Course Contents
Functions of complex variable. The exponential, logarithmic, trigonometric and hyperbolic functions. Infinite series with complex terms, power series, infinite products. General properties of analytic functions. rational functions. Cauchy’s theorem, Cauchy’s integral formula. Fundamental theorem of integral calculus, Laurent’s expansions, zeroes and singularities of analytic functions, mapping by rational functions, maximum and minimum principles, definite integrals, partial fractions, expansion of cot 2z, Riemann Surfaces. Contour integration.
Course Learning Outcomes
Objective of this course is to enable the students to learn methods and technique with necessary analysis which are based on Complex system and useful while exploring the real life problem and studying universe.
The Cauchy-Riemann Equations
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The Complex Exponential Function
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The Complex Derivative
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INTRODUCTION TO COMPLEX ANALYSIS
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Algebra and Geometry in the Complex Plane
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Polar Representation of Complex Numbers
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Roots of Complex Numbers
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Topology in the Plane
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Complex Functions
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Sequences and Limits of Complex Numbers
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Complex Trigonometric Functions
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First Properties of Analytic Functions
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Inverse Functions of Analytic Functions
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Complex Integration
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Complex Integration - Examples and First Facts
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The Fundamental Theorem of Calculus for Analytic Functions
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Cauchy’s Theorem and Integral Formula
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Consequences of Cauchy’s Theorem and Integral Formula
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Infinite Series of Complex Numbers
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Power Series
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The Radius of Convergence of a Power Series
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The Riemann Zeta Function And The Riemann Hypothesis
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The Prime Number Theorem
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Laurent Series
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Isolated Singularities of Analytic Functions
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The Residue Theorem
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Finding Residues
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Evaluating Integrals via the Residue Theorem
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Evaluating an Improper Integral via the Residue Theorem
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Book Title : Complex Variables and Applications
Author : J. W. Brown and R. V. Churchill
Edition :
Publisher : McGraw Hill
Book Title : Elements of Complex Variables
Author : Louis L. Pennisi
Edition :
Publisher : Linehart and Winston
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