I have the following functions:
a) $\displaystyle f:\mathbb{C}\backslash\{0\}\rightarrow\mathbb{C},\ f(z)=\frac{1}{e^{\frac{1}{z}}-1}$
b) $\displaystyle f:\mathbb{C}\backslash\{0,2\}\rightarrow\mathbb{C},\ f(z)=\frac{\sin z ^2}{z^2(z-2)}$
c) $\displaystyle f:\mathbb{C}\backslash\{0\}\rightarrow\mathbb{C},\ f(z)=\cos\left(\frac{1}{z}\right)$
d) $\displaystyle f:\mathbb{C}\backslash\{0\}\rightarrow\mathbb{C},\ f(z)=\frac{1}{1-\cos\left(\frac{1}{z}\right)}$
e) $\displaystyle f:\mathbb{C}\backslash\{0\}\rightarrow\mathbb{C},\ f(z)=\frac{1}{\sin\left(\frac{1}{z}\right)}$
What would the quickest approach to determine if $f$ has a removable singularity, a pole or an essential singularity? What would be the thinking $behind$ the approach?
Edit:
What I know/ What I have tried:
I know that if we have an open set $\Omega \subseteq \mathbb{C}$, then we call an isolated singularity, a point, where $f$ is not analytic in $\Omega$ ($f \in H(\Omega \backslash \{a\}$).
The functions in (a)-(e) are not defined on some values. So I suspect, that these are the first candidates for singularities. For instance in (a), it would be 0. In (b), it would be 0 and 2.
Question: Could there be any other points where these functions are not analytic?
Let's call our isolated singularity $a$. Furthermore I know that we have 3 types of singularities:
1) removable
This would be the case when $f$ is bounded on the disk $D(a,r)$ for some $r>0$.
2) pole
There is $c_1, ... , c_m \in \mathbb{C},\ m\in\mathbb{N}$ with $c_m \neq 0$, so that:
$f(z)-\sum\limits_{k=1}^m c_k\cdot\frac{1}{(z-a)^k},\ z \in \Omega \backslash \{a\})$
has a removable singularity in $a$, then we call $a$ a pole.
We also know that in this case:
$|f(z)|\rightarrow \infty$ when $z\rightarrow a$.
3) essential
If the disk $D(a,r) \subseteq \Omega$, then $f(D(a,r)\backslash\{a\})$ is dense in $\mathbb{C}$ and we call $a$ essential singularity.
The books that I have been using (Zill - Complex Analysis and Murray Spiegel - Complex Analysis) both expand the function as a Laurent series and then check the singularities. But how do I do this, if I use the definitions above? It doesn't seem to me to be so straight forward...
What I would want to learn a method which allows me to do the following:
I look at the function and the I try approach X to determine if it has a removable singularity. If not continue with approach Y to see if we have a pole and if not Z, to see if we have an essential singularity. An algorithmic set of steps so to speak, to check such functions as presented in (a) to (e).
Edit 2: This is not homework and I would start a bounty if I could, because I need to understand how this works by tommorow. Unfortunately I can start a bounty only tommorow...
Edit 3: Is this so easy? Because using the definitions, I am getting nowhere in determing the types of singularities...