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.
Course Outline:
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.