Chester–Friedman–Ursell method

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1: 2.4 Contour Integrals

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§2.4(iii) Laplace’s Method

… ►For this reason the name method of steepest descents is often used. … ►

§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

… ►For examples, proofs, and extensions see Olver (1997b, Chapter 9), Wong (1989, Chapter 7), Olde Daalhuis and Temme (1994), Chester et al. (1957), Bleistein and Handelsman (1975, Chapter 9), and Temme (2015). ►For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …

2: Bibliography C

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Numerical Grid Methods and Their Application to Schrödinger’s Equation. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Vol. 412, Kluwer Academic Publishers, Dordrecht.

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External Links:

ISBN 0-7923-2423-4, Document, Link, MathReview (M. Znojil)

Referenced by:

§18.38(i).

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An extension of the method of steepest descents. Proc. Cambridge Philos. Soc. 53, pp. 599–611.

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External Links:

MathReview (P. Henrici), ZentralBlatt

Referenced by:

§2.4(v).

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A numerical method for generalized exponential integrals. Comput. Math. Appl. 14 (4), pp. 261–268.

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ISSN 0898-1221, MathReview (R. P. Boas), ZentralBlatt

Referenced by:

§8.25(v).

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A method for numerical calculation of Fourier integrals. Numer. Math. 8 (5), pp. 422–436.

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ISSN 0029-599X, Document, MathReview (M. P. S. Madan), ZentralBlatt

Referenced by:

§3.5(vii).

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Methods of mathematical physics. Vol. I. Interscience Publishers, Inc., New York, N.Y..

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External Links:

MathReview (J. B. Diaz), ZentralBlatt

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(14.30.8_5).

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3: 19.13 Integrals of Elliptic Integrals

… ►For definite and indefinite integrals of complete elliptic integrals see Byrd and Friedman (1971, pp. 610–612, 615), Prudnikov et al. (1990, §§1.11, 2.16), Glasser (1976), Bushell (1987), and Cvijović and Klinowski (1999). ►For definite and indefinite integrals of incomplete elliptic integrals see Byrd and Friedman (1971, pp. 613, 616), Prudnikov et al. (1990, §§1.10.2, 2.15.2), and Cvijović and Klinowski (1994). … ►Various integrals are listed by Byrd and Friedman (1971, p. 630) and Prudnikov et al. (1990, §§1.10.1, 2.15.1). …

4: Bibliography B

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Asymptotic methods in enumeration. SIAM Rev. 16 (4), pp. 485–515.

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Document, MathReview (Bernard Harris), ZentralBlatt

Referenced by:

Ch.2.

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Overlapping Stokes smoothings: Survival of the error function and canonical catastrophe integrals. Proc. Roy. Soc. London Ser. A 444, pp. 201–216.

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ISSN 0962-8444, FullText, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§36.12(iii).

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Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-360-9, MathReview (Hongyuan Zha), ZentralBlatt

Referenced by:

§3.11(v).

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Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents. Methods Appl. Anal. 2 (4), pp. 475–489.

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ISSN 1073-2772, Pdf, MathReview (Dan Polis̆evski), ZentralBlatt

Referenced by:

§2.4(iv).

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Handbook of Elliptic Integrals for Engineers and Scientists. 2nd edition, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York.

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Notes:

Table errata: Math. Comp. v. 66 (1997), no. 220, p. 1767, v. 36 (1981), no. 153, p. 317,319, v. 26 (1972), no. 118, p. 597.

External Links:

MathReview, ZentralBlatt

Referenced by:

§19.1, §19.13(i), §19.13(i), §19.13(ii), §19.14(ii), §19.29(i), §19.30(ii), §19.37(ii), §19.37(ii), §19.37(ii), §19.37(ii), §19.37(iii), §19.4(ii), §19.4(ii), §19.5, §19.6(i), §19.6(iv), Ch.19, §22.11.

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5: Bibliography Q

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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.

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ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§2.4(vi).

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Asymptotic expansion of the Krawtchouk polynomials and their zeros. Comput. Methods Funct. Theory 4 (1), pp. 189–226.

