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.