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1: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
2: 36.12 Uniform Approximation of Integrals
§36.12 Uniform Approximation of Integrals
§36.12(i) General Theory for Cuspoids
For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).
3: Bibliography U
  • 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.
  • 4: 12.16 Mathematical Applications
    PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …
    5: 2.4 Contour Integrals
    §2.4(iv) Saddle Points
    §2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method
    For two coalescing saddle points and an endpoint see Leubner and Ritsch (1986). For two coalescing saddle points and an algebraic singularity see Temme (1986), Jin and Wong (1998). …For many coalescing saddle points see §36.12. …
    6: 7.20 Mathematical Applications
    For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). …
    7: Bibliography V
  • B. L. van der Waerden (1951) On the method of saddle points. Appl. Sci. Research B. 2, pp. 33–45.
  • 8: Bibliography L
  • C. Leubner and H. Ritsch (1986) A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points. J. Phys. A 19 (3), pp. 329–335.
  • J. L. López and P. J. Pagola (2011) A systematic “saddle point near a pole” asymptotic method with application to the Gauss hypergeometric function. Stud. Appl. Math. 127 (1), pp. 24–37.
  • 9: Bibliography T
  • N. M. Temme (1978) The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.
  • 10: 3.5 Quadrature
    To avoid cancellation we try to deform the path to pass through a saddle point in such a way that the maximum contribution of the integrand is derived from the neighborhood of the saddle point. … with saddle point at t = 1 , and when c = 1 the vertical path intersects the real axis at the saddle point. … If f is meromorphic, with poles near the saddle point, then the foregoing method can be modified. …