Stokes' phenomenon; smoothing a Victorian discontinuity
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)].
scientific article; zbMATH DE number 4149180