The Concept of Analytic Functions: The complex numbers and the complex plane<, Functions of a complex variable, General properties of analytic functions, Linear transformations, Basic properties of linear transformation, mapping for problems, stereographic projections, Basic concepts of conformal mapping, The exponential and the logarithmic functions, the trigonometric functions, Taylor’s series, Laurent’s series, infinite series with complex terms, power series, infinite products.
Integration in the Complex Domain: Cauchy’s theorem, Cauchy’s integral formula and its applications, Laurent’s expansion, isolated singularities of analytic functions, the residue theorem and its applications.
Contour Integration: Definite integrals, partial fraction, expansion of $\cot 2z$,
The arguments principle theorem and its applications: Rouche’s theorem,
Analytic Continuation: The principle of Analytic Continuation.