Midterm I Review for Math 425, Spring 2010

Most basic definitions and computations

You must know all definitions, computations, be able to do straigthforward computational tasks, and be able to give examples of the following:
  1. absolute fundamentals with complex numbers: Re(z), Im(z), z, |z|, eiθ, vector interpretation, polar coordinates, arg(z) and Arg(z), geometric descriptions of regions in C in terms of complex equations or inequalities
  2. roots of complex numbers, computationally and geometrically -- nth roots of unity
  3. basic functions f from C to C: examples, in terms of real and imaginary parts (the famous u and v), in terms of geometric mappings from regions in the plan to other regions
  4. limits of complex functions: definition and the big idea (esp. some examples of limits which don't exist)
  5. the point at inifinity, the Riemann sphere, limits at or equalling infinity
  6. C derivatives: definition, rules, examples and counterexamples
  7. the Cauchy-Riemann equations
  8. the words analytic and entire
  9. what is a harmonic function, how to check if a function is harmonic, a common source of harmonic functions
  10. harmonic conjugates: definition and computing them
  11. the complex exponential and logarithm functions (log(z) and Log(z)): branches, branch cuts, algebraic identities, derivatives
  12. complex exponents
  13. complex trigonometric functions, definitions

Theorems/Results you must know.

This means you must be able to state them, to give examples of them being true and examples of how the conclusion is false if individual hypothesese are dropped, and to have some idea of how they are proved.
  1. a theorem relating the CR equations to analyticity
  2. a theorem about analytic functions which agree on a "thin" set agreeing everywhere
  3. the Reflection Principle
  4. a theorem on the relationship between harmonic functions and the real and imaginary parts of analytic functions
  5. a theorem which relates limits of or equal to infinity to limits of or equal to zero (using 1/z)

Format of the test

The test will be roughly one third (in terms of points) of each of the following:
  1. direct, purely computational exercises, or stating definitions of basic terms.
  2. slightly more involved calcuations and simple proofs
  3. a few (with lots of room for you to choose which ones you like) more involved calculations and proofs