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1: 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). …
2: 2.8 Differential Equations with a Parameter
The form of the asymptotic expansion depends on the nature of the transition points in D , that is, points at which f ( z ) has a zero or singularity. … In Case I there are no transition points in D and g ( z ) is analytic. …
§2.8(ii) Case I: No Transition Points
For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978).
§2.8(vi) Coalescing Transition Points
3: Bibliography O
  • F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.
  • 4: Bibliography T
  • S. A. Tumarkin (1959) Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.
  • 5: Bibliography W
  • Z. Wang and R. Wong (2005) Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.
  • 6: 33.12 Asymptotic Expansions for Large η
    §33.12(i) Transition Region
    When = 0 and η > 0 , the outer turning point is given by ρ tp ( η , 0 ) = 2 η ; compare (33.2.2). …
    7: 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. …
    8: Bibliography S
  • J. Segura (2002) The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40 (1), pp. 114–133.
  • P. N. Shivakumar and J. Xue (1999) On the double points of a Mathieu equation. J. Comput. Appl. Math. 107 (1), pp. 111–125.
  • J. H. Silverman and J. Tate (1992) Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, Springer-Verlag, New York.
  • D. M. Smith (1991) Algorithm 693: A FORTRAN package for floating-point multiple-precision arithmetic. ACM Trans. Math. Software 17 (2), pp. 273–283.
  • I. I. Sobelman (1992) Atomic Spectra and Radiative Transitions. 2nd edition, Springer-Verlag, Berlin.