MATH 3364: Introduction to Complex Analysis

Prerequisites: MATH 3331.
Text: Fundamentals of Complex Analysis with Applications to Engineering and Science,
3rd Edition, by E.B. Saff and A.D. Snider, Prentice-Hall, 2003.

Chapter 1: Complex Numbers
1.1 The Algebra of Complex Numbers
1.2 Point Representation of Complex Numbers
1.3 Vectors and Polar Forms
1.4 The Complex Exponential
1.5 Powers and Roots
1.6 Planar Sets
1.7 The Riemann Sphere and Stereographic Projection

Chapter 2: Analytic Functions
2.1 Functions of a Complex Variable
2.2 Limits and Continuity
2.3 Analyticity
2.4 The Cauchy-Riemann Equations
2.5 Harmonic Functions

Chapter 3: Elementary Functions
3.1 Polynomials and Rational Functions
3.2 The Exponential, Trigonometric and Hyperbolic Functions
3.3 The Logarithmic Function
3.4 Washers, Wedges, and Walls
3.5 Complex Powers and Inverse Trigonometric Functions

Chapter 4: Complex Integration
4.1 Contours
4.2 Contour Integrals
4.3 Independence of Path
4.4 Cauchy's Integral Theorem
4.5 Cauchy's Integral Formula and Its Consequences
4.6 Bounds for Analytic Functions

Chapter 5: Series Representations for Analytic Functions
5.1 Sequences and Series
5.2 Taylor Series
5.3 Power Series
5.4 Mathematical Theory of Convergence
5.5 Laurent Series
5.6 Zeros and Singularities
5.7 The Point at Infinity

Chapter 6: Residue Theory
6.1 The Residue Theorem
6.2 Trigonometric Integrals
6.3 Improper Integrals of Certain Functions
6.4 Improper Integrals Involving Trigonometric Functions
6.5 Indented Contours
6.6 Integrals Involving Multiple-Value Functions
6.7 The Argument Principle and Rouche's Theorem
At the instructor's discretion, other topics as time permits.