#### Specific Objectives of course:

This is an introductory course in complex analysis, giving the basics of the theory along with applications, with an emphasis on applications of complex analysis and especially conformal mappings. Students should have a background  in  real  analysis  (as  in  the  course  Real  Analysis  I), including the ability to write a simple proof in an analysis context.

#### Introduction:

The algebra of complex numbers, Geometric representation of complex numbers, Powers and roots of complex numbers.

#### Functions of  Complex  Variables:

Definition,  limit  and  continuity,Branches of functions, Differentiable and analytic functions. The Cauchy-Riemann equations, Entire functions, Harmonic functions, Elementary functions: The exponential, Trigonometric, Hyperbolic, Logarithmic   and   Inverse   elementary   functions,   Open   mapping theorem. Maximum modulus theorem.

#### Complex Integrals:

Contours and contour integrals, Cauchy-Goursattheorem, Cauchy integral formula, Lioville’s theorem, Morerea’s theorem.

#### Series:

Power series, Radius of convergence and analyticity, Taylor’s and Laurent’s series, Integration and differentiation of power series. Singularities,  Poles  and  residues:  Zero,  singularities,  Poles  and Residues,  Types  of  singular  points,  Calculus  of  residues,  contour integration, Cauchy’s residue theorem with applications.
Mobius transforms, Conformal mappings and transformations.