# Background definitions for complex analysis

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.

## References

[1]
Petra Bonfert-Taylor. Introduction to complex analysis. Course on Coursera.