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21: Bibliography P

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Statistical Field Theory. Addison-Wesley, Reading, MA.

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ISBN 0-201-05985-1, MathReview (Tony C. Dorlas), ZentralBlatt

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§22.19(ii).

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La Théorie des Fonctions de Bessel Exposée en vue de ses Applications à la Physique Mathématique. Centre National de la Recherche Scientifique, Paris (French).

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MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.63(ii).

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Gauge Field Theories. Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.

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ISBN 0-521-26537-1, MathReview (G. R. Allcock), ZentralBlatt

Referenced by:

§22.19(ii).

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String Theory: An Introduction to the Bosonic String, Vol. I. Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.

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ISBN 0-521-63303-6, MathReview (Angel Maria Uranga), ZentralBlatt

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3rd item.

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Catastrophe Theory and its Applications. Pitman, London.

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Notes:

Reprinted by Dover in 1996.

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ISBN 0-273-01029-8, MathReview (George K. Francis), ZentralBlatt

Referenced by:

§36.4(i), Ch.36.

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22: 2.9 Difference Equations

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§2.9(iii) Other Approximations

►For asymptotic approximations to solutions of second-order difference equations analogous to the Liouville–Green (WKBJ) approximation for differential equations (§2.7(iii)) see Spigler and Vianello (1992, 1997) and Spigler et al. (1999). … ►For an introduction to, and references for, the general asymptotic theory of linear difference equations of arbitrary order, see Wimp (1984, Appendix B). …

23: 18.39 Applications in the Physical Sciences

… ►Shizgal (2015) gives a broad overview of techniques and applications of spectral and pseudo-spectral methods to problems arising in theoretical chemistry, chemical kinetics, transport theory, and astrophysics. … ►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). … ►

Discretized and Continuum Expansions of Scattering Eigenfunctions in terms of Pollaczek Polynomials: J-matrix Theory

… ►The equivalent quadrature weight, $w_i/w^{\mathrm{CP}}(x_i)$, also forms the foundation of a novel inversion of the Stieltjes–Perron moment inversion discussed in §18.40(ii). ►The fact that non-$L^2$ continuum scattering eigenstates may be expressed in terms or (infinite) sums of $L^2$ functions allows a reformulation of scattering theory in atomic physics wherein no non-$L^2$ functions need appear. …

24: Bibliography D

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Multiplicative Number Theory. 3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.

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Notes:

Revised and with a preface by Hugh L. Montgomery

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ISBN 0-387-95097-4, MathReview, ZentralBlatt

Referenced by:

§27.12.

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Pólya’s Theory of Counting. In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.), pp. 144–184.

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ISBN 0898741726

Referenced by:

§26.20.

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Elements of the Theory of Numbers. Harcourt/Academic Press, San Diego, CA.

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ISBN 0-12-209130-2, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

Ch.24.

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History of the Theory of Numbers (3 volumes). Carnegie Institution of Washington, Washington, D.C..

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Notes:

Reprinted by Chelsea Publishing Co., New York, 1966.

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MathReview, ZentralBlatt

Referenced by:

§27.21.

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Uniform asymptotic approximation of Mathieu functions. Methods Appl. Anal. 1 (2), pp. 143–168.

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ISSN 1073-2772, Pdf, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§28.8(iv).

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25: Bibliography J

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REMES2 — a Fortran program to calculate rational minimax approximations to a given function. Technical Report Technical Report AECL-4210, Atomic Energy of Canada Limited. Chalk River Nuclear Laboratories, Chalk River, Ontario.

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Referenced by:

§3.11(iii).

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Atomic Structure Theory: Lectures on Atomic Physics. Springer, Berlin and Heidelberg.

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ISBN 978-3-540-68010-9

Referenced by:

§33.22(iv).

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The Theory of Electromagnetism. International Series of Monographs on Pure and Applied Mathematics, Vol. 47. A Pergamon Press Book, The Macmillan Co., New York.

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MathReview (W. E. Williams), ZentralBlatt

Referenced by:

§23.21(iii).

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Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, MA.

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ISBN 0-201-13510-8, MathReview (E. Frank), ZentralBlatt

Referenced by:

§1.12(ii), §1.12(iii), §1.12(iv), §1.12(v), §13.17, §13.17, §13.5, §13.5, §4.25, §5.10.

