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1: 36.14 Other Physical Applications
Diffraction catastrophes describe the (linear) wave amplitudes that smooth the geometrical caustic singularities and decorate them with interference patterns. …
2: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)2.11(v)) see Wong and Zhao (1999b) when ρ > 0 , and Wong and Zhao (1999a) when - 1 < ρ < 0 . … This reference includes exponentially-improved asymptotic expansions for E a , b ( z ) when | z | , together with a smooth interpretation of Stokes phenomena. …
3: 1.6 Vectors and Vector-Valued Functions
The surface is smooth at this point if T u × T v 0 . A surface is smooth if it is smooth at every point. … The area A ( S ) of a parametrized smooth surface is given by … Suppose S is a piecewise smooth surface which forms the complete boundary of a bounded closed point set V , and S is oriented by its normal being outwards from V . …
4: 7.20 Mathematical Applications
The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv). …
5: 8.22 Mathematical Applications
plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon. …
6: Bibliography P
  • R. B. Paris (1992a) Smoothing of the Stokes phenomenon for high-order differential equations. Proc. Roy. Soc. London Ser. A 436, pp. 165–186.
  • R. B. Paris (1992b) Smoothing of the Stokes phenomenon using Mellin-Barnes integrals. J. Comput. Appl. Math. 41 (1-2), pp. 117–133.
  • 7: Mathematical Introduction
    Lastly, users may notice some lack of smoothness in the color boundaries of some of the 4D-type surfaces; see, for example, Figure 10.3.9. This nonsmoothness arises because the mesh that was used to generate the figure was optimized only for smoothness of the surface, and not for smoothness of the color boundaries. …
    8: Bibliography D
  • L. Dekar, L. Chetouani, and T. F. Hammann (1999) Wave function for smooth potential and mass step. Phys. Rev. A 59 (1), pp. 107–112.
  • G. C. Donovan, J. S. Geronimo, and D. P. Hardin (1999) Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. SIAM J. Math. Anal. 30 (5), pp. 1029–1056.
  • T. M. Dunster (1996b) Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities. Proc. Roy. Soc. London Ser. A 452, pp. 1351–1367.
  • 9: Bibliography W
  • R. Wong and Y.-Q. Zhao (1999a) Smoothing of Stokes’s discontinuity for the generalized Bessel function. II. Proc. Roy. Soc. London Ser. A 455, pp. 3065–3084.
  • R. Wong and Y.-Q. Zhao (1999b) Smoothing of Stokes’s discontinuity for the generalized Bessel function. Proc. Roy. Soc. London Ser. A 455, pp. 1381–1400.
  • 10: 36.12 Uniform Approximation of Integrals
    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. The function g has a smooth amplitude. …