Suppose that a complex valued functionoperating on complex numbers has a singularity atThen

  1. has a removable singularity atif there is a functionanalytic atsuch thatfor

  2. has a pole of orderatif there is a functionanalytic atwith and a positive numbersuch thatfor

  3. has an essential singularity atif the singularity atis neither removable or a pole.

Example:has a singularity at 0 since the denominator =0 atbut the singularity is removable since

is analytic atandon

Example:has two poles, one atand one at

so the pole athas order 1.

so the pole athas order 2

Example:has an essential singularity atTo see this, write

Because the powers ofare negative and becoming larger without limit, the singularity is essential. We cannot multiplybyfor anyto makeanalytic.