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