phase $0$ to red, $\pi /2$ to yellow, $\pi$ to cyan and $3\pi /2$ to blue. …

38: Bibliography Z

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Algorithm 916: computing the Faddeyeva and Voigt functions. ACM Trans. Math. Software 38 (2), pp. Art. 15, 22.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

2nd item, 5th item.

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Numerical analysis of Struve functions with applications to other special functions. Ann. Numer. Math. 2 (1-4), pp. 199–208.

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External Links:

ISSN 1021-2655, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§11.7(ii).

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Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.

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External Links:

ISSN 0024-6115, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

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Error bounds for asymptotic solutions of second-order linear difference equations. J. Comput. Appl. Math. 71 (2), pp. 191–212.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§2.9(i), §2.9(ii).

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The summation of series of hyperbolic functions. SIAM J. Math. Anal. 10 (1), pp. 192–206.

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External Links:

ISSN 0036-1410, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§4.41.

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39: 3.3 Interpolation

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Linear Interpolation

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Example

… ►For interpolation of a bounded function $f$ on $ℝ$ the cardinal function of $f$ is defined by …where …is called the Sinc function. …

40: 1.9 Calculus of a Complex Variable

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Analyticity

… ►The linear transformation $f(z)=az+b$, $a\ne 0$, has $f^{\prime }(z)=a$ and $w=f(z)$ maps $ℂ$ conformally onto $ℂ$. ►

Bilinear Transformation

… ►Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …