behavior at singularities
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11—12 of 12 matching pages
11: 2.3 Integrals of a Real Variable
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►Other types of singular behavior in the integrand can be treated in an analogous manner.
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►Without loss of generality, we assume that this minimum is at the left endpoint .
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►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.
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►For extensions to oscillatory integrals with more general -powers and logarithmic singularities see Wong and Lin (1978) and Sidi (2010).
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►it is free from singularity at
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12: 2.4 Contour Integrals
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►with known asymptotic behavior as .
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►with and their derivatives evaluated at
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►For integral representations of the and their asymptotic behavior as see Boyd (1995).
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►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.
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►For two coalescing saddle points and an algebraic singularity see Temme (1986), Jin and Wong (1998).
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