Complex Integration: Line Integrals Rectifiable Arcs – Line Integrals as Functions of Arcs – Cauchy’s theorem for a rectangle - Cauchy’s theorem in a disk, Cauchy’s Integral formula: The Index of a point with respect to a closed curve – The Integral formula – Higher derivatives
Removable singularities, Taylor’s Theorem – Zeros and Poles – The Local Mapping – The Maximum principle – chains and cycles.
The Residue theorem – The Argument principle – Evaluation of definite integrals (type I and type II)only. The Definitions and basic Properties – Mean value property – Poisson’s Formula.
Power series and expansion:Weierstrass’s Theorem – The Taylor Series – The Laurent Series – Partial fractions and Factorization: Partial Fractions – Infinite Products – Canonical Products.
Statement and Proof – Boundary Behavior – Use of the reflection principle – Analytic arcs – Conformal mapping of Polygons: The Behavior at an angle – The Schwarz – Christoffel Formula – Mapping on a rectangle.
Reference Book:
1. Real and Complex Analysis by W.Rudin, 3rd Edition, McGraw-Hill, New York, 1986. 2. A First Course in Complex Analysis with Applications by Dennis.G. Zill, Patrick.D. Shanahan, Jones &Bartlet, 2nd Edition. 3. Complex Analysis For Mathematics And Engineering by Mathew Howell, 5th Edition , Narosa Publications,2011
Text Book:
Complex Analysis by L.V. Ahlfors, McGraw Hill, New York, International Edition, 1979. Unit I : Chapter – 4 Sections 1.1 – 1.5, and 2.1 – 2.3. Unit II : Chapter – 4 Sections 3.1 - 3.4 and 4.1. Unit III : Chapter – 4 Sections 5.1 – 5.3, 6.1 – 6.3. Unit IV : Chapter – 5 Sections 1.1 – 1.3, 2.1 – 2.3. Unit V : Chapter – 6 Sections 1.1 – 1.4, 2.1 – 2.3.
