Those Blair students who have finished Magnet Analysis II are eligible to take the semester-long Complex Analysis. This course, taught by Mr. Walstein, examines various mathematical concepts as they apply to funcations of a complex variable.

Topics include:

• Complex Number System
  • Definition of the Complex Number System
  • Fundamental Operations with Complex Numbers
  • Algebraic Properties of Complex Numbers
  • Geometric Interpretation of Complex Numbers
  • Further Properties of Moduli
  • Polar Form
  • Exponential Form
  • Powers and Roots
  • Regions in the Complex Plane
  • Spherical Representation of Complex Numbers
  • Dot and Cross Products
• Analytic Functions
  • Functions of a Complex Variable
  • Mappings
  • Limits
  • Theorems on Limits
  • Continuity
  • Derivatives
  • Differentiation Formulas
  • Cauchy-Riemann Equations
  • Sufficient Conditions for Analycity
  • Polar Coordinates
  • Analytic Functions
  • Harmonic Functions
• Elementary Functions
  • The Exponential Function
  • Other Properties of exp z
  • Trigonometric Functions
  • Hyperbolic Functions
  • The Logarithmic Functions and Its Branches
  • Further Properties of Logarithms
  • Complex Exponents
  • Inverse Trigonometric and Hyperbolic Functions
• Complex Integration
  • Definite Integrals of w(t)
  • Contours
  • Line Integrals and Examples
  • Cauchy-Goursat Theorem
  • Preliminary Lemma
  • Proof of the Cauchy-Goursat Theorem
  • Simply and Multiply Connected Domains
  • Antiderivatives and Independence of Path
  • Cauchy Integral Formula
  • Derivatives of Analytic Functions
  • Morera's Theorem
  • Maximum Moduli of Functions
  • Liouville's Theorem and the Fundamental Theorem of Algebra
• Infinite Series
  • Convergence of Sequences and Series
  • Taylor Series
  • Observations and Examples
  • Laurent Series
  • Further Properties of Series
  • Uniform Convergence
  • Integration and Differentiation of Power Series
  • Uniqueness of Series
  • Representations
  • Multiplication and Division of Series
  • Examples
  • Zeros of Analytic Functions
• Residues and Poles
  • Residues
  • Calculation of Residues
  • The Residue Theorem
  • Principal Part of a Function
  • Residues at Poles
  • Quotients of Analytic Functions
  • Evaluation of Improper Real Integrals
  • Improper Integrals Involving Sines and Cosines
  • Definite Integrals Involving Sines and Cosines
  • Integration through a Branch Cut
• Mappings By Elementary Functions
  • Linear Functions
  • The Function 1/z
  • Linear Fractional Transformations
  • Special Linear Fractional Transformations
  • The Function z²
  • The Function square root of z
  • Related Functions
  • The Transformation of w = exp z
  • The Transformation of w = sin z
  • Successive Transformations
  • Table of Transformations of Regions
• Conformal Mappings
  • Basic Properties
  • Further Properties and Examples
  • Harmonic Conjugates
  • Transformations of Harmonic Functions
  • Transformations of Boundary Condition