How to recognize removable singularity and how to remove it

Question

I don't understand the idea of a removable singularity yet. Can someone explain me how to recognize a removable singularity and how to remove it?

Example: $g(z)=f(z)/z$. Is $z=0$ then a removable singularity and if yes, how would I remove it?

Answer 1

If both $f$ and $g$ are holomorphic at $z_0$, they have Taylor expansions $$f(z)=\sum_{k=0}^\infty a_k(z-z_0)^k,\quad g(z)=\sum_{k=0}^\infty b_k(z-z_0)^k$$ Let $m$ be the smallest index for which $a_k\ne 0$, and $n$ be the smallest index for which $b_k\ne 0$. Then $$ \frac{f(z)}{g(z)} = z^{m-n}\frac{a_{m }+a_{m+1}(z-z_0)+\dots}{b_{n}+b_{n+1}(z-z_0)+\dots} \tag{1}$$ where the fraction on the right is holomorphic in a neighborhood of $z_0$ (because the denominator does not vanish), and takes on the nonzero value $a_m/b_n$ at $z_0$. The behavior of $f/g$ at $z_0$ is determined by $m-n$. If $m\ge n $, you have removable singularity, since the right-hand side of (1) is holomorphic. If $m<n $, you get a pole.