Singularities

A singularity of is a point where is not analytic.
Singularity checklist:

• Is a branch-point singularity?
• Is it a non-isolated singularity?
• If neither, find Laurent series around and check:
  • If we have a Taylor series (i.e. for ), then it is a removable singularity
  • If we have an s.t. for and then we have a pole of order
  • If there is no such then we have an essential singularity.

Let have an essential singularity at . Then in any neighbourhood of , takes all possible complex values except at most one.