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)Complexnumbers 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.