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ISSN 1617-9447, FullText, MathReview (C. M. Joshi), ZentralBlatt (N. M. Temme)

Referenced by:

§18.24.

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6: 19 Elliptic Integrals

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7: 19.14 Reduction of General Elliptic Integrals

… ►The classical method of reducing (19.2.3) to Legendre’s integrals is described in many places, especially Erdélyi et al. (1953b, §13.5), Abramowitz and Stegun (1964, Chapter 17), and Labahn and Mutrie (1997, §3). …It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. …A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges. If no such branch point is accessible from the interval of integration (for example, if the integrand is ${(t(3-t)(4-t))}^{-3/2}$ and the interval is [1,2]), then no method using this assumption succeeds. …

8: 19.37 Tables

… ►Tabulated for $k^2=0(.01)1$ to 6D by Byrd and Friedman (1971), to 15D for $K\left(k\right)$ and 9D for $E\left(k\right)$ by Abramowitz and Stegun (1964, Chapter 17), and to 10D by Fettis and Caslin (1964). … ►Tabulated for $\arcsin k=0(1^{\circ }){90}^{\circ }$ to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17). … ►Tabulated for $k^2=0(.01)1$ to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17). ►Tabulated for $\arcsin k=0(1^{\circ }){90}^{\circ }$ to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17). … ►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $\arcsin k=0(1^{\circ }){90}^{\circ }$ to 6D by Byrd and Friedman (1971), for $\varphi =0(5^{\circ }){90}^{\circ }$, $\arcsin k=0(2^{\circ }){90}^{\circ }$ and $5^{\circ }({10}^{\circ }){85}^{\circ }$ to 8D by Abramowitz and Stegun (1964, Chapter 17), and for $\varphi =0({10}^{\circ }){90}^{\circ }$, $\arcsin k=0(5^{\circ }){90}^{\circ }$ to 9D by Zhang and Jin (1996, pp. 674–675). …

9: Bibliography U

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On Kelvin’s ship-wave pattern. J. Fluid Mech. 8 (3), pp. 418–431.

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External Links:

Document, MathReview (R. C. MacCamy), ZentralBlatt

Referenced by:

§36.13.

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Integrals with a large parameter. Several nearly coincident saddle-points. Proc. Cambridge Philos. Soc. 72, pp. 49–65.

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§36.12(iii).

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Integrals with a large parameter: A double complex integral with four nearly coincident saddle-points. Math. Proc. Cambridge Philos. Soc. 87 (2), pp. 249–273.

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External Links:

ISSN 0305-0041, Document, MathReview (F. I. Ibragimov), ZentralBlatt

Referenced by:

§36.12(iii).

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Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.

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External Links:

ISSN 0305-0041, Document, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§14.26.

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Ship Hydrodynamics, Water Waves and Asymptotics. Collected works of F. Ursell, 1946-1992, Vol. 2, World Scientific, Singapore.

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External Links:

ZentralBlatt

Referenced by:

§36.13.

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10: Bibliography F

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Algebraic methods to compute Mathieu functions. J. Phys. A 34 (17), pp. 3541–3551.

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ISSN 0305-4470, Document, Link, MathReview (Aleksander Denisiuk), ZentralBlatt

Referenced by:

§28.6(i), §28.6(i), §28.6(ii), §28.6(ii), §28.8(i), §28.8(i), §28.8(ii), §28.8(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii).

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Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions. In Progress in Optics, E. Wolf (Ed.), Vol. 9, pp. 311–407.

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Referenced by:

§30.15(i), §30.15(ii), §30.15(iii), §30.15(iv), §30.15(v), §30.15(v).

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Principles and Techniques of Applied Mathematics. Dover, New York.

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Notes:

Reprint of First Edition, 1956, John-Wiley: New York

External Links:

ISBN 0-486-66444-9, MathReview Entry

Referenced by:

§1.16(ix), §1.17(ii), §1.17(ii), §1.17(i), §1.17(i), §1.17(iv), §1.18(ix), §1.18(iii), §1.18(vi), §1.18(i), §1.18(x).

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