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Angular Momentum Theory for Diatomic Molecules. Academic Press, New York.

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ISBN 0-12-391950-9

Referenced by:

§30.12, §34.12, Profile Brian R. Judd.

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26: Bibliography

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Theory of Incomplete Cylindrical Functions and Their Applications. Springer-Verlag, Berlin.

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Notes:

Translated from the Russian by H. E. Fettis, J. W. Goresh and D. A. Lee, Die Grundlehren der mathematischen Wissenschaften, Band 160

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MathReview, ZentralBlatt

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1st item.

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Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable. 2nd edition, McGraw-Hill Book Co., New York.

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Notes:

A third edition was published in 1978 by McGraw-Hill, New York.

External Links:

MathReview (P. Lelong), ZentralBlatt

Referenced by:

§1.9(iii), §23.18.

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Unsteady lifting-line theory as a singular-perturbation problem. J. Fluid Mech 153, pp. 59–81.

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Document, ZentralBlatt

Referenced by:

§11.12.

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Elements of the Theory of Elliptic Functions. Translations of Mathematical Monographs, Vol. 79, American Mathematical Society, Providence, RI.

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Notes:

Translated from the second Russian edition by H. H. McFaden

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ISBN 0-8218-4532-2, MathReview (Glenn Stevens), ZentralBlatt

Referenced by:

§22.18(ii), §22.18(iv).

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Number Theory. In The New Encyclopaedia Britannica, Vol. 25, pp. 14–37.

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Referenced by:

§27.13(i), §27.15, §27.16, Ch.27.

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27: Bibliography L

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Essai sur la Théorie des Nombres. 2nd edition, Courcier, Paris.

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ZentralBlatt

Referenced by:

§27.2(i).

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Zur Theorie der Gaußschen Summen. Math. Ann. 57 (4), pp. 554–567 (German).

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ISSN 0025-5831, Document, MathReview Entry, ZentralBlatt

Referenced by:

§20.11(i).

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Kinetic Theory: Classical, Quantum, and Relativistic Descriptions. third edition, Springer, New York.

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ISBN 0387955518

Referenced by:

§18.39(iii).

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Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.

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FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

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Further approximations for elliptic integrals. Math. Comp. 24 (109), pp. 191–198.

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FullText, MathReview, ZentralBlatt

Referenced by:

§19.38.

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28: Frank W. J. Olver

… ►In April 2011, NIST co-organized a conference on “Special Functions in the 21st Century: Theory & Application” which was dedicated to Olver. … ►

2 Asymptotic Approximations

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29: 18.40 Methods of Computation

… ►The theory behind these remarks is in Shohat and Tamarkin (1970), Akhiezer (2021), Chihara (1978). … ►These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to $w(x)$, as will be considered in the following paragraphs. … ►

Stieltjes Inversion via (approximate) Analytic Continuation

… ►Equation (18.40.7) provides step-histogram approximations to ${\int }_a^{x}d\mu (x)$, as shown in Figure 18.40.1 for $N=12$ and $120$, shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein. … ►In Figure 18.40.2 the approximations were carried out with a precision of 50 decimal digits.

30: Bibliography V

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Quantum Theory of Angular Momentum. World Scientific Publishing Co. Inc., Singapore.

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External Links:

ISBN 9971-50-107-4, MathReview (M. A. Rashid), ZentralBlatt

Referenced by:

§16.24(iii), §34.1, §34.11, §34.13, §34.13, §34.14, §34.2, §34.3(iii), §34.3(vi), §34.4, §34.4, §34.5(iii), §34.5(vi), §34.7(ii), §34.7(iii), §34.8, §34.9, Ch.34.

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Rational approximations for exponential integrals $E_n(x)$ . Acad. Roy. Belg. Bull. Cl. Sci. (5) 56, pp. 1064–1072.

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MathReview (A. Magnus), ZentralBlatt

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3rd item.

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Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, RI.

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MathReview, ZentralBlatt

Referenced by:

§13.27.

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. J. Comput. Appl. Math. 213 (2), pp. 488–500.

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ISSN 0377-0427, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

item (f), §28.34(ii), §29.20(i).

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