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21: 2.6 Distributional Methods

… ►This leads to integrals of the form … ►The distribution method outlined here can be extended readily to functions $f(t)$ having an asymptotic expansion of the form …An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). … ►It is easily seen that $K^{+}$ forms a commutative, associative linear algebra. … ►On inserting this identity into (2.6.54), we immediately encounter divergent integrals of the form …

22: Errata

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The following additions were made in Chapter 18:

• In Subsection 18.2(i), Equation (18.2.1_5); the paragraph title “Orthogonality on Finite Point Sets” has been changed to “Orthogonality on Countable Sets”, and there are minor changes in the presentation of the final paragraph, including a new equation (18.2.4_5). The presentation of Subsection 18.2(iii) has changed, Equation (18.2.5_5) was added and an extra paragraph on standardizations has been included. The presentation of Subsection 18.2(iv) has changed and it has been expanded with two extra paragraphs and several new equations, (18.2.9_5), (18.2.11_1)–(18.2.11_9). Subsections 18.2(v) (with (18.2.12_5), (18.2.14)–(18.2.17)) and 18.2(vi) (with (18.2.17)–(18.2.20)) have been expanded. New subsections, 18.2(vii)–18.2(xii), with Equations (18.2.21)–(18.2.46),

• A new introduction, minor changes in the presentation, and three new paragraphs.

• Extra details for Chebyshev polynomials, and Equations (18.5.4_5), (18.5.11_1)–(18.5.11_4), (18.5.17_5).

• Line numbers and two extra rows were added to Table 18.8.1.

• Subsection 18.9(i) has been expanded, and 18.9(iii) has some additional explanation. Equations (18.9.2_1), (18.9.2_2), (18.9.18_5) and Table 18.9.2 were added.

• Three extra generating functions, (18.12.2_5), (18.12.3_5), (18.12.17).

• Equation (18.14.3_5). New subsection, 18.14(iv), with Equations (18.14.25)–(18.14.27).

• Equation (18.15.4_5).

• The title of Subsection 18.16(iii) was changed from “Ultraspherical and Legendre” to “Ultraspherical, Legendre and Chebyshev”. New subsection, 18.16(vii) Discriminants, with Equations (18.16.19)–(18.16.21).

• Extra explanatory text at many places and seven extra integrals (18.17.16_5), (18.17.21_1)–(18.17.21_3), (18.17.28_5), (18.17.34_5), (18.17.41_5).

• Extra explanatory text at several places and the title of Subsection 18.18(iv) was changed from “Connection Formulas” to “Connection and Inversion Formulas”.

• A new introduction.

• Equation (18.21.13).

• Extra explanatory text in Subsection 18.25(i) and the title of Subsection 18.25(ii) was changed from “Weights and Normalizations: Continuous Cases” to “Weights and Standardizations: Continuous Cases”.

• In Subsection 18.26(i) an extra paragraph on dualities has been included, with Equations (18.26.4_1), (18.26.4_2).

• Extra text at the start of this section and twenty seven extra formulas, (18.27.4_1), (18.27.4_2), (18.27.6_5), (18.27.9_5), (18.27.12_5), (18.27.14_1)–(18.27.14_6), (18.27.17_1)–(18.27.17_3), (18.27.20_5), (18.27.25), (18.27.26), (18.28.1_5).

• A big expansion. Six extra formulas in Subsection 18.28(ii) ((18.28.6_1)–(18.28.6_5)) and three extra formulas in Subsection 18.28(viii) ((18.28.21)–(18.28.23)). New subsections, 18.28(ix)–18.28(xi), with Equations (18.28.23)–(18.28.34).

• Originally this section did not have subsections. The original seven formulas have now more explanatory text and are split over two subsections. New subsections 18.30(iii)–18.30(viii), with Equations (18.30.8)–(18.30.31).

• This short section has been expanded, with Equation (18.32.2).

• Additional references and a new large subsection, 18.33(vi), including Equations (18.33.17)–(18.33.32).

• This section has been expanded, including an extra orthogonality relations (18.34.5_5), (18.34.7_1)–(18.34.7_3).

