picture of Stokes set
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11: 1.6 Vectors and Vector-Valued Functions
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►Note: The terminology open and closed sets and boundary
points in the plane that is used in this subsection and §1.6(v) is analogous to that introduced for the complex plane in §1.9(ii).
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►and be the closed and bounded point set in the plane having a simple closed curve as boundary.
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►with , an open set in the plane.
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Stokes’s Theorem
…12: 18.39 Applications in the Physical Sciences
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►The properties of determine whether the spectrum, this being the set of eigenvalues of , is discrete, continuous, or mixed, see §1.18.
Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being and forming a complete set.
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►with an infinite set of orthonormal eigenfunctions
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►A relativistic treatment becoming necessary as becomes large as corrections to the non-relativistic Schrödinger picture are of approximate order , being the dimensionless fine structure constant , where is the speed of light.
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►These, taken together with the infinite sets of bound states for each , form complete sets.
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13: Bibliography B
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Stokes surfaces of diffraction catastrophes with codimension three.
Nonlinearity 3 (2), pp. 281–291.
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Overlapping Stokes smoothings: Survival of the error function and canonical catastrophe integrals.
Proc. Roy. Soc. London Ser. A 444, pp. 201–216.
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Uniform asymptotic smoothing of Stokes’s discontinuities.
Proc. Roy. Soc. London Ser. A 422, pp. 7–21.
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Infinitely many Stokes smoothings in the gamma function.
Proc. Roy. Soc. London Ser. A 434, pp. 465–472.
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Stieltjes transforms and the Stokes phenomenon.
Proc. Roy. Soc. London Ser. A 429, pp. 227–246.
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14: 8.22 Mathematical Applications
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8.22.1
►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.
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15: 6.12 Asymptotic Expansions
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►For re-expansions of the remainder term leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv), with .
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16: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
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►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 , and Wong and Zhao (1999a) when .
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►This reference includes exponentially-improved asymptotic expansions for when , together with a smooth interpretation of Stokes phenomena.
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17: 3.10 Continued Fractions
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►can be converted into a continued fraction of type (3.10.1), and with the property that the th convergent to is equal to the th partial sum of the series in (3.10.3), that is,
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►A more stable version of the algorithm is discussed in Stokes (1980).
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►This forward algorithm achieves efficiency and stability in the computation of the convergents , and is related to the forward series recurrence algorithm.
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►The recurrences are continued until is within a prescribed relative precision.
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18: Bibliography K
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Essential singularity of the Painlevé function of the second kind and the nonlinear Stokes phenomenon.
Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.
(LOMI) 187, pp. 139–170 (Russian).
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Quasi-linear Stokes phenomenon for the Painlevé first equation.
J. Phys. A 37 (46), pp. 11149–11167.
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A Program for Computing the Conical Functions of the First Kind for and
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Comput. Phys. Comm. 23 (1), pp. 51–61.
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19: Bibliography H
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On an “equivalent quadrature” calculation of matrix elements of using an expansion technique.
J. Comput. Phys. 13, pp. 536–550.
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Algorithm 571: Statistics for von Mises’ and Fisher’s distributions of directions: , and their inverses [S14].
ACM Trans. Math. Software 7 (2), pp. 233–238.
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On the higher-order Stokes phenomenon.
Proc. Roy. Soc. London Ser. A 460, pp. 2285–2303.
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20: Bibliography P
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Stokes phenomenon demystified.
Bull. Inst. Math. Appl. 31 (1-2), pp. 21–28.
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Smoothing of the Stokes phenomenon for high-order differential equations.
Proc. Roy. Soc. London Ser. A 436, pp. 165–186.
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Smoothing of the Stokes phenomenon using Mellin-Barnes integrals.
J. Comput. Appl. Math. 41 (1-2), pp. 117–133.
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The Stokes phenomenon associated with the Hurwitz zeta function
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Proc. Roy. Soc. London Ser. A 461, pp. 297–304.
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