COURSE NUMBER AND TITLE: MATH 4510 Complex Variables

CREDIT HOURS: 3

CATALOG DESCRIPTION: A study of the field of complex numbers, elementary functions of a complex variable, limits, derivatives, analytic functions, mapping by elementary functions, integrals, power series, residues and poles.

PREREQUISITE(S): MATH 2012 or permission of instructor.

SUGGESTED TEXT(S): Complex Variables and Applications, by Brown and Churchill

COURSE OUTLINE:

1. Complex Numbers
   • Sums and Products
   • Algebraic Properties
   • Moduli
   • Complex Conjugates
   • Exponential Form
   • Products and Quotients in Exponential Form
   • Roots of Complex Numbers
   • Regions in the Complex Plane
2. Analytic Functions
   • Functions of a Complex Variable
   • Mappings
   • Mappings by the Exponential Function
   • Limits
   • Continuity
   • Derivatives
   • Differentiation Formulas
   • Cauchy-Riemann Equations
   • Polar Coordinates
   • Analytic Functions
   • Harmonic Functions
3. Elementary Functions
   • The Exponential Function; the Logarithmic Function
   • Branches and derivatives of Logarithmic Functions
   • Complex Exponents
   • Trigonometric Functions
   • Hyperbolic Functions
   • Inverse Trigonometric and Inverse Hyperbolic Functions
4. Integrals
   • Definite Integrals of Functions
   • Contours; contour Integrals
   • Upper Bounds for Moduli of Contour Integrals
   • Antiderivatives
   • Cauchy-Goursat Theorem
   • Simply and Multiply Connected Domains
   • Cauchy Integral Formula
   • Derivatives of analytic Functions
   • Liouville’s Theorem and the Fundamental Theorem of Algebra
   • Maximum Modulus Principle
5. Series
   • Convergence of Sequences
   • Convergence of Series
   • Taylor Series; Laurent Series
   • Absolute and Uniform Convergence of Power Series
   • Continuity of Sums of Power Series
   • Integration and Differentiation of Power Series
   • Multiplication and Division of Power Series
6. Residues and Poles
   • Cauchy’s Residue Theorem
   • The Three Types of Isolated Singular Points
   • Zeros of analytic Functions