School of Mathematics School of Mathematics
Course 214 - Complex Variable 2007-08 (SF Mathematics, SF Theoretical Physics, optional JS Two-subject Moderatorship )
Lecturer: Dr. D.R. Wilkins
Requirements/prerequisites:
Duration: 12 weeks

Number of lectures per week: 3

Assessment: Two assignments, providing 10% of the credit for the course

End-of-year Examination: One 2-hour examination

Description: See http://www.maths.tcd.ie/~dwilkins/Courses/214/ for more detailed information.

Section 1: Functions of a Complex Variable.
The complex plane; definition and basic properties of limits of infinite sequences of complex numbers; basic definitions of limits and continuity for functions of a complex variable; basic theorems concerning limits and continuity.
Section 2: Infinite Series.
Definition of convergence for infinite series; the Comparison and Ratio Tests; absolute convergence; Cauchy products; uniform convergence; power series; the exponential function.
Section 3: Winding Numbers of Closed Paths in the Complex Plane.
The Path Lifting Theorem; winding numbers; path-connected and simply-connected subsets of the complex plane; the Fundamental Theorem of Algebra.
Section 4: Path Integrals in the Complex Plane.
The definition of the path integral; path integrals and boundaries.
Section 5: Holomorphic Functions.
The definition of holomorphic functions and their derivatives; the Cauchy-Riemann equations; the Chain Rule for holomorphic functions; differentiation of power series.
Section 6: Cauchy's Theorem.
Path integrals of polynomial functions; winding numbers and path integrals; Cauchy's Theorem for a triangle; Cauchy's Theorem for star-shaped domains; more general forms of Cauchy's Theorem; residues; Cauchy's Residue Theorem.
Section 7: Basic Properties of Holomorphic Functions.
Taylor's Theorem for holomorphic functions; Liouville's Theorem; Laurent's Theorem; Morera's Theorem; meromorphic functions; the Maximum Modulus Principle; the Argument Principle.
Section 8: Examples of Contour Integration.
Section 9: The Gamma Function.
Section 10: Elliptic Functions.

Oct 3, 2007

File translated from TEX by TTH, version 2.70.
On 3 Oct 2007, 17:05.