1)      Complex numbers and functions.

2)      Topology in the complex plane.

3)      Power series and Operations.

4)      Exponential, Logarithmic, trigonometric and hyperbolic functions.

5)      Curvilinear Integrals.

6)      Homotopy and its impact on integration.

7)      Conformal transformation.

8)      Bilinear transformation.

9)      Cauchy theory.

10)   Morera Theorem and The Continuation Principle.

11) Mean Value Property, Maximum Principle and their connection.Schwarz Lemma.

12) Cauchy’s Inequality and Liouville’s Theorem.

13) Application to the Fundamental Theorem of Algebra and others.

14) Singularities and Weierstrass Theorem.

15) Laurent Series.

16) Residue Theory.

17) Residue Shortcuts with examples.

18) Applications of the Residue Theorem on Integration.

19) More Applications of the Residue Theorem.

20) Rouche’s Theorem with Applications.