Theorem 7.14  (Division of Power Series).  Suppose f(z) and g(z) are analytic at with power series representations  

               and      for all   .

If  ,  then the quotient    has the power series representation  

               for all   ,

where the coefficients satisfy the equations  .  

In other words, the series for the quotient    can be obtained by the familiar process of dividing the series for f(z) by the series for g(z) using the standard long division algorithm.

Proof.

Proof of Theorem 7.14 is an exercise in the book.

Complex Analysis for Mathematics and Engineering