School of Mathematics, Trinity College

Course 414 - Complex Analysis 2005-06 (click for more information)

Optional JS & SS Mathematics, SS Two-subject Moderatorship

Lecturer Dmitri Zaitsev

Problem Sheets in PDF: Sheet 1 Sheet 2 Sheet 3 Sheet 4 Sheet 5 Sheet 6

Course outline:

Elementary functions of one complex variable: polynomials, exponential, logarithmic and trigonometric functions, their inverses. Real and complex differentiability. Holomorphic functions. Conformal mappings.

Piecewise smooth and rectifiable paths and curves. Complex integration along curves. Antiderivatives. Cauchy's theorem: Goursat's version for a triangle, for star-shaped regions and their unions, homotopy version. Elements of homology and homological version of Cauchy's theorem.

Cauchy's integral formula. Power series expansion of holomoprhic functions. Mean value property. Maximum modulus principle. Radius of convergence of power series. Cauchy-Hadamard formula. Differentiation of power series. Theorem of Morera. Cauchy's estimates. Liouville's theorem. Compact convergence and Weierstrass theorem.

Order of zeroes. The identity principle. Laurent series expansion in a ring. Isolated singularities. Removable singularities, poles, essential singularities. Riemann extension theorem. Meromorphic functions. Casorati-Weierstrass theorem.

Residues, their caculation. Residue theorem: for unions of star-shaped regions and the winding number version. Applications to calculation of integrals.

The argument principle. Rouché's theorem. Open mapping theorem. The univalence theorem (local injectivity criterion). Inverse function theorem. Branched covering structure theorem.

Spaces of holomorphic functions. Seminorms. Montel's theorem. Biholomphic maps between open sets. The Riemann mapping theorem.

Schwarz Lemma. Automorphisms of the disk. Homogeneity of the disk. Cayley transform. Automorphisms of the upper half-plane. Möbius transformations. Riemann sphere. Holomorphic and meromorphic functions on the Riemann sphere. Automorphisms of the Riemann sphere.

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 414 web page for 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.

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.