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Chester–Friedman–Ursell method

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1: 2.4 Contour Integrals
§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
  • C. Chester, B. Friedman, and F. Ursell (1957) An extension of the method of steepest descents. Proc. Cambridge Philos. Soc. 53, pp. 599–611.
  • C. Chiccoli, S. Lorenzutta, and G. Maino (1987) A numerical method for generalized exponential integrals. Comput. Math. Appl. 14 (4), pp. 261–268.
  • W. W. Clendenin (1966) A method for numerical calculation of Fourier integrals. Numer. Math. 8 (5), pp. 422–436.
  • C. W. Clenshaw and A. R. Curtis (1960) A method for numerical integration on an automatic copmputer. Numer. Math. 2 (4), pp. 197–205.
  • R. Courant and D. Hilbert (1953) Methods of mathematical physics. Vol. I. Interscience Publishers, Inc., New York, N.Y..
  • 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
  • E. A. Bender (1974) Asymptotic methods in enumeration. SIAM Rev. 16 (4), pp. 485–515.
  • M. V. Berry and C. J. Howls (1994) Overlapping Stokes smoothings: Survival of the error function and canonical catastrophe integrals. Proc. Roy. Soc. London Ser. A 444, pp. 201–216.
  • Å. Björck (1996) Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • W. G. C. Boyd (1995) Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents. Methods Appl. Anal. 2 (4), pp. 475–489.
  • P. F. Byrd and M. D. Friedman (1971) Handbook of Elliptic Integrals for Engineers and Scientists. 2nd edition, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York.
  • 5: 19 Elliptic Integrals
    6: Bibliography Q
  • W.-Y. Qiu and R. Wong (2000) Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.
  • W. Qiu and R. Wong (2004) Asymptotic expansion of the Krawtchouk polynomials and their zeros. Comput. Methods Funct. Theory 4 (1), pp. 189–226.
  • 7: 19.37 Tables
    Tabulated for k 2 = 0 ( .01 ) 1 to 6D by Byrd and Friedman (1971), to 15D for K ( k ) and 9D for E ( k ) by Abramowitz and Stegun (1964, Chapter 17), and to 10D by Fettis and Caslin (1964). … Tabulated for arcsin k = 0 ( 1 ) 90 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 ) 90 to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17). … Tabulated for ϕ = 0 ( 5 ) 90 , arcsin k = 0 ( 1 ) 90 to 6D by Byrd and Friedman (1971), for ϕ = 0 ( 5 ) 90 , arcsin k = 0 ( 2 ) 90 and 5 ( 10 ) 85 to 8D by Abramowitz and Stegun (1964, Chapter 17), and for ϕ = 0 ( 10 ) 90 , arcsin k = 0 ( 5 ) 90 to 9D by Zhang and Jin (1996, pp. 674–675). …
    8: 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. …
    9: Bibliography U
  • F. Ursell (1960) On Kelvin’s ship-wave pattern. J. Fluid Mech. 8 (3), pp. 418–431.
  • F. Ursell (1972) Integrals with a large parameter. Several nearly coincident saddle-points. Proc. Cambridge Philos. Soc. 72, pp. 49–65.
  • F. Ursell (1980) 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.
  • F. Ursell (1984) Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
  • F. Ursell (1994) Ship Hydrodynamics, Water Waves and Asymptotics. Collected works of F. Ursell, 1946-1992, Vol. 2, World Scientific, Singapore.
  • 10: 34.9 Graphical Method
    §34.9 Graphical Method
    The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. …For an account of this method see Brink and Satchler (1993, Chapter VII). For specific examples of the graphical method of representing sums involving the 3 j , 6 j , and 9 j symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).