AM 205 : ADVANCED COMPLEX ANALYSIS (Common with paper MA 205 of Branch I(A) Mathematics) UNIT : –I Laurent Series-Singular Points: Laurent Series-Laurent’s and Theorem-Poles Essential Singular points-Behavior at an Essential Singular point. Picard’s Behavior at infinity. UNIT II: – The Residue Theorem and its Applications: The Residue Theorem-Residues at infinity-Jordan’s of Definite Integrals The Argument Lemma-Evaluation – Theorems of Rouche and Hurwitz-Local of Analytic Mappings-The Maximum Modulus principle and Schwarz’s Lemma. UNIT III: – Harmonic Functions: Equations-Conjugate Harmonic Functions-Poisson’s Laplace’s integral. Formula-The Dirichlet problem. Schwarz’s Conformal Mapping: General Principles of Conformal Mapping of the –Mapping Upper Half-Plane onto a Rectangle Schwarz-Christoffel Transformation. –The UNIT-IV: Infinite product and Partial Fraction Expansions: Preliminary Results- Infinite Products-Weierstrass’ Theorem Theorem The gamma Functions –Mittage – Leffer’s – Theorem on Partial Fraction Expansions. –Cauchy’s Scope and Standard as in Introductory Complex by Richard A. “ Analysis” Silverman, Dover Publications, Inc. New York (1972) Chapter 11 to 15. References : 1. Fundamentals of Complex Analysis- Edward B. Saff, Arthur David Snider, Pearson Education 2. Foundations of Complex Analysis by S. Ponnusamy- Narosa Publications
AM 205 : ADVANCED COMPLEX ANALYSIS (Common with paper MA 205 of Branch I(A) Mathematics) UNIT –I : Laurent Series-Singular Points: Laurent Series-Laurent’s Theorem-Poles and Essential Singular points-Behavior at an Essential Singular point. Picard’s at infinity. UNIT – II: The Residue Theorem and its Applications: The Residue Theorem-Residues at infinity-Jordan’s Lemma-Evaluation of Definite Integrals – The Argument Theorems of Rouche and Hurwitz-Local Behavior of Analytic Mappings-The Maximum Modulus principle and Schwarz’s Lemma. UNIT – III: Harmonic Functions: Laplace’s Equations-Conjugate Harmonic Functions-Poisson’s integral. Schwarz’s Formula-The Dirichlet problem. Conformal Mapping: General Principles of Conformal Mapping –Mapping of the Upper Half-Plane onto a Rectangle –The Schwarz-Christoffel Transformation. UNIT-IV: Infinite product and Partial Fraction Expansions: Preliminary Results- Infinite Products-Weierstrass’ Theorem –Mittage – Leffer’s Theorem – The gamma Functions –Cauchy’s Theorem on Partial Fraction Expansions. Scope and Standard as in “ Introductory Complex Analysis” by Richard A. Silverman, Dover Publications, Inc. New York (1972) Chapter 11 to 15. References : 1. Fundamentals of Complex Analysis- Edward B. Saff, Arthur David Snider, Pearson Education 2. Foundations of Complex Analysis by S. Ponnusamy- Narosa Publications