TITLE: What is the difference between a singularity and a pole?
QUESTION [11 upvotes]: From what I could find, a singularity is a point at which an equation, surface, etc., blows up or becomes degenerate.
And a pole of a function is an isolated singular point a of single-valued character of an analytic function $f(z)$ of the complex variable $z$ for which $|f(z)|$ increases without bound when $z$ approaches $a$: $\lim_{z\rightarrow a}f(z) = \infty.$
I really don't fully understand this definition of a pole, like (what is an isolated singular point) and the limit says for $\lim_{z\rightarrow a}f(z) = \infty.$ What is $a$ that $z$ should approach?
$f = 1/(z-1) \ e^{z}$ 
Can I say $f$ has a singularity at $z = 1$ because we get $1/0$ at that point i.e. blows up and gives $\infty$?

REPLY [5 votes]: One says that $z_0$ is an isolated singularity of $f$ if $f$ is defined in a punctured neighborhood $D\setminus\{z_0\}$ of $z_0$.
One says $z_0$ is a removable singularity of $f$ if there exists a holomorphic function $F(z)$ defined on $D$ which extends $f$.
Suppose $f$ is nonvanishing in a punctured neighborhood $D\setminus\{z_0\}$ of $z_0$.
Define $F(z)$ on $D$ by $F(z) = 1/f(z)$ if $z \neq z_0$ and $F(z_0)=0$.  Then $z_0$ is a pole of $f$ if $F$ is holomorphic at $z_0$.
A singularity $z_0$ is an essential singularity of $f$ if $z_0$ is neither a pole nor a removable singularity.