How to locate and name the singularities of a complex - valued function, $f(z) $

Question

When is a point $z = z_0$ said to be a singularity of a function $f(z)$? Hence, locate and name all the singularities of $$f(z) = \cfrac{(z^2 + z^3)}{(z + i)^{3}(2z - 3i)^{2}} $$

Source: MTH301 - Functions of Complex Variables/OAU Harmattan Mid-Semester Examination/2017 - 2018 Academic Session/Q1. (b)

Answer 1

This is a rational function so the answer is easy: the singularities are the points where its value is not defined, that is where the denominater is equal to $0$ (since we cannot divide by $0$). We have $$ (z+i)^3(2z-3i)^2 = 0 \\ \Leftrightarrow \qquad z+i=0 \quad \lor \quad 2z-3i =0 \\ \Leftrightarrow \qquad z=-i \quad \lor \quad z = \frac32 i $$ As for naming the singularities, there are several types of possible categories, for example a removable singularity, a pole, and an essential singularity. They can be defined in several equivalent ways, for example as follows:

Function $f$ has a removable singularity at $z_0$, if it's not defined at $z_0$ but there exists $c\in\mathbb C$ such that function $$ \tilde f(z) = \left\{\begin{array}{ll} f(z) & \text{for }z\neq z_0 \\ c & \text{for }z= z_0 \end{array}\right.$$

In case of rational functions it is enough to check that there exists limit $\lim_{z\rightarrow z_0} f(z) = c$ to be able to tell that $z_0$ is a removable singularity.

Function $f$ has a pole of order $n$, $n\in\mathbb N$ at $z_0$ if $n$ is the lowest natural number such that $z_0$ is a removable singularity of $f(z)\cdot(z-z_0)^n$.

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Function $f$ has an essential singularity at $z_0$ if for any $n\in\mathbb N$ point $z_0$ is a not-removable singualrity of $f(z)\cdot(z-z_0)^n$.

In your case point $z_1=-i$ will be a pole of order 3, and $z_2= \frac32 i$ will be a pole of order $2$; I'll leave you to check it.