Suppose that a complex valued functionoperating on complex numbers has a singularity atThen
has a removable singularity atif there is a functionanalytic atsuch thatfor
has a pole of orderatif there is a functionanalytic atwith and a positive numbersuch thatfor
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.