Syllabus for 534

The complex exponential and Polar form
Complex logarithms and roots
Complex differentiation and the Cauchy-Riemann equations
Harmonic functions
The Riemann sphere
Linear fractional transformations
Contours and line integrals
Cauchy's theorem and Cauchy's formula
Power series expansions of analytic functions
Zeroes of analytic functions
Laurent expansions
The structure of isolated singularities
The residue theorem
Integrals over the real line and trigonometric integrals
The argument principle
Theorems of Rouche and Hurwitz
Local inverse theorem
Simple connectivity
Maximum modulus theorem and Schwarz's lemmma
Conformal automorphisms of D and H.