The classical analysis mainly deals with convergent series in spite of the fact that an attempt to solve a problem using series often leads to divergence. If treated in a consistent way, a divergent solution may provide even more information about the original problem than a convergent one. Asymptotic series has been a very successful tool to understand the structure of solutions of ordinary and partial differential equations.
Divergent series: summation of divergent series, divergent power series, analytic continuation of a convergent series outside the disk of convergence, asymptotic series, an application to ODEs.
Laplace transform: basic properties, Borel transform, Gevrey-type series, Borel sums, Watson theorem.
Stokes phenomenon: examples, asymptotics in sectors of a complex plane, an application - asymptotic of Airy function.
Multivalued analytic functions: analytic continuation, multivalued functions, introduction to Riemann surfaces.
Formal convergence: space of formal series, formal convergence, an application to ODEs.
Rapidly oscillating integrals: asymptotics of rapidly oscillating integrals, method of stationary phase, examples.
To introduce a systematic approach to analysis of divergent series, their interpretation as asymptotic series, and application of these methods to study of ordinary differential equations and integrals.
At the end of the module the student should be familiar with the methods involving analysis of asymptotic series and to acquire basic techniques in studying asymptotic problems. The student should be able to perform analysis of divergent series and to be able to correctly interpret them as asymptotic series.
We will not follow any particular book, but most of the material can be found in:
C.F. Carrier, M. Krook and C.E. Pearson, Functions of a Complex Variable: theory and technique, Hodbooks.
N.G. De Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing co. (3d ed.) (1970).
P.P.G. Dyke, An Introduction to Laplace Transforms and Fourier Series, Springer Undergraduate Mathematics Series (2000).
G. Hardy, Divergent Series, Clarendon Press, 1963/American Mathematical Society, 2000.
R.B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press (1973).