FoldUnfold Table of Contents The Casorati-Weierstrass Theorem The Casorati-Weierstrass Theorem Theorem 1 (The Casorati-Weierstrass Theorem): Let $U \subseteq \mathbb{C}$ be open and let $z_0 \in U$. If $f$ is holomorphic on $U \setminus \{ z_0 \}$ and if $f$ has an essential singularity at $z_0$ then for any open set $V \subset U$ containing $z_0$, the image $f(V … [Read more...]

## Rouche’s Theorem

FoldUnfold Table of Contents Rouche's Theorem Rouche's Theorem We will now look at a very important and relatively simple theorem in complex analysis known as Rouche's theorem. This theorem gives us a method to determine the number of roots of a function (counting multiplicities) in a region under certain conditions. Theorem 1 (Rouche's Theorem): Let $A \subseteq … [Read more...]

## Evaluating Definite Integrals of Type 3

FoldUnfold Table of Contents Evaluating Definite Integrals of Type 3 Evaluating Definite Integrals of Type 3a Evaluating Definite Integrals of Type 3 We will continue to apply some of results we have recently looked at regarding residues of functions at points to solve definite integrals of real-valued functions that would otherwise be difficult to compute. There are … [Read more...]

## Evaluating Definite Integrals of Type 2

FoldUnfold Table of Contents Evaluating Definite Integrals of Type 2 Evaluating Definite Integrals of Type 2a Evaluating Definite Integrals of Type 2b Evaluating Definite Integrals of Type 2 We will continue to apply some of results we have recently looked at regarding residues of functions at points to solve definite integrals of real-valued functions that would … [Read more...]

## Evaluating Definite Integrals of Type 1

FoldUnfold Table of Contents Evaluating Definite Integrals of Type 1 Example 1 Evaluating Definite Integrals of Type 1 We will now begin to apply some of results we have recently looked at regarding residues of functions at points to solve definite integrals of real-valued functions that would otherwise be difficult to compute. There are three main types of definite … [Read more...]

## The Residue Theorem

FoldUnfold Table of Contents The Residue Theorem The Residue Theorem Theorem 1 (The Residue Theorem): Let $U \subseteq \mathbb{C}$ be a simply connected and open set and let $\{ a_1, a_2, ..., a_n \} \subset U$ be a finite collection of points in $U$. Let $f$ be a complex function that is holomorphic on $U \setminus \{ a_1, a_2, ..., a_n \}$ and let $\gamma$ be a … [Read more...]

## The Residue of an Analytic Function at a Pole Singularity

FoldUnfold Table of Contents The Residue of an Analytic Function at a Pole Singularity The Residue of an Analytic Function at a Pole Singularity Recall from The Residue of an Analytic Function at a Point page that if $f$ is analytic on the annulus $A(z_0, 0, r)$ and if $\displaystyle{\sum_{n=-\infty}^{\infty} a_n(z - z_0)^n}$ is the Laurent series expansion of $f$ on … [Read more...]

## The Residue of an Analytic Function at a Point

FoldUnfold Table of Contents The Residue of an Analytic Function at a Point The Residue of an Analytic Function at a Point Consider the annulus $A(z_0, 0, r)$ (where $r > 0$), and suppose that $f$ is analytic on $A$. Then the Laurent series expansion of $f$ on $A(z_0, 0, r)$ is: (1) \begin{align} \quad f(z) = \sum_{n=-\infty}^{\infty} a_n(z - z_0)^n \end{align} Where for … [Read more...]

## Laurent’s Theorem for Analytic Complex Functions

FoldUnfold Table of Contents Laurent's Theorem for Analytic Complex Functions Laurent's Theorem for Analytic Complex Functions Recall from the Laurent Series of Analytic Complex Functions page that if $f$ is an analytic function on the annulus $A(z_0, r_1, r_2)$ then the Laurent series of $f$ centered at $z_0$ on $A(z_0, r_1, r_2)$ is defined as the following … [Read more...]

## Laurent Series of Analytic Complex Functions

FoldUnfold Table of Contents Laurent Series of Analytic Complex Functions Laurent Series of Analytic Complex Functions So far we have looked at Taylor series of analytic complex functions. We are about to look at a more general type of series expansion for a complex analytic function known as a Laurent series. We will first need to define a special type of domain called … [Read more...]