Complex Numbers (1.1-1.6),
Analytic Functions (2.1-2.5),
Elementary Functions (3.1-3.2)
Math 407, EXAM 1

Definitions, Examples, Facts, Background:

Complex numbers , as a vector, where and $\theta =\arg z$
no order on $\QTR{Bbb}{C}$ , etc. and all associated facts
and so on
Definition of $\arg z$ , , properties of principal value, and so on.
Definition of $e^{z}$ , including power series representation
DeMoivre's formula, $z^{n},z^{1/n},$ the n $^{th}$ roots of unity , various properties and geometry
All definitions of the geometry of the Complex plane, interior points, open sets, closed sets, boundary points, connected sets, domains, regions, bounded, unbounded, disks, circles, etc.
Complex-valued functions of a complex variable , graphing by -plane images to -plane images,
Sequences and limits of sequences, limits of functions, continuity,
Derivative of a complex function and what is different
Analytic function (major definition), entire functions
Cauchy-Riemann equations (rectangular and polar forms)
Laplace equation, harmonic functions, harmonic conjugates
Polynomial functions, various formulations using zeros, Taylor/Maclaurin form
Rational functions, zeros, poles
More on the exponential function $e^{z}$ , periodic and fundamental regions
$\sin z,\cos z$ definitions in terms of $e^{iz}$ , when zero, the differences

To be able to Prove or Do:

Write any complex number in any standard form; find any roots
Graph functions from -plane to -plane
Solve complex equations for
Prove various limits exist or not, or functions are continuous or not, differentiable, analytic, etc.
Show that a function is differentiable, is analytic, is entire, satisfies the Cauchy-Riemann equtions, etc.
Theorem 6, p.76, with example that shows connectedness is essential.
Finding analytic functions with specified real or imaginary parts
Theorem 3, p.111

To be able to State, Understand, or Use:

Theorem 4, p.73: differentiable at $z_{0}$ implies the Cauchy Riemann equations hold at $z_{0}$ . (Converse is false.)
Theorem 5, p.74: Modified converse of Theorem 4.
Theorem 7, p.79: Source of harmonic functions
Theorem 2, p.105: Partial fraction decomposition

Other Questions

Items appearing in any collected homework problems