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11: 36 Integrals with Coalescing Saddles
Chapter 36 Integrals with Coalescing Saddles
12: 31.13 Asymptotic Approximations
For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). …
13: 36.13 Kelvin’s Ship-Wave Pattern
There are two stationary points, given by …These coalesce when … The disturbance z ( ρ , ϕ ) can be approximated by the method of uniform asymptotic approximation for the case of two coalescing stationary points (36.12.11), using the fact that θ ± ( ϕ ) are real for | ϕ | < ϕ c and complex for | ϕ | > ϕ c . …
14: 9.16 Physical Applications
The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other. The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. … These examples of transitions to turbulence are presented in detail in Drazin and Reid (1981) with the problem of hydrodynamic stability. The investigation of the transition between subsonic and supersonic of a two-dimensional gas flow leads to the Euler–Tricomi equation (Landau and Lifshitz (1987)). … This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. …
15: Christopher J. Howls
16: 8.13 Zeros
As x increases the positive zeros coalesce to form a double zero at ( a n , x n ). …When x > x n a pair of conjugate trajectories emanate from the point a = a n in the complex a -plane. …
17: Bibliography M
  • P. Martín, R. Pérez, and A. L. Guerrero (1992) Two-point quasi-fractional approximations to the Airy function Ai ( x ) . J. Comput. Phys. 99 (2), pp. 337–340.
  • T. Masuda (2003) On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46 (1), pp. 121–171.
  • Maxima (free interactive system)
  • D. S. Meek and D. J. Walton (1992) Clothoid spline transition spirals. Math. Comp. 59 (199), pp. 117–133.
  • MPFR (free C library)
  • 18: 2.9 Difference Equations
    For discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005). …
    19: Bibliography D
  • T. M. Dunster (1990b) Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6), pp. 1594–1618.
  • T. M. Dunster (1994b) Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane. SIAM J. Math. Anal. 25 (2), pp. 322–353.
  • T. M. Dunster (1996a) Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.
  • T. M. Dunster (2001a) Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.
  • T. M. Dunster (2014) Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.
  • 20: 32.2 Differential Equations
    An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. …
    §32.2(vi) Coalescence Cascade
    P I P V  are obtained from P VI  by a coalescence cascade: …