UNIVERSITY OF MADRAS
B.Sc. DEGREE COURSE IN MATHEMATICS
SEMESTER SYSTEM WITH CREDITS
(Effective from the Academic Year 2003-2004)

SYLLABUS

Semester VI - Paper XV - Complex Analysis

Duration of Examination: 3 hrs
Maximum Marks: 100
Credits: 4

Content and treatment as in "Complex Variables and Applications"-Ruel V.Churchill, James W.Brown and Roger F.Verhey-McGrawhill International student edition.

Complex Numbers:

Point at infinity,Stereographic projection.

Analytic Functions:

Functions of a complex variable, mappings, limits,theorems of limits without proof, continuity, derivatives, differentiation formula,Cauchy-Riemann equations, sufficient conditions, Cauchy-Riemann equations in Polar form, analytic functions, harmonic functions.

Mappings by elementary functions:

linear functions,the function 1/Z, linear fractional transformations, the functions w = zⁿ,w =exp(Z), special linear fractional transformations.

Integrals:

definite integrals, contours, line integrals, Cauchy-Goursat theorem(without proof), Cauchy integral formula, derivatives of analytic functions, maximum moduli of functions.

Series:

convergence of sequences and series (theorems without proofs), Taylor's series, Laurent's series, zero's of analytic functions.

Residues and Poles:

residues,the residue theorem, the principle part of a function, poles, evaluation of improper real integrals, improper integrals. integrals involving trigonometric functionc, definite integrals of trigonometric functions.

Reference Books:-

1. Theory and Problems of Complex Variables-Murray.R.Spiegel,Schaum outline series.

2. Complex Analysis-P. Duraipandian.

3. Introduction To Complex Analysis.S. Ponnuswamy, Narosa publishers 1993.