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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An extension of the method of steepest descents.
Proc. Cambridge Philos. Soc. 53, pp. 599–611.
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A numerical method for generalized exponential integrals.
Comput. Math. Appl. 14 (4), pp. 261–268.
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A method for numerical calculation of Fourier integrals.
Numer. Math. 8 (5), pp. 422–436.
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Methods of mathematical physics. Vol. I.
Interscience Publishers, Inc., New York, N.Y..
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3: 19.13 Integrals of Elliptic Integrals
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►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).
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►Various integrals are listed by Byrd and Friedman (1971, p. 630) and Prudnikov et al. (1990, §§1.10.1, 2.15.1).
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4: Bibliography B
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Asymptotic methods in enumeration.
SIAM Rev. 16 (4), pp. 485–515.
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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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Numerical Methods for Least Squares Problems.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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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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Handbook of Elliptic Integrals for Engineers and Scientists.
2nd edition, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York.
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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Asymptotic expansion of the Krawtchouk polynomials and their zeros.
Comput. Methods Funct. Theory 4 (1), pp. 189–226.
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6: 19 Elliptic Integrals
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7: 19.14 Reduction of General Elliptic Integrals
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►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 and the interval is [1,2]), then no method using this assumption succeeds.
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8: 19.37 Tables
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►Tabulated for to 6D by Byrd and Friedman (1971), to 15D for and 9D for by Abramowitz and Stegun (1964, Chapter 17), and to 10D by Fettis and Caslin (1964).
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►Tabulated for to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17).
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►Tabulated for to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17).
►Tabulated for to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17).
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►Tabulated for , to 6D by Byrd and Friedman (1971), for , and to 8D by Abramowitz and Stegun (1964, Chapter 17), and for , to 9D by Zhang and Jin (1996, pp. 674–675).
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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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Integrals with a large parameter. Several nearly coincident saddle-points.
Proc. Cambridge Philos. Soc. 72, pp. 49–65.
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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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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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Ship Hydrodynamics, Water Waves and Asymptotics.
Collected works of F. Ursell, 1946-1992, Vol. 2, World Scientific, Singapore.
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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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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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Principles and Techniques of Applied Mathematics.
Dover, New York.
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