Complex numbers in the form $x+\ii y$ and $r \e^{\ii\theta}$;
complex plane and representation by 2D vectors

Fundamental Theorem of Algebra: algebraic equation of degree
$n$ as $(z-z_1)(z-z_2)\ldots(z-z_n)=0$

The $n$th roots of unity: $\e^{2\pi\ii k/n}$ for $0\le k < n$

Sets of points; interior, boundary and exterior points;
open sets; curves and regions of the complex plane.

Spiegel 1.1-1.5, 1.7, 1.8, 1.10-1.14, 1.18

Riley 2; Boas 2; Arfken 6.1

Functions of complex variables

Real and imaginary parts of functions: $f(z)=u(x,y)+\ii v(x,y)$

Standard functions - ratios of polynomials, exponential
and log, trig and hyperbolic trig functions

Multiple-valued functions; non-integer powers and log;
principal values of functions, e.g. $\Arg z$ versus $\arg z$ and
$\Ln z$ versus $\ln z$; branches and branch cuts

Spiegel 2.1-2.7

Riley 18.1, 18.5; Boas 14.1; Arfken 6.1

Functions as mappings

Mappings of points, curves and regions from the $z$ plane to
the plane of $w=f(z)$

Curves which circle $z_0$ in the $z$-plane map to curves which
circle the origin in the $w$ plane if $f(z_0)=0$

The argument theorem: $\Delta\phi=2\pi N$, where $\phi=\arg w$
and $N$ is the number of simple zeros of $f$ that are enclosed by
a curve in the $z$-plane

In the diagrams above the blue and red lines are the mappings of
the lines $x=$ const and $y=$ const, and the black dot is the mapping
of the point $z=0$.

Spiegel 2.4-2.7

(Riley 14.8); (Boas 14.9); Arfken 6.6

Differentiation and Cauchy-Riemann equations

Definition of derivative: $$\dby fz=\lim_{\delta z\to
0}\frac{f(z+\delta z)-f(z)}{\delta z}$$

Derivative must be finite and independent of direction