Time: Monday, Thursday, 12:00 to 1:50 PM
Room: Science Center 1W01
Instructor: Gregor Kovacic
Office: 419 Amos Eaton
E-mail: kovacg at rpi dot edu
Office Hours: Click here.
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.
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
Comments and recommendations about textbooks
I ordered Ablowitz and Fokas, which you can buy at the bookstore if you feel you need to have a book. It covers much of the material that will be presented in this class. This book, and also Dettman, have excellent applications, the first more modern and the second more classical. Carrier, Krook, and Pearson is very applied with many excellent examples. Ahlfors is much more rigorous and has been the classic textbook on the subject for more than half a century. Titchmarsh is a classic British-style textbook on complex analysis, covering many topics not found elsewhere. The books by Henrici and Markushevich are extensive monographs, covering large numbers of topics. Evgrafov and Nehari are other standard textbooks. Rubel and Colliander covers some less well-known but fascinating properties of complex functions. Hille gives an excellent account of complex-analytic aspects of ODE's. Whittaker and Watson is a great classic on applications of complex analysis to the theory of special functions. Erdelyi was the first of the asymptotic expansion books, it's rigorous and concise. Olver is a rigorous reference monograph, and covers many topics beside the asymptotic expansions of integrals. Miller is a very modern book with good topics and examples. The rest of the lot contain standard topics on asymptotic expansions of integrals that we will cover here. Sneddon is a nice book on Fourier transforms with many applications.
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