School of Mathematics, Trinity College

Course 214 - Complex Variable 2009 (click for more information)

Optional JS & SS Mathematics, SS Two-subject Moderatorship

Lecturer Dmitri Zaitsev

Examinations 2009: The format will be the same as in the last year 214 exam (which can be considered a sample paper). The theoretical questions will be within the scope of the current course and the practical problems within the scope of the current homework. The exam will count for 100% of the grade.

Problem Sheets in PDF (to appear): Sheet 1 Sheet 2 Sheet 3

Solutions to most problems are analogous to those in Course 414 - Complex Analysis 2007-08. Solutions to different problems can be found here.

Course outline:

Complex numbers, elementary operations: addition, multiplication, their properties. The conjugate, the absolute value and their behaviour with respect to addition and multiplication. Elementary functions of one complex variable: polynomials, exponential, logarithmic and trigonometric functions, their inverses.

Open, closed, connected sets. Limits of sequences and functions, their behaviour with respect to addition, multiplication, division. Cauchy's criterion for convergence. Continuous functions. Continuity of sums, products, ratios, compositions. Definitions of continiuty using open and closed sets. Compactness and connectedness, its preservation under continuous maps. Uniform convergence and continuity of uniform limits of continuous functions. Branches of multi-valued functions. Examples of branches of the argument function and the logarithm.

Infinite series of complex numbers. The comparison test. Absolute convergence. Infinite function series and their uniform convergence. Weierstrass test. Power series. Abel's Lemma. Radius of convergence.

Complex-differentiable and holomorphic functions. Differentiability of sums, products, ratios, composition and inverse functions. Real-differentiable functions. Cauchy-Riemann equations. Complex differentiability of polynomials, rational functions, exponential, logarithm and trigonometric functions.

Path integrals. Independence of parametrization. Length of a path and estimates for path integrals. Antiderivatives. Calculation of path integrals using antiderivatives.

Cauchy's theorem: Goursat's version for a triangle, generalization for polygonal regions and simple bounded regions with piecewise smooth boundary. Cauchy's integral formula. Residue theorem. Taylor series expansion. Calculation of residues for ratios of holomorphic functions.

Applications of Residue theorem: Trigonometric integrals, Improper integrals, Fourier transform type integrals etc.

Laurent series expansion. Poles. Calculation of residues using Laurent series expansion. Order of zeroes. Identity principle. Maximum modulus principle.

R. V. Churchill, J. W. Brown, Complex Variables and Applications, Fourth edition. McGraw-Hill Book Co., New York, 1984.
L. V. Ahlfors, Complex Analysis, Third Edition, McGraw-Hill, New York, 1978.
J. B. Conway, Functions of One Complex Variable, Second Edition, Graduate Texts in Mathematics 11, Springer-Verlag, New York, 1978.
R. Remmert, Theory of Complex Functions, Graduate Texts in Mathematics 122, Springer-Verlag, New York, 1991.
B. P. Palka, An Introduction to Complex Function Theory, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1991.

Some links.
Complex Analysis Project by John H. Mathews.
Graphics for Complex Analysis by Douglas N. Arnold.
A Complex Function Viewer by The University of British Columbia SunSITE.
Wolfram Mathworld Pages on Complex Analysis
Wikipedia Pages on Complex Analysis
Conformal Projections in Cartography by Carlos A. Furuti

Old courses homepages:
Course 214 - Complex Variable 2008 by David Wilkins with Lecture Notes and other information.
Course 414 - Complex Analysis 2007-08 with Problem Sheets and Solutions.
Course 414 - Complex Analysis 2005-06 with Problem Sheets.
Course 414 - Complex Analysis 2003-04 by Richard M. Timoney with Lecture Notes and Problem Sheets.

For exam-related problems look in TCD past examination papers and Mathematics department examination papers.

Student Counselling Service

I will appreciate any (also critical) suggestions that you may have for the course. Let me know your opinion, what can/should be improved, avoided etc. and I will do my best to follow them. Feel free to come and see me if and when you have a question about anything in this course. Or use the feedback form from where you can also send me anonymous messages.