Complex Analysis

(Last revised: September 2005)

Qualifier topics:

Topics with * denote advanced topics

References:

		E. Stein, R. Shakarchi, Complex Analysis (PUP)
		G. Jones, D. Singerman, Complex Functions: An Algebraic and Geometric Viewpoint (Cambridge)
		L. V. Ahlfors, Complex Analysis (McGraw-Hill)
		J. B. Conway, Functions of One Complex Variable I (Springer-Verlag)
		S. Fisher, Complex Variables, second edition.
		S. Lang, Complex Analysis (4th ed., Springer)

Also, for background:

		Rudin, Real Analysis (McGraw-Hill);
		Brown and Churchill, Complex Variables (McGraw-Hill);
		Spiegel, Theory and Problems of Complex Variables (Schaum's Outline Series)

Analytic functions, basic properties:

		the derivative:
		Cauchy-Riemann equations
		conformal mapping
		harmonic conjugates

power series representation of analytic functions:

		uniform convergence
		radius of convergence

elementary examples of analytic functions and their mapping properties:

		z^n and polynomials
		rational functions
		Moebius transformations
		exp z and log z and trig functions

Complex integration:

		the complex line integral
		Cauchy integral formula and theorem
		Estimates of the absolute value of the complex integral
		Liouville's Theorem
		Fundamental Theorem of Algebra
		winding number
		simple connectedness and the existence of the antiderivative
		Morera's theorem
		Fourier transform

Singularities:

		three types of singularities
		Laurent series
		residues
		the argument principle
		Rouche's Theorem
		Casorati-Weierstrass Theorem (concerning essential singularities)
		evaluation of real integrals using residues

Conformal Maps:

		Examples
		Angles

The Riemann Sphere:

		Point at infinity
		Rational functions and meromorphic functions
		Residue theorem

Moebius Transformations:

		Group structure, PSL(2,C) and SL(2,C), same with R.
		Properties of the cross-ratio.
		Classification.

Other theorems and concepts:

		*Schwarz lemma
		*Open mapping theorem
		*Maximum Modulus Theorem
		*Mean value property for harmonic functions
		*Poisson kernel
		*the Riemann Mapping Theorem
		*Weierstrass products
		*the Gamma function, product representation
		*Elliptic functions, Weierstrass P function