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behavior at singularities

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11: 2.3 Integrals of a Real Variable
Other types of singular behavior in the integrand can be treated in an analogous manner. … Without loss of generality, we assume that this minimum is at the left endpoint a . … For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint. … For extensions to oscillatory integrals with more general t -powers and logarithmic singularities see Wong and Lin (1978) and Sidi (2010). … it is free from singularity at t = α . …
12: 2.4 Contour Integrals
with known asymptotic behavior as t + . … with p , q and their derivatives evaluated at t 0 . … For integral representations of the b 2 s and their asymptotic behavior as s see Boyd (1995). … For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions. … For two coalescing saddle points and an algebraic singularity see Temme (1986), Jin and Wong (1998). …