##
**Stokes’ phenomenon; smoothing a Victorian discontinuity.**
*(English)*
Zbl 0701.58012

Stokes’ phenomenon concerns the behaviour of small exponentials whilst hidden behind large ones. A simple context in which it arises is the approximation of integrals
\[
y(k,X)=\int_{C}dsg(s;X)\exp (k\Phi (s,X))
\]
as \(k\to \infty\). Here C is an infinite contour in the complex s-plane and \(\Phi\) is analytic in s and X. Asymptotically, contributions can come from critical points (saddles) of \(\Phi\), i.e. \(s=s_ j(X)\), where \(\delta_ s\Phi (s_ j(X),X)=0.\)

The complex heights of the critical points are defined by \(\phi_ j(X)=\Phi (s_ j(X);X)\). By varying X the critical values Im \(\phi\) \({}_ j\) can coalesce; this happens on the Stokes set in the X-space.

In general one has, to leading order, \[ y(k,X)=M_+(k;X)\exp (k\Phi_+(X))+iS(k;X)M_ -(k;M)\exp (k\Phi_ -(X))+... \] Here \(+\) and - denote the dominant exponential and the principal subdominant one (i.e. Re \(\phi\) \({}_+>Re \phi_ -)\), the prefactors \(M_+\) and \(M_ -\) are slowly-varying functions of k and X. The quantity S is called the Stokes multiplier.

Stokes analyzed the divergence of \[ y(k;X)=M_+(k;X)\exp (k\Phi_+(X))\sum^{\infty}_{r=0}a_ r. \] The coefficients \(a_ r\) first decrease and then increase. Away from the Stokes set Stokes could resum the divergent tail of the series, but on the set he was unable to do so.

The author shows that it is possible to resum the divergent series of \(a_ r\) beyond the least term, even on the Stokes set, and thereby control the asymptotics of y to an exponential accuracy in k. Moreover the Stokes multiplier is universal, that is the same for all functions in a wide class. The variation of S across the Stokes set is not discontinuous, but smooth. [Details appeared in the author, Proc. R. Soc. Lond., Ser. A 422, No.1862, 7-21 (1989; Zbl 0683.33004)].

The complex heights of the critical points are defined by \(\phi_ j(X)=\Phi (s_ j(X);X)\). By varying X the critical values Im \(\phi\) \({}_ j\) can coalesce; this happens on the Stokes set in the X-space.

In general one has, to leading order, \[ y(k,X)=M_+(k;X)\exp (k\Phi_+(X))+iS(k;X)M_ -(k;M)\exp (k\Phi_ -(X))+... \] Here \(+\) and - denote the dominant exponential and the principal subdominant one (i.e. Re \(\phi\) \({}_+>Re \phi_ -)\), the prefactors \(M_+\) and \(M_ -\) are slowly-varying functions of k and X. The quantity S is called the Stokes multiplier.

Stokes analyzed the divergence of \[ y(k;X)=M_+(k;X)\exp (k\Phi_+(X))\sum^{\infty}_{r=0}a_ r. \] The coefficients \(a_ r\) first decrease and then increase. Away from the Stokes set Stokes could resum the divergent tail of the series, but on the set he was unable to do so.

The author shows that it is possible to resum the divergent series of \(a_ r\) beyond the least term, even on the Stokes set, and thereby control the asymptotics of y to an exponential accuracy in k. Moreover the Stokes multiplier is universal, that is the same for all functions in a wide class. The variation of S across the Stokes set is not discontinuous, but smooth. [Details appeared in the author, Proc. R. Soc. Lond., Ser. A 422, No.1862, 7-21 (1989; Zbl 0683.33004)].

Reviewer: D.Siersma

### Keywords:

Stokes set### Citations:

Zbl 0683.33004
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\textit{M. V. Berry}, Publ. Math., Inst. Hautes Étud. Sci. 68, 211--221 (1988; Zbl 0701.58012)

### References:

[1] | M. V. Berry, Uniform asymptotic smoothing of Stokes’s discontinuities,Proc. Roy. Soc. Lond.,A422 (1989), 7–21. · Zbl 0683.33004 |

[2] | T. Poston andI. N. Stewart,Catastrophe theory and its applications, London, 1978. · Zbl 0382.58006 |

[3] | F. J. Wright, The Stokes set of the cusp diffraction catastrophe,J. Phys.,A13 (1980), 2913–2928. · Zbl 0514.58009 |

[4] | G. G. Stokes, On the discontinuity of arbitrary constants that appear as multipliers of semi-convergent series,Acta. Math.,26 (1902), 393–397, reprinted inMathematical and Physical Papers by the late Sir George Gabriel Stokes, Cambridge University Press, 1905, vol. V, p. 283–287. · JFM 33.0261.01 |

[5] | G. G. Stokes, On the discontinuity of arbitrary constants which appear in divergent developments,Trans. Camb. Phil. Soc.,10 (1864), 106–128, reprinted inMathematical and Physical Papers... (ref. [4]), vol. IV, p. 77–109. |

[6] | J. Larmor (ed.),Sir George Gabriel Stokes: Memoir and Scientific Correspondence (Cambridge University Press, 1907), vol. 1, p. 62. · JFM 38.0024.04 |

[7] | G. G. Stokes, On the critical values of the sums of periodic series,Trans. Camb. Phil. Soc.,8 (1847), 533–610, reprinted inMathematical and Physical Papers... (ref. [4]), vol. I, p. 236–313. |

[8] | R. B. Dingle,Asymptotic Expansions: their Derivation and Interpretation, New York and London, Academic Press, 1973. · Zbl 0279.41030 |

[9] | M. V. Berry andC. Upstill, Catastrophe optics: morphologies of caustics and their diffraction patterns,Progress in Optics,18 (1980), 257–346. |

[10] | G. B. Airy, On the intensity of light in the neighbourhood of a caustic,Trans. Camb. Phil. Soc.,6 (1838), 379–403. |

[11] | G. G. Stokes, On the numerical calculation of a class of definite integrals and infinite series,Trans. Camb. Phil. Soc.,9 (1847), 379–407, reprinted inMathematical and Physical Papers... (ref. [4]), vol. II, p. 329–357. |

[12] | R. Balian, G. Parisi andA. Voros, Discrepancies from asymptotic series and their relation to complex classical trajectories,Phys. Rev. Lett.,41 (1978), 1141–1144. |

[13] | J. Ecalle,Les fonctions résurgentes (3 vol.), Publ. Math. Université de Paris-Sud, 1981, andCinq applications des fonctions résurgentes, preprint 84T62, Orsay, 1984. |

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