Midterm II Review for Math 425, Spring 2010

Most basic definitions and computations

Here are basic words and topics. You should be able to state the definitions, to do computations, and to give examples of the following concepts:
  1. steps toward defining and computing contour integrals:
    1. how to differentiate a complex-valued function of a real variable
    2. how to integrate a complex-valued function of a real variable
    3. words describing curves in the complex plane:
      1. arc
      2. parametrized arc (or parametrization of an arc)
      3. simple arc
      4. simple closed curve (or Jordan curve)
      5. positively oriented simple closed curve (or simple closed curve with counterclockwise orientation)
      6. differentiable arc
      7. smooth arc
      8. contour (or piecewise smooth arc)
      9. simple closed contour
    4. definition and examples of the complex contour integral
  2. two bounds on the modulus of a contour integral:
    1. the (infinite-dimensional) triangle inequality, thinking of the integral as a kind of sum
    2. another coming from a bound on the modulus of the integrand and the length of the contour
  3. complex antiderivatives:
    1. definition
    2. a condition in terms of integrals around closed contours for the existence of anti-derivatives
  4. a simply or multiply connected domain
  5. computing contour integrals:
    1. open contours — using the antiderivative
    2. closed contours:
      1. deform the contour to several separate contours, each going around one of the singularities of the integrand inside the original contour; then do the following for each smaller contour:
      2. if the singularity is of the form 1/(z-z0), use the CIF
      3. if the singularity is of the form 1/(z-z0)n for n>1, use the xCIF
  6. using Liouville's Theorem or the Maximum Modulus Principle to prove some function is constant, given some conditions it satisfies

Theorems/Results you must know.

This means you must be able to state them and to give examples of them being true and examples of how the conclusion is false if individual hypothesese are dropped. When possible, it is also good to have some idea of how they are proved.
  1. the Jordan Curve Theorem
  2. the theorem in §43 giving a bound on the modulus of a contour integral
  3. the theorem in §44 relating the existence of an antiderivative of a complex function to values of various contour integrals of that function
  4. Cauchy-Goursat — proof in a (very) simple case (such as if the domain is a square and the real and imaginary parts of the function have continuous partial derivatives)
  5. the theorem in §48, very like Cauchy-Goursat but with a hypothesis about the domain and for general paths (instead of a hypothesis about a particular curve and its interior)
  6. the corollary in §48 guaranteeing the existance of antiderivatives of analytic functions in simply connected domains
  7. the theorem in §49 on contour integrals in multiply connected integrals
  8. the Cauchy Integral Formula — outline of the proof
  9. the Extended Cauchy Integral Formula
  10. the three theorems in §52, all consequences of the xCIF
  11. Liouville's Theorem
  12. the Fundamental Theorem of Algebra and its proof
  13. the Maximum Modulus Principle

Format of the test

The test will be roughly one third (in terms of points) of each of the following:
  1. doing basic computations, typically of contour integrals or related expressions
  2. stating definitions or theorems
  3. proving (or giving the key idea of the proof of) either big, famous theorems or smaller, specific exercise-type statements