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1: 5.21 Methods of Computation
Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. …
2: 9.17 Methods of Computation
As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation. … In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent2.4(iv)). …
3: 2.4 Contour Integrals
Let 𝒫 denote the path for the contour integral … Additionally, it may be advantageous to arrange that ( z p ( t ) ) is constant on the path: this will usually lead to greater regions of validity and sharper error bounds. Paths on which ( z p ( t ) ) is constant are also the ones on which | exp ( z p ( t ) ) | decreases most rapidly. For this reason the name method of steepest descents is often used. However, for the purpose of simply deriving the asymptotic expansions the use of steepest descent paths is not essential. …
4: 3.5 Quadrature
For these cases the integration path may need to be deformed; see §3.5(ix). …
§3.5(ix) Other Contour Integrals
For example, steepest descent paths can be used; see §2.4(iv). … The steepest descent path is given by ( t 2 t ) = 0 , or in polar coordinates t = r e i θ we have r = sec 2 ( 1 2 θ ) . … A special case is the rule for Hilbert transforms (§1.14(v)): …
5: Bibliography T
  • N. M. Temme (1994c) Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Methods Appl. Anal. 1 (1), pp. 14–24.
  • 6: Bibliography P
  • R. B. Paris (2004) Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.
  • R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.
  • 7: Bibliography B
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • W. G. C. Boyd (1993) Error bounds for the method of steepest descents. Proc. Roy. Soc. London Ser. A 440, pp. 493–518.
  • W. G. C. Boyd (1994) Gamma function asymptotics by an extension of the method of steepest descents. Proc. Roy. Soc. London Ser. A 447, pp. 609–630.
  • 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.
  • 8: 36.5 Stokes Sets
    where j denotes a real critical point (36.4.1) or (36.4.2), and μ denotes a critical point with complex t or s , t , connected with j by a steepest-descent path (that is, a path where Φ = constant ) in complex t or ( s , t ) space. …
    36.5.4 80 x 5 40 x 4 55 x 3 + 5 x 2 + 20 x 1 = 0 ,
    36.5.7 X = 9 20 + 20 u 4 Y 2 20 u 2 + 6 u 2 sign ( z ) ,
    9: Bibliography C
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • C. Chester, B. Friedman, and F. Ursell (1957) An extension of the method of steepest descents. Proc. Cambridge Philos. Soc. 53, pp. 599–611.
  • M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.
  • 10: 13.20 Uniform Asymptotic Approximations for Large μ