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?
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 $\lvert z - a\rvert &lt; 1/n$ 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 $\lvert f(z_n) - w\rvert &lt; 1/n$ for all $n\in \mathbb{N}$. Since $z_n \in U_n$ implies $\lvert z_n - a\rvert &lt; 1/n$, 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 $\mathbb{C}$.
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 $f(z_n) \to w$.