The topics to be covered (numbers in the bracket indicate the the number of lectures allotted):

I- week
1.1 Basic properties of complex numbers:historical remarks, why the name complex etc.(1)
2.1 Review of Diff. Calculus of 2-variables.(1)
1.2 Geometry of complex numbers: rigid motions of the plane.(1)
2.2 Complex differentiability; CR equations.(1)
1.3 Sequences and series:uniform convergence, Weierstrass' M-test.(1)
3.1 Line integrals, M-L inequality; diff. under integral sign(1)
1.4 Formal power series, convergent power series, analytic functions; Exponential, trigonometric, and logarithmic functions; complex exponents.(1)
3.2 Path-independence; exactness, existence of primitives.(1)
1.5 Extended complex plane: FLTs.(1)
3.3 Cauchy-Goursat for triangles; Cauchy's theorem, Integral formula, estimates, Liouville's theorem. (1)
1.6 Conformality: applications.(1)
3.4 FTA, Taylor's theorem and analyticity, Zeros of holomorphic functions; Identity theorem.(1)

II-week
3.5 Isolated singularities.(1)
4.1 Winding number.(1)
3.6 Laurent series and residues.(1)
4.2 Argument principle, local solutions, open mapping theorem.(1)
2.3+2.4 Simple connectivity, homology and homotopy form of Cauchy's theorem.(2) 4.3 Inverse function theorem, branch theorem, Maximum Modulus Principle, Schwartz lemma.(1)
4.4 Residues, GAP, Rouche's theorem.(1)
4.5+4.6 Applications to improper integrals.(2)
2.5+2.6 Convergence in function theory, Runge's theorem with some applications. (2)

Special Lectures:
1. Application of Complex Analysis in Fluid Dynamics. (VDS)
2. Gamma function, Riemann zeta function. (GKS)
3. Gaussian Sums. (MN)