### My summary of principal subject matter for the first exam in Math 403:1, spring 2008

1. Computation with complex numbers and algebra of complex numbers (DeMoivre, etc.)
2. Properties of modulus, including estimation with triangle and reverse triangle inequalities, and other tricks.
3. Vocabulary: open and closed and connected.
4. Mapping properties of complex functions (especially powers, exp, log, and conjugate).
5. Sequences and series with complex numbers
6. Computations of functions (such as powers, trig, hyperbolic, exp, log); also, Arg and arg, and Log and log.
7. Limits and continuity including where is ?
8. Differentiability and analyticity of complex functions
1. The Cauchy-Riemann equations
2. Derivative=0 implies the function is constant.
3. Other conditions imply the function is constant.
4. Connections with harmonicity, including harmonic conjugates.
9. Line integrals
1. The definition.
2. Using parameterizations, especially for line segments and circular arcs.
3. Green's Theorem and early examples of deformations of contours.
4. Computation via antidiffentiation; why 1/z around the unit circle can't be done this way.
5. Estimation via the ML inequality.
10. Power series
1. Radius of convergence, including computation with examples.
2. Continuity and differentiability of power series.
3. Using tricks with geometric series to create power series, and doing this in reverse, that is, using tricks to create "closed forms" for power series.