saddle%20points
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1: 36.5 Stokes Sets
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►where denotes a real critical point (36.4.1) or (36.4.2), and denotes a critical point with complex or , connected with by a steepest-descent path (that is, a path where ) in complex or space.
►In the following subsections, only Stokes sets involving at least one real saddle are included unless stated otherwise.
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►This part of the Stokes set connects two complex saddles.
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►In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles.
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.
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2: 3.4 Differentiation
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Two-Point Formula
… ►Three-Point Formula
… ►Four-Point Formula
… ►Five-Point Formula
… ►The choice is motivated by saddle-point analysis; see §2.4(iv) or examples in §3.5(ix). …3: Bibliography O
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On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles.
Methods Appl. Anal. 7 (4), pp. 727–745.
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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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Error bounds for asymptotic expansions in turning-point problems.
J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.
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Second-order linear differential equations with two turning points.
Philos. Trans. Roy. Soc. London Ser. A 278, pp. 137–174.
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Second-order differential equations with fractional transition points.
Trans. Amer. Math. Soc. 226, pp. 227–241.
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