Background definitions used in my personal notes for complex analysis.

Notes taken while taking a course [1].

## Disk

\(B_r(z_0) = \{ z \in \Complex \mid |z-z_0| < r \}\)

### Circle

\(K_r(z_0) = \{ z \in \Complex \mid |z-z_0| = r \}\)

## Points

### Interior point

For \(E \subset \Complex\), \(z_0\) is an *interior point* of \(E\)
if there is \(r > 0\) such that \(B_r(z_0) \subset E\).

### Boundary point

For \(E \subset \Complex\), \(b\) is a *boundary point* of \(E\)
if for all \(r>0\), \(E \cap B_r(b) \neq \emptyset\)
and \(E^\complement \cap B_r(b) \neq \emptyset\).

The set of all boundary points of \(E\) is the *boundary set* of \(E\),
denoted \(\partial E\).

## Sets

### Open set

A set in \(\Complex\) is *open* if all of its points are interior points.

### Closed set

A set \(E\) in \(\Complex\) is *closed* if \(\partial E \subset E\).

### Closure

The *closure* of \(E\) is \(\overline{E} = E \cup \partial E\).

### Interior

The *interior* of \(E\) is the set \(\overset{\circ}{E}\) of all interior points.

## Path

A *path* in the complex plane is a continuous function
from a real number interval \([a,b]\) to complex numbers.

### Curve

A *curve* is a smooth path or a piecewise smooth path.

### Smooth path

A path is *smooth* if the function can be differentiated an arbitrary number of times.

### Piecewise smooth path

A path is *piecewise smooth* if it is the concatenation of a finite number of smooth paths.