**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 T

On 3 Oct 2007, 17:05.