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1: 36.4 Bifurcation Sets
§36.4(i) Formulas
Critical Points for Cuspoids
Critical Points for Umbilics
This is the codimension-one surface in x space where critical points coalesce, satisfying (36.4.1) and … This is the codimension-one surface in x space where critical points coalesce, satisfying (36.4.2) and …
2: 36.12 Uniform Approximation of Integrals
§36.12 Uniform Approximation of Integrals
Correspondence between the u j ( y ) and the t j ( x ) is established by the order of critical points along the real axis when y and x are such that these critical points are all real, and by continuation when some or all of the critical points are complex. …In (36.12.10), both second derivatives vanish when critical points coalesce, but their ratio remains finite. The square roots are real and positive when y is such that all the critical points are real, and are defined by analytic continuation elsewhere. … 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: 36.15 Methods of Computation
Direct numerical evaluation can be carried out along a contour that runs along the segment of the real t -axis containing all real critical points of Φ and is deformed outside this range so as to reach infinity along the asymptotic valleys of exp ( i Φ ) . … This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of Φ , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …
4: 36.5 Stokes Sets
Stokes sets are surfaces (codimension one) in x space, across which Ψ K ( x ; k ) or Ψ ( U ) ( x ; k ) acquires an exponentially-small asymptotic contribution (in k ), associated with a complex critical point of Φ K or Φ ( U ) . …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. … Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions. …
5: 36.11 Leading-Order Asymptotics
With real critical points (36.4.1) ordered so that …
6: 9.16 Physical Applications
Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. …
7: 2.4 Contour Integrals
§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method
§2.4(vi) Other Coalescing Critical Points
8: Bibliography
  • V. I. Arnol’d (1972) Normal forms of functions near degenerate critical points, the Weyl groups A k , D k , E k and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).
  • V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
  • V. I. Arnol’d (1975) Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).
  • 9: 31.15 Stieltjes Polynomials
    The zeros z k , k = 1 , 2 , , n , of the Stieltjes polynomial S ( z ) are the critical points of the function G , that is, points at which G / ζ k = 0 , k = 1 , 2 , , n , where …
    10: 10.72 Mathematical Applications
    §10.72(i) Differential Equations with Turning Points
    Simple Turning Points
    Multiple or Fractional Turning Points
    §10.72(iii) Differential Equations with a Double Pole and a Movable Turning Point
    In (10.72.1) assume f ( z ) = f ( z , α ) and g ( z ) = g ( z , α ) depend continuously on a real parameter α , f ( z , α ) has a simple zero z = z 0 ( α ) and a double pole z = 0 , except for a critical value α = a , where z 0 ( a ) = 0 . …