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...]