Complex Analysis I and II
Qualifier Topics
topics with * denote advanced topics

References:
L. V. Ahlfors, Complex Analysis (McGraw-Hill)
J. B. Conway, Functions of One Complex Variable I (Springer-Verlag)
S. Fisher, Complex Variables, second edition.

Also, for background:
Jones and Singerman, Complex Functions (Cambridge)
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

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

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