• This section on Pollaczek polynomials has been significantly updated with much more explanations and as well to include the Pollaczek polynomials of type 3 which are the most general with three free parameters. The Pollaczek polynomials which were previously treated, namely those of type 1 and type 2 are special cases of the type 3 Pollaczek polynomials. In the first paragraph of this section an extensive description of the relations between the three types of Pollaczek polynomials is given which was lacking previously. Equations (18.35.0_5), (18.35.2_1)–(18.35.2_5), (18.35.4_5), (18.35.6_1)–(18.35.6_6), (18.35.10).

• This section on miscellaneous polynomials has been expanded with new subsections, 18.36(v) on non-classical Laguerre polynomials and 18.36(vi) with examples of exceptional orthogonal polynomials, with Equations (18.36.1)–(18.36.10). In the titles of Subsections 18.36(ii) and 18.36(iii) we replaced “OP’s” by “Orthogonal Polynomials”.

• The paragraphs of Subsection 18.38(i) have been re-ordered and one paragraph was added. The title of Subsection 18.38(ii) was changed from “Classical OP’s: Other Applications” to “Classical OP’s: Mathematical Developments and Applications”. Subsection 18.38(iii) has been expanded with seven new paragraphs, and Equations (18.38.4)–(18.38.11).

• This section was completely rewritten. The previous 18.39(i) Quantum Mechanics has been replaced by Subsections 18.39(i) Quantum Mechanics and 18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom, containing the same essential information; the original content of the subsection is reproduced below for reference. Subsection 18.39(ii) was moved to 18.39(v) Other Applications. New subsections, 18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences, 18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods; Equations (18.39.7)–(18.39.48); and Figures 18.39.1, 18.39.2.

  The original text of 18.39(i) Quantum Mechanics was:

  “Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. Consider, for example, the one-dimensional form of this equation for a particle of mass $m$ with potential energy $V(x)$:

  errata.1 $$\left(\frac{-{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}+V(x)\right)\psi (x,t)=i\mathrm{\hslash }\frac{\partial }{\partial t}\psi (x,t),$$

  where $\mathrm{\hslash }$ is the reduced Planck’s constant. On substituting $\psi (x,t)=\eta (x)\zeta (t)$, we obtain two ordinary differential equations, each of which involve the same constant $E$. The equation for $\eta (x)$ is

  errata.2 $$\frac{d^2\eta }{{dx}^2}+\frac{2m}{{\mathrm{\hslash }}^2}\left(E-V(x)\right)\eta =0.$$

  For a harmonic oscillator, the potential energy is given by

  errata.3 $$V(x)=\frac{1}{2}m{\omega }^2{x}^2,$$

  where $\omega$ is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval , the constant $E$ (the total energy of the particle) must satisfy

  errata.4 $$E=E_n=\left(n+\frac{1}{2}\right)\mathrm{\hslash }\omega ,$$ $n=0,1,2,\mathrm{\dots }$.

  The corresponding eigenfunctions are

  errata.5 $${\eta }_n(x)={\pi }^{-\frac{1}{4}}{2}^{-\frac{1}{2}n}{(n!b)}^{-\frac{1}{2}}{H}_n\left(x/b\right){\mathrm{e}}^{-x^2/2b^2},$$

  where $b={(\mathrm{\hslash }/m\omega )}^{1/2}$, and $H_n$ is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 71-72).

  A second example is provided by the three-dimensional time-independent Schrödinger equation

  errata.6 $${\nabla }^2\psi +\frac{2m}{{\mathrm{\hslash }}^2}\left(E-V(𝐱)\right)\psi =0,$$

  when this is solved by separation of variables in spherical coordinates (§1.5(ii)). The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. See Seaborn (1991, pp. 69-75).

  For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).”

• The old section is now Subsection 18.40(i) and a large new subsection, 18.40(ii), on the classical moment problem has been added, with formulae (18.40.1)–(18.40.10) and Figures 18.40.1, 18.40.2.

…

23: 10.25 Definitions

… ►

§10.25(ii) Standard Solutions

… ►The defining property of the second standard solution $K_{\nu }\left(z\right)$ of (10.25.1) is …

24: Bibliography C

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Three improvements in reduction and computation of elliptic integrals. J. Res. Nat. Inst. Standards Tech. 107 (5), pp. 413–418.

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External Links:

FullText

Referenced by:

§19.22(ii), §19.29(ii), §19.36(i), §19.8(i).

Encodings:

BibTeX

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Intégrandes à deux formes quadratiques. C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).

