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.

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.

Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations

2001 ◽  
Vol 12 (4) ◽  
pp. 433-463 ◽  
Author(s):  
J. R. KING ◽  
S. J. CHAPMAN
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Difference Equations ◽  
Singularly Perturbed ◽  
Time Dependent ◽  
Stokes Line ◽  
Linear Ordinary Differential Equations ◽  
Stokes Lines ◽  
Asymptotics Beyond All Orders

A technique for calculating exponentially small terms beyond all orders in singularly perturbed difference equations is presented. The approach is based on the application of a WKBJ-type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines, and is closely related to the well-known Stokes line smoothing phenomenon in linear ordinary differential equations. The method is illustrated by application to examples and the results extended to time-dependent differential-difference problems.

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.

Gamma function asymptotics by an extension of the method of steepest descents

1994 ◽  
Vol 447 (1931) ◽  
pp. 609-630 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Recent Work ◽  
Gamma Function ◽  
Error Bounds ◽  
Stokes Line ◽  
Asymptotic Estimates ◽  
Asymptotic Representations

Recent work of Berry & Howls, which reformulated the method of steepest de­scents, is exploited to derive a new representation for the gamma function. It is shown how this representation can be used to derive a number of properties of the asymptotic expansion of the gamma function, including explicit and realistic error bounds, the Berry transition between different asymptotic representations across a Stokes line, and asymptotic estimates for the late coefficients.

scholarly journals Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal

2015 ◽  
Vol 145 (3) ◽  
pp. 571-596 ◽  
Author(s):  
Gergő Nemes
Keyword(s):  
Asymptotic Expansion ◽  
Error Estimates ◽  
Gamma Function ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Asymptotic Series ◽  
Smooth Transition ◽  
Error Terms

In 1994 Boyd derived a resurgence representation for the gamma function, exploiting the 1991 reformulation of the method of steepest descents by Berry and Howls. Using this representation, he was able to derive a number of properties of the asymptotic expansion for the gamma function, including explicit and realistic error bounds, the smooth transition of the Stokes discontinuities and asymptotics for the late coefficients. The main aim of this paper is to modify Boyd’s resurgence formula, making it suitable for deriving better error estimates for the asymptotic expansions of the gamma function and its reciprocal. We also prove the exponentially improved versions of these expansions complete with error terms. Finally, we provide new (formal) asymptotic expansions for the coefficients appearing in the asymptotic series and compare their numerical efficacy with the results of earlier authors.

scholarly journals ON THE ACCURACY OF ASYMPTOTIC APPROXIMATIONS TO THE LOG-GAMMA AND RIEMANN–SIEGEL THETA FUNCTIONS

2018 ◽  
Vol 107 (3) ◽  
pp. 319-337
Author(s):  
RICHARD P. BRENT
Keyword(s):  
New York ◽  
Half Plane ◽  
Gamma Function ◽  
Asymptotic Approximation ◽  
Asymptotic Series ◽  
Theta Functions ◽  
Stirling’S Formula ◽  
Stokes Phenomenon ◽  
Principal Branch ◽  
The Right

We give bounds on the error in the asymptotic approximation of the log-Gamma function $\ln \unicode[STIX]{x1D6E4}(z)$ for complex $z$ in the right half-plane. These improve on earlier bounds by Behnke and Sommer [Theorie der analytischen Funktionen einer komplexen Veränderlichen, 2nd edn (Springer, Berlin, 1962)], Spira [‘Calculation of the Gamma function by Stirling’s formula’, Math. Comp.25 (1971), 317–322], and Hare [‘Computing the principal branch of log-Gamma’, J. Algorithms25 (1997), 221–236]. We show that $|R_{k+1}(z)/T_{k}(z)|<\sqrt{\unicode[STIX]{x1D70B}k}$ for nonzero $z$ in the right half-plane, where $T_{k}(z)$ is the $k$th term in the asymptotic series, and $R_{k+1}(z)$ is the error incurred in truncating the series after $k$ terms. We deduce similar bounds for asymptotic approximation of the Riemann–Siegel theta function $\unicode[STIX]{x1D717}(t)$. We show that the accuracy of a well-known approximation to $\unicode[STIX]{x1D717}(t)$ can be improved by including an exponentially small term in the approximation. This improves the attainable accuracy for real $t>0$ from $O(\exp (-\unicode[STIX]{x1D70B}t))$ to $O(\exp (-2\unicode[STIX]{x1D70B}t))$. We discuss a similar example due to Olver [‘Error bounds for asymptotic expansions, with an application to cylinder functions of large argument’, in: Asymptotic Solutions of Differential Equations and Their Applications (ed. C. H. Wilcox) (Wiley, New York, 1964), 16–18], and a connection with the Stokes phenomenon.

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 Error bounds and exponential improvement for Hermite's asymptotic expansion for the Gamma function

2013 ◽  
Vol 7 (1) ◽  
pp. 161-179 ◽  
Author(s):  
Gergő Nemes
Keyword(s):  
Asymptotic Expansion ◽  
Error Estimates ◽  
Gamma Function ◽  
Error Bounds ◽  
Asymptotic Series

In this paper we reconsider the asymptotic expansion of the Gamma function with shifted argument, which is the generalization of the well-known Stirling series. To our knowledge, no explicit error bounds exist in the literature for this expansion. Therefore, the first aim of this paper is to extend the known error estimates of Stirling?s series to this general case. The second aim is to give exponentially-improved asymptotics for this asymptotic series.

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.

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

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