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

- Computation with complex numbers and algebra of complex numbers
(DeMoivre, etc.)
- Properties of modulus, including estimation with triangle and
reverse triangle inequalities, and other tricks.
- Vocabulary: open and closed and connected.
- Mapping properties of complex functions (especially powers, exp,
log, and conjugate).
- Sequences and series with complex numbers
- Computations of functions (such as powers, trig,
hyperbolic, exp, log); also, Arg and arg, and Log and log.
- Limits and continuity including
where is ?
- Differentiability and analyticity of complex functions
- The Cauchy-Riemann equations
- Derivative=0 implies the function is constant.
- Other conditions imply the function is constant.
- Connections with harmonicity, including harmonic conjugates.

- Line integrals
- The definition.
- Using parameterizations, especially for line segments and circular
arcs.
- Green's Theorem and early examples of deformations of contours.
- Computation via antidiffentiation; why 1/z around the unit circle
*can't* be done this way.
- Estimation via the ML inequality.

- Power series
- Radius of convergence, including computation with examples.
- Continuity and differentiability of power series.
- 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.

**
Maintained by **`
greenfie@math.rutgers.edu` and last modified 2/26/2008.