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External Links:

MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§19.31.

Encodings:

BibTeX

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Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.16.

Encodings:

BibTeX

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Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.

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External Links:

ISBN 0-387-94609-8; 0-387-98998-6, MathReview (M. Ram Murty), ZentralBlatt

Referenced by:

§23.20(v).

Encodings:

BibTeX

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Notes:

Complex interval standard function library. For details see Neher (2007)

External Links:

CoStLy

Referenced by:

§ ‣ Software Cross Index, M. Neher (2007).

Encodings:

BibTeX

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25: Bibliography G

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A table of integrals of the exponential integral. J. Res. Nat. Bur. Standards Sect. B 73B, pp. 191–210.

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§6.14(iii), §6.17, §6.7(iv).

Encodings:

BibTeX

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A table of integrals of the error function. II. Additions and corrections. J. Res. Nat. Bur. Standards Sect. B 75B, pp. 149–163.

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Notes:

See for the first part same journal, Sect. B 73, 1-20 (1969)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii).

Encodings:

BibTeX

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Definite integrals of the complete elliptic integral $K$ . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.

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External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.13(i).

Encodings:

BibTeX

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Tables of binomial coefficients and Stirling numbers. J. Res. Nat. Bur. Standards Sect. B 80B (1), pp. 99–171.

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External Links:

MathReview (M. S. Cheema), ZentralBlatt

Referenced by:

§26.21.

Encodings:

BibTeX

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General form of the quantum-defect theory. Phys. Rev. A 19 (4), pp. 1485–1509.

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External Links:

Document

Referenced by:

item Greene et al. (1979):, Notations F, Notations F, Notations G.

Encodings:

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26: 14.2 Differential Equations

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§14.2(i) Legendre’s Equation

… ►Standard solutions: ${𝖯}_{\nu }\left(\pm x\right)$, ${𝖰}_{\nu }\left(\pm x\right)$, ${𝖰}_{-\nu -1}\left(\pm x\right)$, $P_{\nu }\left(\pm x\right)$, $Q_{\nu }\left(\pm x\right)$, $Q_{-\nu -1}\left(\pm x\right)$. … ►

§14.2(ii) Associated Legendre Equation

… ►Standard solutions: ${𝖯}_{\nu }^{\mu }\left(\pm x\right)$, ${𝖯}_{\nu }^{-\mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{\mu }\left(\pm x\right)$, ${𝖰}_{-\nu -1}^{\mu }\left(\pm x\right)$, $P_{\nu }^{\mu }\left(\pm x\right)$, $P_{\nu }^{-\mu }\left(\pm x\right)$, ${𝑸}_{\nu }^{\mu }\left(\pm x\right)$, ${𝑸}_{-\nu -1}^{\mu }\left(\pm x\right)$. … ►When $\mu -\nu =0,-1,-2,\mathrm{\dots }$, or $\mu +\nu =-1,-2,-3,\mathrm{\dots }$, ${𝖯}_{\nu }^{-\mu }\left(x\right)$ and ${𝖯}_{\nu }^{-\mu }\left(-x\right)$ are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair. …

27: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►These are based on the Liouville normal form of (1.13.29). … ► … ► … ►compare (1.18.30) and (1.18.45), and the eigenfunction expansions are of the form … ►Consider formally self-adjoint operators of the form …

28: 3.5 Quadrature

… ►Similar results hold for the trapezoidal rule in the form … ►If $f\in C^{2m+2}[a,b]$, then the remainder $E_n(f)$ in (3.5.2) can be expanded in the form … ►Integrals of the form … ►Other contour integrals occur in standard integral transforms or their inverses, for example, Hankel transforms (§10.22(v)), Kontorovich–Lebedev transforms (§10.43(v)), and Mellin transforms (§1.14(iv)). … ►The standard Monte Carlo method samples points uniformly from the integration region to estimate the integral and its error. …

29: 13.14 Definitions and Basic Properties

… ►

Standard Solutions

►Standard solutions are: … ►Although $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$, many formulas containing $M_{\kappa ,\mu }\left(z\right)$ continue to apply in their limiting form. … ►

§13.14(iii) Limiting Forms as $z\to 0$

… ►

§13.14(iv) Limiting Forms as $z\to \mathrm{\infty }$

…