Online Course on Complex Analysis



1. Welcome

Professor Badih Ghusayni, Department of Mathematics, Faculty of Science-1, Lebanese University.

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This online course is about Complex Variables, one of the very important subjects of mathematics.   This course serves as a basis and provides a solid background rich in both theory and applications.  At the end of this course the student should be able to understand the theory of complex functions in one variable and its applications. It should provide a solid teaching background if the student goes to secondary teaching.  It should prepare the student for graduate studies as well as in later interdisciplinary courses like Analytic Number Theory, where Complex Variables plays a major role in handling problems in Number Theory.

2. Description

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.