Terminology and Generalisations
If the first derivative of a function f has a simple pole at a, then a is a branch point of f. (The converse need not be true).
A non-removable singularity that is not a pole or a branch point is called an essential singularity.
A complex function which is holomorphic except for some isolated singularities and whose only singularities are poles is called meromorphic.
Read more about this topic: Pole (complex Analysis)