By Boyd J.P.
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Extra resources for Asymptotic, superasymptotic and hyperasymptotic series
Still, one can give a little of the flavor of resurgence. One key concept is what one might call “saddle point democracy”. Instead of focusing in quickly on one or two dominant saddle points (on the steepest descent path), resurgence treats all saddle points on an equal footing. One may define an integral passing through an arbitrary saddle point; the coefficients of the steepest descent expansion about that point encodes the expansions about all the other saddle points. Furthermore, the late terms in the asymptotic expansion about a dominant saddle point can be expressed in terms of the early terms of a subdominant series, and vice-versa.
For the inverse power series to be an accurate approximation to the inner solution, we must have | y |>> 1. It follows that the inverse power series is a good approximation only in the annulus 1 <<| y |<< 1/ (96) It is fair to dub this annulus the “matching region” because it will turn out that the inner limit of the outer expansion will also be legitimate in this annulus. (95) to a sector of the annulus. To sort out Stokes phenomenon, it is helpful to sum the divergent series by Borel summation.
Matched Asymptotic Expansions in the Complex Plane: The PKKS Method In “above-the-barrier” quantum scattering, there are no turning points where the coefficient of the undifferentiated term in the Schr¨ odinger equation is zero except at complex values of the spatial coordinate. When there are real-valued turning points, it was discovered in the 1920s that the scattering — including the exponentially small transmission through the barrier – can be computed by means of the so-called turning point connection formulas.