Complex analysis

Basic reference on complex analysis.

Notes taken while taking a course [1].


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

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 \(\partial E \subset E\).
The closure of \(E\) is \(\overline{E} = E \cup \partial E\).
The interior of \(E\) is the set \(\overset{\circ}{E}\) of all interior points.


If for all \(\varepsilon > 0\) there is \(\delta > 0\) such that \(|f(z) - c| < \varepsilon\) whenever \(|z-z_0| < \delta\), then

\[ \lim_{z \rightarrow z_0} f(z) = c \]

\(f\) is continuous at \(z_0\) if

\[ \lim_{z \rightarrow z_0} f(z) = f(z_0) \]


\[ \frac{d}{dz} f(z_0) = \lim_{z \rightarrow z_0} f(z) \]

\(f\) is analytic in an open set \(U \subset \Complex\) if \(f\) is differentiable for every \(z \in U\). A function which is analytic in \(\Complex\) is an entire function.

Cauchy-Riemann equations

If \(f(z) = u(x,y) + i v(x,y)\) for \(z = x+iy\) and real functions \(u\) and \(v\),

\[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \] \[ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \]


\[ \frac{d f}{d z} = \frac{\partial f}{\partial x} = -i\frac{\partial f}{\partial y} \]

\(f=u+iv\) is analytic in \(D\) if and only if for any \(z \in D\), \(\frac{\partial u}{\partial x}\) and \(\frac{\partial v}{\partial y}\) exist, are continuous, and satisfy the Cauchy-Riemann equations.

Fundamental theorem of algebra

If \(a_0\), \(\ldots\), \(a_n\) are complex numbers with \(a_n \neq 0\), then the polynomial

\[ p(z) = \sum_{k=0}^n a_k z^k \]

has \(n\) complex roots \(z_1\), \(\ldots\), \(z_n\), where

\[ p(z) = a_n \prod_{k=1}^n (z - z_k) \]

See also


Petra Bonfert-Taylor. Introduction to complex analysis. Retrieved from