1. TEXTBOOK AND OTHER BOOK REFERENCES:
   
   
1. Complex Variables and Applications, 6th ed.,
   Ruel Churchill, James Brown McGraw Hill Site
2.  Complex Variables (Schaum's Outline), M. Spiegel - lots of examples and cheap
3.  Basic Complex Analysis, 3rd ed., J. Marsden and M. Hoffman (harder/theoretical)


2. INSTRUCTOR: Mark Pernarowski
3. PREREQUISITES: Math 181, Math 182 (especially series), Math 224 (Partials and Green's Theorem)
4. OFFICE: Wilson 2-236
5. OFFICE HOURS: Schedule
6. PHONE: 994-5356
7. GRADING SYSTEM:
The course % will determined as follows:

		Total Points
Midterm	M	25%
Final	F	25%
Homework	HW	50%
		100

Official Textbook Syllabus :  (What parts of the text we are going to cover listed by chapter number)


1.  Chapter one: 1-11
2.  Chapter two: 12-14,16,18-26
3.  Chapter three: 29-36
4.  Chapter four: 37-43, 46, 44, 50-52
5.  Chapter five: 56-60, 62, 66-67 (techniques only)
6.  Chapter six: 68-70, 72-73
7.  Chapter seven: 78-80, TBA



MIDTERM Wednesday March 5 (before Spring Break) 

In Class, Wednesday, March 5, 2008. You may bring any hand written 
notes with you. Calculators and the text are not permitted. The exam will 
cover the first 5 material on the first 5 homework assignments. The sections 
from the textbook cover

• Chapter One: 1-11
• Chapter Two: 12-14, 16, 19-26
• Chapter Three: 29-36

Homework Assignments

1. Homework 1 Due Monday January 28, 2008 (in class).
• Basic algebra
• Sketching regions
• Re(z), Im(z), |z|, Arg(z), arg(z)
• Polar representations via DeMoivre's formula
• Simple roots
2. Homework 2 Due Monday February 4, 2008 (in class).
• Computing Roots
• Triangle Inequality
• Quadratic Formula
• Trigonometric Identities and de Moivre's formula
• Topology definitions
3. Homework 3 Due Due Friday February 15, 2008 (in class).
• Real and Imaginary parts of a complex valued function
• Images of Sets
• Maps and Mapping
• Formal Proofs of Continuity, Differentiablity
• Stereographic Projection
4. Homework 4 Due Wednesday February 27, 2008 (in class)
• Cauchy Riemann Equations
• Harmonic Conjugates
• Analytic and Entire functions
• exp(z),  and branches of log(z)
• Trigonometric functions
• Hyperbolic Trigonometric functions
5. Homework 5 Due Monday March 3, 2008 (in class)
   • Power Functions z^c
   • Inverse Trig and Inverse hyperbolic Trig Functions
   • Branch Cuts
6. Midterm Wednesday March 5 in class (open notes). For a summary of the topics conside the bulleted items in Homework assignments 1-5. The exam will cover anything we have done in the first three chapters (which includes problems on the homework that are not like questions in the book. A noteable example is the alternate "radical" form for square roots).
7. Homework 6 Due Wednesday March 19, 2008 (in class)
• Contour Integrals by direct calculation
• Placing upper bounds on contour integrals
• Contour integrals via antiderivatives
• Counter integrals of functions with branch cuts.


Topics out of the text to be chosen from:

In reality we will only be able to cover about half of this  so see the
constantly updated "official textbook syllabus" above.


• Basic Properties of Complex Numbers
• Complex numbers as a field
• Algebraic Properties
• Geometry, Modulii, Arguments, polar representation, conjugates
• Regions and elementary topology
• Analytic Functions
• Elementary Functions
• Limits
• Continuity and Derivatives
• Analytic Function Definition(s)
• Practical calculations of derivatives
• Cauchy-Riemann Equations
• Harmonic Functions
• exp(z), sin(z), cos(z), tan(z), Log(z)
• Branches of the Logarithm Function
• Complex Integration
• Contour integrals, Path Independence, Line integrals
• Cauchy Goursat Theorem
• Simply and Multiply Connect Domains
• Cauchy's integral formula (for the function and derivatives)
• Morera's Theorem
• Maximum Moduli
• Liouville's Theorem and the Fundamental Theorem of Algebra
• Analytic Series
• Taylor Series, Binomial Series
• Convergence issues and Tests
• Laurent Series
• Uniqueness of Series Representation
• Series Manipulations
• Zeros of analytic functions
• Casorati-Weierstrass and Picard Theorems (optional)
• Calculus of Residues and Integrals
• Residue Theorem and Principal Parts of Functions
• Partial Fraction Theorem (optional)
• Residues and Poles
• Simple, Multiple and Essential Poles
• Evaluation of Definite Integrals using Residue Calculations
• Introductory Examples
• Integrals involving sin(x) and cos(x)
• Integration around a branch cut
• Other Integrals
• Elementary Mappings
• Linear Functions
• Linear Fractional Transformations
• Simple Polynomial Transformations
• Inverse Functions
• Mappings involving exp(z), sin(z), Log(z)
• Conformal Maps
• Defintion
• Harmonic Functions
• Transformation of Boundary Conditions
• Applications of Conformal Mapping (overview)
• Overview of Additional Topics
• Fourier Transforms
• Laplace Transforms
• Series Summation
• Mandelbrot sets, Chaos and Alternate Dimensions
• Really weird stuff