MATH-6300, COMPLEX ANALYSIS

MATH-6300

COMPLEX ANALYSIS

Time: Monday and Thursday, 12:00 to 1:50 PM.
Room: Carnegie 206
Instructor: Gregor Kovacic
Office: 419 Amos Eaton
Phone: 276-6908
E-mail: kovacg at rpi dot edu
Office Hours: Click here.

Topics

Topics will include: Cauchy's theorem, residues, branch points, Fourier and Laplace transforms, asymptotic evaluation of integrals, infinite series, partial fractions, infinite products, entire functions, conformal mappings and the Riemann mapping theorem, inverse scattering, Riemann-Hilbert and Wiener-Hopf problems, elliptic and theta functions, analytic continuation and Riemann surfaces, holomorphic differentials on compact Riemann surfaces, Jacobi inversion problem. Examples and applications will include: ordinary differential equations, Sturm-Liouville problems, fluid mechanics, electricity and magnetism, stochastic processes, singular integral equations, integrable partial differential equations and soliton theory.

Notes

Here, you can find PDF files of my lecture notes:

File #1, File #2, File #3, File #4, File #5, File #6, File #7, File #8, File #9, File #10, File #11, File #12, File #13

The list of homework problems

The homework assignments:

# 1, # 2, # 3, # 4, # 5, # 6, # 7, # 8, # 9

Textbooks

The following textbooks on complex analysis and related topics may be useful:

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Ccattering, Cambridge
M. V. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM
L. V. Ahlfors, Complex Analysis, McGraw-Hill
E. D. Belokos, A. I. Bobenko, V. Z. Enol'skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer
G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Variable, Theory and Technique, Hod Books
J. B. Conway, Functions of One Complex Variable I, II, Springer
J. W. Dettman, Applied Complex Variables, 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
G. L. Lamb, Elements of Soliton Theory, Wiley
S. V. Manakov, S. Novikov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons, The Inverse Scattering Method, Plenum
A. I. Markushevich, Theory of Functions of a Complex Variable, 3 Volumes, Chelsea
H. P. McKean and V. Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic, Cambridge
N. I. Muskhelishvili, Singular Integral Equations, Dover
Z. Nehari, Conformal Mapping, Dover
A. C. Newell, Solitons in Mathematics and Physics, SIAM
L. A. Rubel and J. E. Colliander, Entire and Meromorphic Functions, Springer
W. Rudin, Real and Complex Analysis, McGraw-Hill
G. Springer, Introduction to Riemann Surfaces, Chelsea
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

I ordered the textbooks by Ablowitz and Fokas, Dettman, and Ahlfors, all of which you can buy in the bookstore. The first two have excellent applications, the first more modern and the second more classical. The last is much more rigorous and has been the classic textbook on the subject for half a century. The books by Conway, Dettman, Evgrafov, Nehari, Rudin, and Titchmarsh are other standard textbooks. The book by Carrier, Krook, and Pearson is very applied with many excellent examples. The books by Henrici and Markushevich are extensive monographs. The book by Springer is not only an excellent introduction to Riemann surfaces, but also to algebraic topology in a concrete setting. The book by Hille gives an excellent account of complex-analytic aspects of ODE's. The book by Whittaker and Watson is a great classic on applications of complex analysis to the theory of special functions. Other books on the list present specialized topics or applications.

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