Corollary 7.8.  Let f(z) and g(z) be analytic with  zeros of orders  m  and  n,  respectively at  .  Then their quotient   has the following behavior:

(i)  If  ,  then h(z) has a removable singularity at  .   If we define  ,  then h(z) has a zero of order  .

(ii)  If  ,  then h(z) has a pole of order  .

(iii)  If  ,  then h(z) has a removable singularity  at  ,  and can be defined so that h(z) is analytic at  ,  by  .  

Proof.

Proof of Corollary 7.8 is an exercise in the book.

Complex Analysis for Mathematics and Engineering