Algebra of complex numbers, Analytic functions, C-R equations, Harmonic functions, Elementary functions, Branches of log z, Complex exponents, Contours, Cauchy-Goursat theorem, Cauchy integral formula, Morera’s theorem, Maximum moduli of functions, Liouville’s theorem, Fundamental theorem of algebra, Convergence of sequences and series, Taylor series, Laurent series, Power series, Residues and poles, Residues theorems, Zeros of analytic function, Zeros and poles, Evaluation of improper integrals, Integrals involving trigonometric functions, Integration around a branch point, Definite integrals involving sine and cosine, Argument principle, Rouche’s theorem, Inverse Laplace transform, Mapping of complex functions, Conformal mapping, Analytic continuation.
At the end of this course the students will be able to understand the basic properties of functions of a complex variable with the theory of analytic functions and its applications.