Stokes’s phenomenon for superfactorial asymptotic series

1991 ◽  
Vol 435 (1894) ◽  
pp. 437-444 ◽  
Keyword(s):  
Error Function ◽  
Asymptotic Series ◽  
Numerical Test ◽  
Stokes Phenomenon ◽  
Borel Summation ◽  
Stokes Lines

Superfactorial series depending on a parameter are those whose terms a ( n, z ) grow faster than any power of n !. If the terms get smaller before they increase, the function F ( z ) represented by Ʃ ∞ 0 a ( n, z ) will exhibit a Stokes phenomenon similar to that occurring in asymptotic series whose divergence is merely factorial: across ‘Stokes lines’ in the Z plane, where the late terms all have the same phase, a small exponential switches on in the remainder when the series is truncated near its least term. The jump is smooth and described by an error function whose argument has a slightly more general form than in the factorial case. This result is obtained by a method which is heuristic but applies to superfactorial series where Borel summation fails. Several examples are given, including an analytical interpretation of the sum, and a numerical test of the error-function formula, for the function represented by F ( Z ) = ∞ Ʃ 0 exp { n 2 / A -2 nz }, where A ≫ 1.

Hyperasymptotics

1990 ◽  
Vol 430 (1880) ◽  
pp. 653-668 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Transition Point ◽  
Asymptotic Series ◽  
Stokes Phenomenon ◽  
Numerical Computations ◽  
Borel Summation ◽  
Nested Sequence ◽  
Exponentially Small

We develop a technique for systematically reducing the exponentially small (‘superasymptotic’) remainder of an asymptotic expansion truncated near its least term, for solutions of ordinary differential equations of Schrödinger type where one transition point dominates. This is achieved by repeatedly applying Borel summation to a resurgence formula discovered by Dingle, relating the late to the early terms of the original expansion. The improvements form a nested sequence of asymptotic series truncated at their least terms. Each such ‘hyperseries’ involves the terms of the original asymptotic series for the particular function being approximated, together with terminating integrals that are universal in form, and is half the length of its predecessor. The hyperasymptotic sequence is therefore finite, and leads to an ultimate approximation whose error is less than the square of the original superasymptotic remainder. The Stokes phenomenon is automatically and exactly incorporated into the scheme. Numerical computations confirm the efficacy of the technique.

Infinitely many Stokes smoothings in the gamma function

1991 ◽  
Vol 434 (1891) ◽  
pp. 465-472 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Gamma Function ◽  
Error Function ◽  
Asymptotic Series ◽  
Stokes Line ◽  
Separate Component ◽  
Negative Real Axis ◽  
Stokes Lines ◽  
The Right ◽  
Anti Stokes

The Stokes lines for Г( z ) are the positive and negative imaginary axes, where all terms in the divergent asymptotic expansion for In Г( z ) have the same phase. On crossing these lines from the right to the left half-plane, infinitely many subdominant exponentials appear, rather than the usual one. The exponentials increase in magnitude towards the negative real axis (anti-Stokes line), where they add to produce the poles of Г( z ). Corresponding to each small exponential is a separate component asymptotic series in the expansion for In Г( z ). If each is truncated near its least term, its exponential switches on smoothly across the Stokes lines according to the universal error-function law. By appropriate subtractions from In Г( z ), the switching-on of successively smaller exponentials can be revealed. The procedure is illustrated by numerical computations.

Waves near Stokes lines

1990 ◽  
Vol 427 (1873) ◽  
pp. 265-280 ◽  
Keyword(s):  
Refractive Index ◽  
Error Function ◽  
Divergent Series ◽  
Stokes Line ◽  
Stokes Phenomenon ◽  
Total Change ◽  
Integral Approximation ◽  
Reflected Waves ◽  
Stokes Lines ◽  
Phase Integral

The large- k asymptotics of d 2 u ( z )/d z 2 = k 2 R 2 ( z ) u ( z ) are studied near a Stokes line ( ω ≡ ∫ z z 0 R d z real, where z 0 is a zero of R 2 ( z ), of any order), on which there is greatest disparity between the dominant and subdominant exponential waves in the phase-integral (WKB) approximations. The aim is to establish precisely how the multiplier b _ of the subdominant wave varies across the Stokes line. Although b _ always has a total change proportional to i times the multiplier of the dominant wave (the Stokes phenomenon), the form of the change depends on the convention used to define the two waves. The optimal convention, for which the variation is maximally compact and smooth, is to define them by the phase-integral approximation truncated at its least term, whose order is proportional to k and therefore large (‘asymptotics of asymptotics’). Then the variation of b _ is proportional to the error function of the natural Stokes-crossing variable Im ω √( k /Re ω ). This result is obtained without resumming divergent series (thereby avoiding ‘asymptotics of asymptotics of asymptotics’). An application is given, to the birth of exponentially weak reflected waves in media with smoothly varying refractive index.

scholarly journals Exponential asymptotics and Stokes lines in a partial differential equation

2005 ◽  
Vol 461 (2060) ◽  
pp. 2385-2421 ◽  
Author(s):  
S. Jonathan Chapman ◽  
David B Mortimer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Second Generation ◽  
Saddle Points ◽  
Asymptotic Series ◽  
Perturbation Expansion ◽  
Higher Order ◽  
Stokes Line ◽  
Partial Differential ◽  
Stokes Lines

A singularly perturbed linear partial differential equation motivated by the geometrical model for crystal growth is considered. A steepest descent analysis of the Fourier transform solution identifies asymptotic contributions from saddle points, end points and poles, and the Stokes lines across which these may be switched on and off. These results are then derived directly from the equation by optimally truncating the naïve perturbation expansion and smoothing the Stokes discontinuities. The analysis reveals two new types of Stokes switching: a higher-order Stokes line which is a Stokes line in the approximation of the late terms of the asymptotic series, and which switches on or off Stokes lines themselves; and a second-generation Stokes line, in which a subdominant exponential switched on at a primary Stokes line is itself responsible for switching on another smaller exponential. The ‘new’ Stokes lines discussed by Berk et al . (Berk et al . 1982 J. Math. Phys. 23 , 988–1002) are second-generation Stokes lines, while the ‘vanishing’ Stokes lines discussed by Aoki et al . (Aoki et al . 1998 In Microlocal analysis and complex Fourier analysis (ed. K. F. T. Kawai), pp. 165–176) are switched off by a higher-order Stokes line.

