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coalescing critical points

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1: 36.12 Uniform Approximation of Integrals
§36.12 Uniform Approximation of Integrals
36.12.1 I ( y , k ) = - exp ( i k f ( u ; y ) ) g ( u , y ) d u ,
As y varies as many as K + 1 (real or complex) critical points of the smooth phase function f can coalesce in clusters of two or more. … In (36.12.10), both second derivatives vanish when critical points coalesce, but their ratio remains finite. … 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).
2: 2.4 Contour Integrals
§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method
§2.4(vi) Other Coalescing Critical Points
3: 36.4 Bifurcation Sets
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 …