Unit 1

Complex Numbers: Definition, Algebra of complex numbers, Geometric representation, Conjugates, Absolute values, properties, Polar form. Complex Functions: Introduction, Limits, Continuity, Differentiability, Analytic functions, Cauchy-Riemann Equations in Carteisan and polar coordinates.

Unit 2

Elementary functions, exponential and Logarithmic functions, Branches of logarithm, Trigonometric and Hyperbolic functions.

Unit 3

Complex Integration: Definitions, Line integrals, Cauchy Gorsat theorem, Cauchy’s integral formula, Derivatives of analytic functions, Morera’s theorem, Liouville’s theorem, Fundamental theorem of Algebra, Gauss’ mean value theorem, Maximum modulus principle.

Unit 4

Power Series: Definitions, Taylor’s series, Laurent’s series, circle and radius of convergence. Contour Integration: Zeros and Singularities of Analytic functions, types of singularities, Poles, Meromorphic functions, principle of argument, Rouche’s theorem, Fundamental theorem of Algebra, Residues, Evaluation of residues at a pole, Cauchy’s residue theorem.

Unit 5

Evaluation of Real definite integrals by Contour integration, Evaluation of improper integrals, Jordan’s lemma, Mappings by elementary functions, linear fractional Transformation: Image of a line and circle.

Course Outcome:
CO1: Ability to understand basic concepts of the complex numbers
CO2: Understand about complex integrations
CO3: Understand about the singularities and Residues CO4: Understand the evaluation of different type integrals
CO5: Understand the concept of complex mappings and linear transformations.

Textbook:

R. V. Churchill and J. W. Brown, Complex Variables and Applications, Tata McGraw-Hill.

References:
Dennis Zill, Complex Analysis, Jones and Bartlett. Ahlfors, Complex Analysis, McGraw-Hill.
Anant R Shastri, Basic Complex Analysis of one Variable, Lakshmi Publications.