Zeros, Singularities, Residues: Singularites
Print this page
  
First   |   Last   |   Prev   |   Next

Suppose that $z_{0}$ is a pole of $f(z)$. Consider a neighborhood $N(z_{0})$ of the point $z_{0}$. Take any point $z \in N(z_{0})$ and compute the value of $\vert f(z) \vert$. Now, if we plot the points MATH in the three dimensional space, then the surface obtained in this manner will have a pole structure about the point $z_{0}$. That is why, it is called a pole of $f(z)$.

Note 3: If a point $z_{0}$ is an essential singularity of $f(z)$ then the limit of $f(z)$ does not exist as $z$ appproaches $z_{0}$. The function behaviour as $z$ approaches $z_{0}$ is quiet complicated.

Suppose that $z_{0}$ is an isolated singular point of $f(z)$. Then $f(z)$ has a Laurent expansion in MATH for some $r > 0$. The following theorem characterises the singularities in terms the nature of the Laurent series.

Theorem: Let $f(z)$ have an isolated singularity at the point $z_{0}$.And let

MATH

 
 
First   |   Last   |   Prev   |   Next