Computation of Asymptotic Expansions of Turning Point Problems via Cauchy's Integral Formula: Bessel Functions

Linear second-order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two infinite series, the coefficients of which are usually difficult to compute. By considering the series as asymptotic expansions for two explicitly defined analytic functions, Cauchy's integral formula is employed to compute the coefficient functions to a high order of accuracy. The method employs a certain exponential form of Liouville´Green expansions for solutions of the differential equation, as well as for the Airy function. We illustrate the use of the method with the high accuracy computation of Airy-type expansions of Bessel functions of complex argument.The authors acknowledge support from Ministerio de Economía y Competitividad, project MTM2015-67142-P (MINECO/FEDER, UE). A.G. and J.S. acknowledge support from Ministerio de Economía y Competitividad, project MTM2012-34787. A.G. acknowledges the Fulbright/MEC Program for support during her stay at SDSU. J.S. acknowledges the Salvador de Madariaga Program for support during his stay at SDSU

Analytic Continuation of Liouville Theory

Correlation functions in Liouville theory are meromorphic functions of the Liouville momenta, as is shown explicitly by the DOZZ formula for the three-point function on the sphere. In a certain physical region, where a real classical solution exists, the semiclassical limit of the DOZZ formula is known to agree with what one would expect from the action of the classical solution. In this paper, we ask what happens outside of this physical region. Perhaps surprisingly we find that, while in some range of the Liouville momenta the semiclassical limit is associated to complex saddle points, in general Liouville's equations do not have enough complex-valued solutions to account for the semiclassical behavior. For a full picture, we either must include "solutions" of Liouville's equations in which the Liouville field is multivalued (as well as being complex-valued), or else we can reformulate Liouville theory as a Chern-Simons theory in three dimensions, in which the requisite solutions exist in a more conventional sense. We also study the case of "timelike" Liouville theory, where we show that a proposal of Al. B. Zamolodchikov for the exact three-point function on the sphere can be computed by the original Liouville path integral evaluated on a new integration cycle.Comment: 86 pages plus appendices, 9 figures, minor typos fixed, references added, more discussion of the literature adde