Math 455 -- Complex Analysis -- Spring 2023

Course Policies: We will cover chapters 1-4 in the textbook and 5 if we have time. There will be two in class 50 minute exams. The first will cover chapters 1 & 2, the second will cover chapters 3 & 4. There will be a comprehensive 2 hour final. There will be regular homework assignments. Grades will be based on the following: 20% homework, 25% for each of the two in class exams, and 30% for the final.
Homework Assignments
Lecture Notes 
Chapter 1
Sections 1.1 & 1.2
Section 1.3
    Graphing Complex Polynomials
    Möbius Transformations Revealed (Optional External Link)
    Möbius Transformations (Optional External Link)
    Fourth Roots of 1-2i.
    Extra Example: (1+i)^(sqrt(2) + i).
Section 1.4 Continuity
    Summary of Section 1.4    
Section 1.5 Dervatives
    Example of Harmonic Conjugates
Section 1.6 More Derivatives
    
Chapter 2
Section 2.1
    Extra Example
Section 2.2
Section 2.3
   Cauchy's Theorem On A Rectangle
   Cauchy's Theorem On A Disk
   Homotopy Definitions
   Homotopy Theorem
   Technical Lemmas
   2.3 #8
Section 2.4
   Outline
   Lecture Notes
Section 2.5
   Part 1
   Part 2
   Extra Examples
   
Chapter 3
Section 3.1: Lecture Notes
Section 3.2
Section 3.3: Lecture Notes

Chapter 4
Section 4.1: Lecture Notes
             Complex Residues with Wolfram Alpha
Section 4.2: Lecture Notes
Section 4.3: Integrals of Rational Functions of cosine and sine
	     Integrals over the Real Line
             Principle Value Integrals
	     Mellin Transform Example
	     Extra Example, Integral of sin(x^2) from 0 to infinity (Optional Reading)
             Another Example (Optional Reading)
	     Another Example (Optional Reading)
Section 4.4: Proof of Main Thms 
             Examples 
	     
Chapter 5
Section 5.1: Lecture Notes
Section 5.2: Lecture Notes, Part 1
             Lecture Notes, Part 2
             Computer Experiments:
	        The map T(z) = 1/z.
                Youtube Video on T(z)=1/z.
                The map T(z)=z + 1/z.
                The arcsine function.
		Schwarz-Christoffel Examples
             Optional Reading:
		Chapter 11: The Schwarz-Christoffel Transformation
             (Brown and Churchill's book
	     Complex Variables and Applications, 9th Edition.) 
Section 5.3: Triangle Barrier Example
             Rectangle Barrier Example