Final Review for Math 425, Spring 2010

The final will be approximately 1.5 times as long as the midterms were, so it should be possible to finsh completely (and without wild stress levels). Please prepare to give definitions or theorem statements and proofs from the earlier part of the course (before Midterm II) only of the most important such.

The only different way to prepare those earlier subjects would be please to be ready to give examples, with justification, of definitions and theorems (and of complements of definitions and of false conclusions of theorems when hypotheses are missing.

Definitions/Theorems/Ideas from before Miterm II which you must know:

  1. algebra/arithmetic with complex numbers, particularly powers, roots, exponentials and logarithms (certainly some nice things with the geometric picture of roots of complex numbers and with branches of logarithms)
  2. definition and computation with the words and phrases "analytic", "entire", "harmonic", "harmonic conjugate", "Cauchy-Riemann Equations" (certainly some nice application of the CR equations)
  3. computing contour integrals from the definition, using antiderivatives, the Cauchy-Goursat Theorem, the (Extended) Cauchy Integral Formula
  4. Liouville's Theorem and some nice application, such as the Fundamental Theorem of Algebra.
  5. the Maximum Modulus Principle, some application (?)



Material we have discussed in class since Midterm II will appear in a bit more detail, including (homework-style) computations to do.

Post-Midterm II topics to know about:

  1. computing and manipulating power series, both Taylor and Laurent Series
    1. computing from Taylor's or Laurent's Theorem (ugh)
    2. based off some famous series you should know (less ugh), including
      1. the formula for the sum of a geometric series
      2. ez
      3. sin z and cos z
    3. multiplying, dividing, differentiating, and integrating power series
  2. isolated singular points
    1. the definition
    2. the three types:
      1. definitions and examples (!)
      2. expressing each type as a product of a function which is harmonic and non-zero at z0 and a power of (z-z0)
      3. for each type, a theorem about the behavior of the function as you get closer to the singularity (for poles, the limit is ∞; for essential singularities, there is the Cassorati-Weierstrass Theorem (be able to state and use!))
    3. residues:
      1. definition, examples
      2. computing
      3. using to compute integrals: the Cauchy Residue Theorem
      4. an application to computing improper real integrals (definitely be able to do this!)