AM 105 : COMPLEX ANALYSIS (Common with paper MA 105 of Branch I(A) Mathematics) UNIT : –I Differentiation: Analytic Functions : Derivative Rules for Differentiating Complex Functions- The Cauchy-Riemann Equations Functions-Geometrical –Analytic 1(z) 1(z) Interpretation of Arg and - Conformal Mapping Mapping w = f f –The Mapping of the Extended Plane. –Conformal UNIT II: – Mobius Transformations: The Group Property of Mobius Transformations The – Circle Property of Mobius Transformations-Fixed points of a Mobius –Preserving Transformation-Invariance of Cross Ratio-Mapping of a circle onto a Circle – Symmetry Transformations. UNIT III: – Complex Integrals: Cauchy Integral Theorem: Rectifiable Curves-Complex Integrals-The Case of Smooth Curves-Cauchy’s Integral Theorem-The Key Lemma proof of Integral Theorem-Application to the Evaluation of Definite Cauchy’s Integrals Integral Theorem for a system of Contours. Integral Cauchy’s Cauchy’s Formula Theorem Inequalities. – Morera’s – Cauchy’s UNIT-IV: Power Series: The Cauchy-Hadamard Theorem Taylor Series. The Uniqueness – Theorem for Power series-Expansion of an Analytic Function in a power series – Liouville's Theorem. The Uniqueness Theorem for Analytic functions-A Points and Zeros-Weierstrass’ Double Series Theorem-Substitution of One Power Series into Another- Division of Power series. Scope and Standard as in Chapters 3,5,7,8 and 10 of Introductory Complex “ by Richard A. Silverman Dover Publications, Inc. (1972). New York Analysis” References : 1 Complex Variables - . Schaum outline series, 2/E by Speigel 2. An Introductions to Complex Analysis, by C.L. Siegel : North Holland, 1989
AM 105 : COMPLEX ANALYSIS (Common with paper MA 105 of Branch I(A) Mathematics) UNIT –I : Differentiation: Analytic Functions : Derivative Rules for Differentiating Complex Functions- The Cauchy-Riemann Equations –Analytic Functions-Geometrical Interpretation of Arg f 1(z) and f 1(z) - Conformal Mapping –The Mapping w = –Conformal Mapping of the Extended Plane. UNIT – II: Mobius Transformations: The Group Property of Mobius Transformations – The Circle –Preserving Property of Mobius Transformations-Fixed points of a Mobius Transformation-Invariance of Cross Ratio-Mapping of a circle onto a Circle – Symmetry Transformations. UNIT – III: Complex Integrals: Cauchy Integral Theorem: Rectifiable Curves-Complex Integrals-The Case of Smooth Curves-Cauchy’s Integral Theorem-The Key Lemma proof of Cauchy’s Integral Theorem-Application to the Evaluation of Definite Integrals Cauchy’s Integral Theorem for a system of Contours. Cauchy’s Integral Formula – Morera’s Theorem – Cauchy’s Inequalities. UNIT-IV: Power Series: The Cauchy-Hadamard Theorem – Taylor Series. The Uniqueness Theorem for Power series-Expansion of an Analytic Function in a power series – Liouville's Theorem. The Uniqueness Theorem for Analytic functions-A Points and Zeros-Weierstrass’ Double Series Theorem-Substitution of One Power Series into Another- Division of Power series. Scope and Standard as in Chapters 3,5,7,8 and 10 of “ Introductory Complex Analysis” by Richard A. Silverman Dover Publications, Inc. (1972). New York References : 1 Complex Variables - . Schaum outline series, 2/E by Speigel 2. An Introductions to Complex Analysis, by C.L. Siegel : North Holland, 1989