Possible error at definition of essential singularity Let $\Omega \subseteq \mathbb{C}$ be open and $a\in\Omega$, with $f\in\mathcal{H}(\Omega\backslash\{  a\})$.
A singularity at $a$ is called essential, iff $\forall\, w\in\mathbb{C},\ \exists$ a sequence $(z_n)_{n}$ in $\mathbb{C}$ with $\lim_{n\to\infty}z_n=a$, so that $\lim_{n\to\infty}f(z_n)=w$.
I know that there are quite a few different defintions of an essential singularity, but they all amount to the fact, that with the possible exception of a point, every complex value is adopted in an arbitrary small neighborhood around the singularity. (Great Picard theorem) At least, if I am correct.
Now how does this first definition in my post make sense? This just seems completely wrong to me. I might live with something like 
"A singularity is called essential, iff $\forall\, w\in\mathbb{C},\ \exists$ a sequence $(z_n)_{n}$ in $\mathbb{C}$ with $\lim_{n\to\infty}z_n=c,\, \ c\in\mathbb{C}$, so that $\lim_{n\to\infty}f(z_n)=w$."
But if $a$ is fixed and the singularity, how can this be true?
 A: The usual definition says that an isolated singularity that is neither removable nor a pole is called an essential singularity. (And some authors also call non-isolated singularities, e.g. branch points, essential singularities.)
Then the Casorati-Weierstraß theorem says that the isolated singularity $a$ of $f\in \mathcal{H}(\Omega\setminus \{a\})$ is essential if and only if $f(U\setminus \{a\})$ is dense in $\mathbb{C}$ for every neighbourhood $U \subset \Omega$ of $a$. [It may be stated with only the direction "$a$ essential $\implies f(U\setminus \{a\})$ dense for all $U$", but since the only possibilities for an isolated singularity are to be removable, a pole, or essential, and neither removable singularities nor poles have this property, the other direction follows.]
The definition you were given is an alternative formulation of the characterisation of essential (isolated) singularities by the Casorati-Weierstraß theorem. For, if we choose a sequence $\{U_n : n \in \mathbb{N}\}$ of neighbourhoods of $a$ in $\Omega$ with $\lvert z - a\rvert < 1/n$ for all $z \in U_n$, given an arbitrary $w\in \mathbb{C}$, by the Casorati-Weierstraß theorem there is a $z_n \in U_n\setminus \{a\}$ with $\lvert f(z_n) - w\rvert < 1/n$ for all $n\in \mathbb{N}$. Since $z_n \in U_n$ implies $\lvert z_n - a\rvert < 1/n$, we have $z_n \to a$, and by construction we also have $f(z_n) \to w$. Conversely, if for every $w \in \mathbb{C}$ there is a sequence $(z_n)$ in $\Omega \setminus \{a\}$ with $z_n \to a$ and $f(z_n) \to w$, then $f(U\setminus \{a\})$ is dense in $\mathbb{C}$ for every neighbourhood $U$ of $a$. For, given such a neighbourhood, we have $z_n \in U$ for all large enough $n$ by the definition of $z_n \to a$, and consequently $w \in \overline{f(U\setminus \{a\})}$. Since $w$ was arbitrary, that means $f(U\setminus \{a\})$ is dense in $\mathbb{C}$.
A: If that's actually the definition in the "script" you're using then in my opinion you have a bad script - giving totally non-standard defintions is a bad idea, going to cause confusion when comparing with other references.
The standard story is this:
Defintions: $f$ has an isolated singularity at $a$ if ... . An isolated singularity at $a$ is removable if ...., it is a pole if $\lim_{z\to a}f(z)=\infty$, and otherwise it is an essential singularity.
Theorem (Riemann) If $f$ has an isolated singularity at $a$and $f$ is bounded near $a$ then the singularity is removable.
THeorem (C-W). If $f$ has an essential singularity at $a$ then $f(D'(a,r))$ is dense in $\mathbb C$ for every $r>0$.
(Here $D'(a,r)=\{z:0<|z-a|<r\}$. Of course the conclusion says that for every $w$ there exist $z_n\to a$ with $f(z_n)\to w$.)
Proof: If $f(D'(a,r))$ is not dense there exist $w\in\mathbb C$ and $\rho>0$ such that $f(D'(a,r)))\cap D(w,\rho)=\emptyset$. This says that $$|f(z)-w|\ge\rho\quad(z\in D'(a,r)).$$
So $g=1/(f-w)$ is bounded near $a$, so it has a removable singularity.
And $f=w+1/g$, so $f$ has either a removable singularity or a pole, depending on whether or not $g(a)=0$. QED.
A: If $f\in\mathcal{H}(\Omega\backslash\{  a\})$, then $a$ is called a isolated sinfgularity of $f$.
The singularity is $a$ is called essential, if $a$ is not a pole and $a$ is not a removable singularity of $f$.
THEOREM(Casorati- Weierstraß): The singularity $a$ is essential iff $\forall\, w\in\mathbb{C},\ \exists$ a sequence $(z_n)$ in $ \Omega \setminus \{a\}$ with $z_n \to a$ and $f(z_n) \to w$.
