M 472. Introduction to Complex Analysis (Spring 2016)


      Mandelbrot set
  Mark Pernarowski
  TEXTBOOK:   Complex Variables and Applications, 9th ed., Churchill, Brown
  GRADE:   The course % will determined as follows:
                         Percent       Date 
Wed Mar 9
see below
For both the Midterm and Final exam you will be allowed one sheet (both sides) of notes.
Calculators and electronic devices will not be permitted. The dates and content of each
exam will announced below later on in the semester.
Below the Homework and due dates will be posted.
Currently these are drafts of homework assignments so don't download any yet. 
            Due Date   
      Homework 1
    Jan 29 Chapter 1 (Notes 1-4 below)
      Homework 2
  Feb 17 Chapter 2 (Notes 5-6 below)
        #10 should have u=r cos(theta) + cos(theta)/r
      Homework 3
  Mar 2 Chapter 3 (Notes 6-9 below)
      Homework 4
  Apr 1  Notes 10,12-13 below
      Homework 5
  April 18  Notes 14-17 below
  NOTES     Periodically I will post handwritten and/or typeset lecture notes here.
Topic Description
Definition of a "Field"
Complex Numbers Definition
Moduli, Conjugates, Geometry
Euler, de Moivre, Polar forms, arguments
Square Root, Nth root, applications and quadratic formula             
Topology, functions of z, region mappings, limits and continuity
  Notes_6 Derivatives and Cauchy-Riemann Equations (revised)
  Notes_7 Exponential function e^z  (introduction)
  Notes_8 Harmonic functions, conjugate functions
  Notes_9 sin(z),cos(z),tan(z), Log(z) and Branch Cuts
  Notes_10 Contour Integrals: introduction
  Notes_11 Bounds for Contour Integrals
  Notes_12 Fundamental Theorem Calculus, Cauchy-Goursat Theorem
  Notes_13 Singularities, Multiply Connected Domains, Cauchy Int. Formulae
  Notes_14 Cauchy Inequality, Liouville Thm, Algebra Theorem, Mean Value
  Notes_15 Series, Absolute Convergence, Geometric Series
  Notes_16 Radius of Convergence, Taylor's Theorem and Series examples
  Notes_17 Laurent Series, Residue Theory introduction
  Notes_18 Improper real integrals on real line.
  Misc_Notes Misc: Residue applications and conformal mapping introduction
  Midterm 1  

You may bring one page of notes (two sided) and the handout on trig and hyperbolic

trig functions. The exam will have questions very similar to HW1-HW3. Given the 

exam is 50 min, I can only ask you a few short questions. 


Take home exam. You may use the text and posted class notes. Please staple the 

exam to your work. The Final is due Friday April 29 in class.


                       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


  Julia Sets: start at 3:00

  Mandelbrot sets

  Mandelbrot: Deep Zoom






 Old HW:

    1. Homework 1 January 28, 2008
    2. Homework 2 February 4, 2008
    3. Homework 3 February 15, 2008 
    4. Homework 4 February 27, 2008
    5. Homework 5 March 3, 2008
    6. Homework 6 March 19, 2008