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βΊ
is the denominator of the th approximant to:
…and is the denominator of the th approximant to:
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βΊ
is the denominator of the th approximant to:
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βΊ
is the denominator of the th approximant to:
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βΊ
is the denominator of the th approximant to:
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βΊWe have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials.
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βΊ
Scales were corrected in all figures. The interval
was replaced by and replaced by . All plots and interactive visualizations were regenerated to improve image quality.
(a) Density plot.
(b) 3D plot.
Figure 36.3.9: Modulus of hyperbolic umbilic canonical integral function
.
(a) Density plot.
(b) 3D plot.
Figure 36.3.10: Modulus of hyperbolic umbilic canonical integral function
.
(a) Density plot.
(b) 3D plot.
Figure 36.3.11: Modulus of hyperbolic umbilic canonical integral function
.
(a) Density plot.
(b) 3D plot.
Figure 36.3.12: Modulus of hyperbolic umbilic canonical integral function
.
The scaling error reported on 2016-09-12 by Dan Piponi also applied to contour and density plots for the phase of the hyperbolic umbilic canonical integrals. Scales were corrected in all figures. The interval
was replaced by and replaced by . All plots and interactive visualizations were regenerated to improve image quality.
(a) Contour plot.
(b) Density plot.
Figure 36.3.18: Phase of hyperbolic umbilic canonical integral
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(a) Contour plot.
(b) Density plot.
Figure 36.3.19: Phase of hyperbolic umbilic canonical integral
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(a) Contour plot.
(b) Density plot.
Figure 36.3.20: Phase of hyperbolic umbilic canonical integral
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(a) Contour plot.
(b) Density plot.
Figure 36.3.21: Phase of hyperbolic umbilic canonical integral
.
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βΊThe canonical integrals (36.2.4) provide a basis for uniform asymptotic approximations of oscillatory integrals.
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βΊThis technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes in (36.2.10) away from , in terms of canonical integrals for .
For example, the diffraction catastrophe defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function when is large, provided that and are not small.
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βΊFor 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).
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βΊThis 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 , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints.
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βΊFor numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).
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βΊRed 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.
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βΊ
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βΊThe are also referred to as the numerator polynomials, the then being the denominator polynomials, in that the -th approximant of the continued fraction, ,
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βΊ
Markov’s Theorem
βΊThe ratio , as defined here, thus provides the same statement of Markov’s Theorem, as in (18.2.9_5), but now in terms of differently obtained numerator and denominator polynomials.
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T. Uzer, J. T. Muckerman, and M. S. Child (1983)Collisions and umbilic catastrophes. The hyperbolic umbilic canonical diffraction integral.
Molecular Phys.50 (6), pp. 1215–1230.
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βΊGlaisher (1940) contains four tables: Table I tabulates, for all : (a) the canonical factorization of into powers of primes; (b) the Euler totient ; (c) the divisor function ; (d) the sum of these divisors.
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