MATH-6640, COMPLEX VARIABLES AND INTEGRAL TRANSFORMS

# COMPLEX VARIABLES AND INTEGRAL TRANSFORMS

Time: Monday, Thursday, 12:00 to 1:50 PM
Room: Science Center 1W01
Instructor: Gregor Kovacic
Office: 419 Amos Eaton
Phone: 276-6908
E-mail: kovacg at rpi dot edu

• This document contains the list of topics to be presented in this course, a list of textbooks covering some of the topics, and some comments and recommendations about these textbooks.

## Topics

Topics will be chosen from those listed below.

Basic Complex Variables: Differentiability in complex domains, Cauchy-Riemann equations, Cauchy's theorem and integral formula, Liouville's theorem, Taylor series, analyticity and uniform convergence, Weierstrass M-test, Laurent series, poles and essential singularities, meromorphic functions, partial-fraction expansions and Mittag-Leffler theorem, entire functions and infinite products, residue calculus, multi-valued functions, branch points and branch cuts, analytic continuation, reflection principle, natural boundary, Gamma and Beta functions, Beseel, Hankel, and Airy functions.

Asymptotic Expansions: Big O and little o symbols, the notion of asymptotic expansion, Watson's lemma, Stokes' phenomenon, Laplace's method, saddle-point or steepest-descent method, the method of stationary phase.

Fourier and Laplace Transforms: Fourier and Laplace transforms and inverse transforms, delta functions and distributions, convolutions, analyticity properties, applications to partial differential equations, probability, and difference equation, asymptotic expansions of Fourier and Laplace transforms.

Riemann-Hilbert and Wiener-Hopf problems: singular integral equations, analytic functions with jump conditions, Plemelj's formulae.

Class notes from the spring semester of 2013 are here.

## Textbooks

The following textbooks contain material similar to that presented in this course:

M. V. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge
L. V. Ahlfors, Complex Analysis, McGraw-Hill
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover
G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Variable, Theory and Technique, SIAM
E. T. Copson, Asymptotic Expansions, Cambridge
N. G. de Bruijn, Asymptotic Methods in Analysis, Dover
J. W. Dettman, Applied Complex Variables, Dover
A. Erdelyi, Asymptotic Expansions, Dover
M. A. Evgrafov, Analytic Functions, Dover
P. Henrici, Applied and Computational Complex Analysis, 3 Volumes, Wiley
E. Hille, Ordinary Differential Equations in the Complex Domain, Dover
W. R. LePage, Complex Variables and the Laplace Transform for Engineers, Dover
A. I. Markushevich, Theory of Functions of a Complex Variable, 3 Volumes, Chelsea
P. D. Miller, Applied Asymptotic Analysis, AMS
N. I. Muskhelishvili, Singular Integral Equations, Dover
J.D. Murray, Asymptotic Analysis, Springer
Z. Nehari, Conformal Mapping, Dover
F. Olver, Asymptotics and Special Functions, CRC Press
L. A. Rubel and J. E. Colliander, Entire and Meromorphic Functions, Springer
I. N. Sneddon, Fourier Transforms, Dover
E. C. Titchmarsh, The Theory of Functions, Oxford
L. I. Volkovskii, G. L. Lunts, and I. G. Agramanovich, A Collection of Problems on Complex Analysis, Dover
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge