Summer Session A 2011 Tentative Course Outline
Week 1
- Lecture 1:
- I.1 Complex Numbers
- I.2 Polar Representation
- Lecture 2:
- I.1 Regions in the Complex Plane
- I.2 Finding nth Roots of a Number
- I.3 Stereographic Projection
- Lecture 3:
- I.3 More Stereographic Projection
- I.4 The Square and Square Root Functions
- I.5 The Exponential Function
- I.6 The Logarithm Function
Week 2
- Lecture 4:
- I.6 More on the Logarithm Function
- I.7 Power Functions and Phase Factors
- I.8 Trigonometric and Hyperbolic Functions
- II.1 Review of Basic Analysis
- II.2 Analytic Functions
- II.3 The Cauchy-Riemann Equations
- Lecture 5:
- II.2 More Analytic Functions
- II.4 Inverse Mappings and the Jacobian
- II.5 Harmonic Functions
- Lecture 6:
- II.6 Conformal Mappings
- II.7 Fractional Linear Transformations
Week 3
- Independence Day Holiday: No Lecture
- Lecture 7:
- Midterm I Solution
- III.1 Line Integrals and Green's Theorem
- III.2 Independence of Path
- Lecture 8:
- III.3 Harmonic Conjugates
- IV.1 Complex Line Integrals
- IV.2 Fundamental Theorem of Calculus for Analytic Functions
Week 4
- Lecture 9:
- IV.3 Cauchy's Theorem
- IV.4 The Cauchy Integral Formula
- IV.5 Liouville's Theorem
- Lecture 10:
- IV.6 Morera's Theorem
- V.1 Infinite Series
- V.2 Sequences and Series of Functions
- V.3 Power Series
- Lecture 11:
- V.3 Finding the Radius of Convergence for Power Series
- V.4 Power Series Expansion of an Analytic Function
- V.5 Power Series Expansion at Infinity
Week 5
- Lecture 12:
- V.6 Manipulation of Power Series
- V.7 The Zeros of an Analytic Function
- VI.1 The Laurent Decomposition
- Lecture 13:
- VI.1 More Laurent Decomposition and Laurent Series
- VI.2 Isolated Singularities of an Analytic Function
- Lecture 14:
- VI.3 Isolated Singularity at Infinity
- VI.4 Partial Fractions Decomposition
- VII.1 The Residue Theorem
- VII.2 Integrals Featuring Rational Functions
Week 6
- Lecture 15:
- Midterm II Solution
- VII.3 Integrals of Trigonometric Functions
- VII.4 Integrands with Branch Points
- Lecture 16:
- Midterm Recap
- VII.4 More Integrands with Branch Points
- VIII.1 The Argument Principle
- Lecture 17:
- Review For Final Exam/Course Evals
- Final Exam