1  algebra and analytic geometry of complex numbers, the Riemann sphere  

2  power series and their convergence  

3  differentiability and analyticity, differentiability of power series  

4  elementary functions in the complex domain

5  conformality

6  Cauchy-Riemann equations, partial z and partial z bar

7  Mobius transformations

8  Path integrals, Cauchy's formula and theorem for a disc, analytic implies power 
   series at each point,

9  Cauchy's estimate, Maximum Modulus Principle, Liouville's Theorem, Fundamental 
   Theorem of Algebra, isolation and finite multiplicity of roots  

10 homotopy of paths and Cauchy's theorem, motivation for winding numbers  

11 winding numbers and Cauchy's integral formula, zero counting

12 Morera's Theorem, theorem on existence of primitives, isolated singularities  

13 poles, essential singularities, Casorati-Weierstrass Theorem, Laurent expansion  

14 The Residue Theorem, residue integrals