PG - TRB - MATHS MAJOR - COMPLEX ANALYSIS - FUNCTIONS OF COMPLEX VARIABLE - UNIT - 4

PG - TRB - MATHS MAJOR - COMPLEX ANALYSIS - FUNCTIONS OF COMPLEX VARIABLE - UNIT - 4 SUBJECT : PG.TRB - MATHS MAJOR WHAT'S UP & TELEGRAM : 90420 30163 FACEBOOK : FUTURE VISION STUDY CENTRE INSTAGRAM : future_vision_study_centre WEBSITE : www.futurevisiontnpsc.com Unit IV COMPLEX ANALYSIS Introduction to the concept of analytic function: Limits and continuity – Analytic functions – Polynomials and rational functions – Elementary theory of power series – Maclaurin’s series – Uniform convergence – Power series and Abel’s limit theorem – Analytic functions as mapping – Conformality arcs and Closed curves – Analytical functions in regions – Conformal mapping – Linear transformations – the linear group, the cross ratio and symmetry. Complex integration – Fundamental theorems – line integrals – rectifiable arcs – line integrals as functions of arcs – Cauchy’s theorem for a rectangle – Cauchy’s theorem in a Circular disc – Cauchy’s integral formula: The index of a point with respect to a closed curve – The integral formula – Higher derivatives – Local properties of Analytic functions and removable singularities – Taylor’s theorem – Zeros and Poles – The local mapping – The maximum modulus Principle