Review: Algebra of complex numbers, operations of absolute value and conjugate, standard inequalities for absolute value. 

Limits, Continuity, derivatives and analytic functions, Cauchy-Riemann equations, , Harmonic functions and harmonic conjugates, Power series, Exponential and Logarithmic functions. 

Contour Integrals -Anti derivatives-Cauchy-Goursat theorem-Simply Connected Domains Multiply Connected Domains, Cauchy’s theorem for rectangle – Cauchy’s theorem in a disk, An Extension of the Cauchy Integral Formula.

Taylors series, Laurent series; Isolated singularities: removable singularities, poles and essential singularities; Cauchy’s residue theorem, Residues at Infinity, evaluation of definite integrals usingCauchy’s residue theorem.

Evaluation of Improper Integrals -Improper Integrals from Fourier Analysis – Jordan’s Lemma – Indented Paths – Definite Integrals Involving Sines and Cosines – Argument Principle. Rouche’s theorem. 

Linear Transformations-The Transformation w = 1/z – Mappings by 1/z -Linear Fractional Transformations. (Chapter 8, Sec: 90-94).

1. James ward Brown, Ruel V. Churchill, Complex Variables and Applications, Eighth Edition, McGrawHill. 
2. S. Ponnusamy, Foundations of Complex Analysis, 2nd Edition, Narosa Publishing House, 2005. 
3. Conway, John B., Functions of One Complex Variable, II, Graduate Texts in Mathematics, 159, Springer-Verlag, New York, 1995. 
4. Lars V. Ahlfors, Complex Analysis, 2ndEdition, McGrawHill, New York, 1966.