THEORY OF ANALYTIC CONTINUATION AND APPLICATION TO DYNAMICAL SYSTEMS

Abstract

The first part of the dissertation is a study of the theory of analytic continuation (A.C)of function of complex variable .Theorems and examples are given to investigate the topics of direct( A .C) , (A.C) along a curve, global analytic branch points, Riemann surfaces and their construction . The second part is devoted to application of (A.C) in some problems of quantum mechanic . The generalized classical treatment of langevin equation for a linear oscillator embedded in a bath of harmonic oscillators is solved to measure rate of energy absorption from an external radiation source . Quantum treatment applies (A.C).on the displacement imaginary – time to obtain the real – time correlation.A brief description of the implementation of the maximum entropy inversion method is given . singular value decomposition method and numerical path integral Monte –carol simulation are used to get numerical solution of the same problems a bove. Finally curves obtained by Fourier integrals (using A.C theory ) are compare with carves obtained numerically ( using statistical and simulation methods).