Liouville%E2%80%93Green approximation theorem

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31: Bibliography H

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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MathReview (L. Berg)

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§7.14(iii).

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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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MathReview (V. L. Makarov)

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§7.14(iii).

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Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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MathReview (V. L. Makarov)

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§7.14(iii).

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Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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MathReview (M. Lawrence Glasser)

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§7.14(iii).

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Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.

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ZentralBlatt

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§7.18(iii), §7.18(iv), §7.18(v), §7.18(vi).

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32: 18.39 Applications in the Physical Sciences

… ►An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with $L^2$ eigenfunctions vanishing at the end points, in this case $\pm \mathrm{\infty }$ see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem. …Both satisfy Sturm’s theorem. … ►A relativistic treatment becoming necessary as $Z$ becomes large as corrections to the non-relativistic Schrödinger picture are of approximate order ${\left(\alpha Z\right)}^2\sim {\left(Z/137\right)}^2$, $\alpha$ being the dimensionless fine structure constant $e^2/(4\pi {\epsilon }_0\mathrm{\hslash }c)$, where $c$ is the speed of light. … ►While $s$ in the basis of (18.39.44) is simply a variational parameter, care must be taken, or the relationship between the results of the matrix variational approximation and the Pollaczek polynomials is lost, although this has no effect on the use of the variational approximations Reinhardt (2021a, b). … ►This equivalent quadrature relationship, see Heller et al. (1973), Yamani and Reinhardt (1975), allows extraction of scattering information from the finite dimensional $L^2$ functions of (18.39.53), provided that such information involves potentials, or projections onto $L^2$ functions, exactly expressed, or well approximated, in the finite basis of (18.39.44). …

33: Bibliography J

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Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. D 1 (1), pp. 80–158.

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ISSN 0167-2789, Document, MathReview, ZentralBlatt

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§32.16.

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Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions ${\overline{ps}}_n^r(\eta ,h)$ and ${\overline{qs}}_n^r(\eta ,h)$ for large $h$ . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

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FullText, MathReview, ZentralBlatt

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§30.9(ii).

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On the simple cubic lattice Green function. Philos. Trans. Roy. Soc. London Ser. A 273, pp. 583–610.

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FullText, MathReview, ZentralBlatt

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§31.17(ii).

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On the cubic lattice Green functions. Proc. Roy. Soc. London Ser. A 445, pp. 463–477.

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ISSN 0962-8444, FullText, MathReview (Peter J. Forrester), ZentralBlatt

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§31.17(ii), §31.7(i).

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Modifications of Coulombic interactions by polarizable atoms. Math. Proc. Cambridge Philos. Soc. 80 (3), pp. 535–539.

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ISSN 0305-0041, Document, MathReview (Cataldo Agostinelli)

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§34.12.

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34: Errata

… ►We now include Markov’s Theorem. In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’s theorem. … ►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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A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order $\alpha$ was more precisely identified as the Riemann-Liouville fractional integral operator of order $\alpha$, and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).

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35: Bibliography K

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A proof of the $q$-Macdonald-Morris conjecture for $B{C}_n$ . Mem. Amer. Math. Soc. 108 (516), pp. vi+80.

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ISSN 0065-9266, MathReview (Eric M. Opdam), ZentralBlatt

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§17.14.

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Linear convergence and the bisection algorithm. Amer. Math. Monthly 93 (1), pp. 48–51.

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ISSN 0002-9890, FullText, MathReview (J. G. Herriot), ZentralBlatt

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§3.8(iii).

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An extension of Saalschütz’s summation theorem for the series ${}_{r+3}{}^{}F_{r+2}^{}$ . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.

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ISSN 1065-2469, Document, FullText, MathReview (Christian Lavault), ZentralBlatt

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§16.4(ii), §16.4(ii).

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A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.

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Document, MathReview (J. Meixner), ZentralBlatt

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§30.10.

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A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

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Document, MathReview (G. Gasper), ZentralBlatt

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§18.10(i).

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36: 10.21 Zeros

… ►The approximations that follow in §10.21(viii) do not suffer from this drawback. ►

§10.21(viii) Uniform Asymptotic Approximations for Large Order

►As $\nu \to \mathrm{\infty }$ the following four approximations hold uniformly for $m=1,2,\mathrm{\dots }$: … ►For describing the distribution of complex zeros by methods based on the Liouville–Green (WKB) approximation for linear homogeneous second-order differential equations, see Segura (2013). … ►This information includes asymptotic approximations analogous to those given in §§10.21(vi), 10.21(vii), and 10.21(x). …

37: 8.23 Statistical Applications

… ►In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of $Q(a,x)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …

38: 26.13 Permutations: Cycle Notation

… ►See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. …

39: 22.7 Landen Transformations

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40: Bibliography C

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Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.

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ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt

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§16.24(ii).

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The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.

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MathReview (K. Goldberg), ZentralBlatt

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§24.6.

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Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.

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Document, MathReview (Y. L. Luke), ZentralBlatt

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§19.26(i).

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The universal von Staudt theorems. Trans. Amer. Math. Soc. 315 (2), pp. 591–603.

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ISSN 0002-9947, FullText, MathReview (T. Metsänkylä), ZentralBlatt

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§24.16(iii).

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Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edition, Applied Mathematical Sciences, Vol. 93, Springer-Verlag, Berlin.

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ISBN 3-540-62838-X, MathReview, ZentralBlatt

Referenced by:

§10.73(i).

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