MA465 COMPLEX VARIABLES (Cr. 3)

COURSE DESCRIPTION

Prerequisite:  MA 211 (Introduction to Matrix Theory and Linear Algebra) and MA 265 (Calculus III)

Course Description
Complex numbers, analytic functions, conformal mapping, residues and poles, analytic continuation, and Riemann surfaces.

Goal/Purpose
The purpose of this course is to apply the basic ideas of calculus (limits, differentiation, and integration) to functions of a complex variable.  These notions are studied from both a theoretical and an applied point of view.

Course Outline

1. Complex Number System
   1. Arithmetic of complex numbers
   2. Limits
2. Complex Functions
   1. Examples
      1. exponential functions
      2. trigonometric functions
      3. logarithms
      4. power functions
      5. various inverse functions
   2. Euler's theorem and its consequences
3. Differentiation
   1. Definition
   2. Rules
   3. Cauchy-Riemann Equation
   4. Introduction to conformal mapping
   5. Applications
      1. vector fields
      2. divergence and curl
      3. Laplace's Equation
4. Integration
   1. Definition
   2. Rules
   3. Cauchy's Theorem
   4. Complex line integrals
   5. Indefinite integrals
   6. Cauchy's integral formula
      1. higher derivatives
      2. principle of maximum modulus
   7. Taylor's Theorem
   8. Laurents' Theorem
   9. Residue Theorem
      1. calculation of residues
      2. evaluation of definite integrals
5. Analytic Continuation
   1. Introduction
   2. Schwartz' reflection principle

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