Annual Examinations:
The format will be the same as in the past year modules, which can be considered sample papers. Credit will be given for the best 3 questions out of total 4 questions. The theoretical questions will be within the scope of the current course and the practical problems within the scope of the current homework. Non-programmable calculators are permitted for this examination. The exam will count for 100% of the grade.Lecture Notes:
My Lecture Notes in PDF are only meant to supplement the material and older lecture notes.Module 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 and compact 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. Connectedness and compactness, their preservation under continuous maps. Uniform convergence and continuity of uniform limits of continuous functions. Uniform continuity of continuous functions on compact sets. Branches of multi-valued functions. Examples of branches of the argument and of the logarithm.
Infinite series of complex numbers. Geometric series and its convergence properties. The comparison test. Absolute convergence. Infinite function series and their uniform convergence. Weierstrass (M-)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. Cauchy's theorem for star-shaped and polygonal sets. Winding numbers. General Cauchy's theorem for piecewise continuously differentiable paths. Cauchy's Integral formula. Cauchy's Residue theorem. Calculation of residues for ratios of holomorphic functions.
Applications of Residue theorem: Trigonometric integrals, Improper integrals, Fourier transform type integrals.
Power series expansions of holomorphic functions. Differentiation of power series.
Textbooks(some books are available online, just copy-paste and search):
R. V. Churchill, J. W. Brown, Complex Variables and Applications, Fourth edition. McGraw-Hill Book Co., New York, 1984.Various links:
Graphics for Complex Analysis by Douglas N. Arnold.Old modules pages:
Course 23204 - Introduction to Complex Analysis 2022 with Problem Sheets.Past examinations:
TCD examination papers (2012 - present)Feedback:
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.