Overlapping Stokes smoothings: survival of the error function and canonical catastrophe integrals

1994 ◽  
Vol 444 (1920) ◽  
pp. 201-216 ◽  
Keyword(s):  
Saddle Point ◽  
Catastrophe Theory ◽  
Asymptotic Theory ◽  
Error Function ◽  
Large Parameter ◽  
Stokes Phenomenon ◽  
Numerical Illustration ◽  
Distant Cluster ◽  
Uniform Approximations ◽  
Common Framework

We derive doubly uniform approximations for the remainder in the optimally truncated saddle-point expansion for an integral containing a large parameter. Double uniformity means that the formulae remain valid while distant saddles responsible for the divergence of the expansion coalesce and separate (as described by catastrophe theory) and while the subdominant exponentials they contribute switch on and off (as described by the error-function smoothing of the Stokes phenomenon). Two sorts of asymptotic singularity are thereby united in a common framework. The formula for the remainder incorporates both the Stokes error function and the canonical catastrophe integrals. A numerical illustration is given, in which the distant cluster contains two saddles; the asymptotic theory gives an accurate description of the details of the fractional remainder, even when this is of order exp ( –36).

The Stokes phenomenon associated with the Hurwitz zeta function ζ( s , a )

2005 ◽  
Vol 461 (2053) ◽  
pp. 297-304 ◽  
Author(s):  
R. B. Paris
Keyword(s):  
Asymptotic Expansion ◽  
Zeta Function ◽  
Infinite Number ◽  
Hurwitz Zeta Function ◽  
Stokes Phenomenon ◽  
Large Complex ◽  
Stokes Lines

We examine the exponentially improved asymptotic expansion of the Hurwitz zeta function ζ ( s , a ) for large complex values of a , with s regarded as a parameter. It is shown that an infinite number of subdominant exponential terms switch on across the Stokes lines arg a = ± ½ π .

Smoothing of the Stokes phenomenon for high-order differential equations

1992 ◽  
Vol 436 (1896) ◽  
pp. 165-186 ◽  
Keyword(s):  
Functional Form ◽  
Arbitrary Order ◽  
Error Function ◽  
Smooth Transition ◽  
Stokes Line ◽  
Parabolic Cylinder ◽  
Stokes Phenomenon ◽  
Function Type ◽  
Parabolic Cylinder Functions ◽  
Class Of Functions

We extend the class of functions for which the smooth transition of a Stokes multiplier across a Stokes line can be rigorously established to functions satisfying a certain differential equation of arbitrary order n . The equation chosen admits solutions of hypergeometric function type which, in the case n = 2, are related to the parabolic cylinder functions. In general, the solutions of this equation involve compound asymptotic expansions, valid in certain sectors of the complex z -plane, with more than one dominant and subdominant series. The functional form of the Stokes multipliers, expressed in terms of an appropriately scaled variable describing transition across a Stokes line, is found to obey the error function smoothing law derived by Berry.

scholarly journals Hyperasymptotics and the Stokes' phenomenon

1993 ◽  
Vol 123 (4) ◽  
pp. 731-743 ◽  
Author(s):  
A. B. Olde Daalhuis
Keyword(s):  
Asymptotic Expansions ◽  
Stokes Phenomenon ◽  
Stokes Lines ◽  
Exponentially Small

SynopsisHyperasymptotic expansions were recently introduced by Berry and Howls, and yield refined information by expanding remainders in asymptotic expansions. In a recent paper of Olde Daalhuis, a method was given for obtaining hyperasymptotic expansions of integrals that represent the confluent hypergeometric U-function. This paper gives an extension of that method to neighbourhoods of the so-called Stokes lines. At each level, the remainder is exponentially small compared with the previous remainders. Two numerical illustrations confirm these exponential improvements.

Uniform asymptotic smoothing of Stokes’s discontinuities

1989 ◽  
Vol 422 (1862) ◽  
pp. 7-21 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Airy Function ◽  
Error Function ◽  
Stokes Line ◽  
Borel Summation ◽  
The Difference ◽  
Exponentially Small

Across a Stokes line, where one exponential in an asymptotic expansion maximally dominates another, the multiplier of the small exponential changes rapidly. If the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form. In terms of the singulant F – the difference between the larger and smaller exponents, and real on the Stokes line - the change in the multiplier is the error function π -½ ∫ σ -∞ d t exp (-t 2 ) Where σ = Im F / (2 Re F ) ½ . The derivation requires control of exponentially small terms in the dominant series; this is achieved with Dingle’s method of Borel summation of late terms, starting with the least term. In numerical illustrations the multiplier is extracted from Dawson’s integral (erfi) and the Airy function of the second kind (Bi): the small exponential emerges in the predicted universal manner from the dominant one, which can be 10 10 times larger.

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