1550251515-Classical_Complex_Analysis__Gonzalez_

540 Chapter^8

from which the unknown coefficients can be obtained successively. We find
that

ao

Co=-,

bo

_ ]__ Ibo ao I

C1 - b2 o b 1 ai '

and so on. In fact, the determinant of the system of the first n+ 1 equation is

0

0

bo

bn bn-1 bn-2 bo

so Cramer's rule gives (8.4-4).
Alternatively, the quotient series can be obtained by the ordinary pro-
cess of long division in ascending powers of z, as if the given series were
polynomials. To prove this, let

f(z) = P(z) + zn+lp(z), g(z) = Q(z) + zn+lq(z)

where P(z) = L:~=oakzk, Q(z) = L:~=obkzk, ~nd zn+lp(z), zn+^1 q(z)
stand for the remaining terms. By long division we may determine the

coefficients Ck of a polynomial R(z) = Z::~=O ckzk such that P(z) =

Q(z)R(z) + zn+lr(z). These coefficients are the same as those given by

(8.4-6). Since f(z) - g(z)R(z) = zn+ls(z) and g(O) = b 0 #-0, we obtain

f(z)/g(z) = R(z) + zn+lt(z).

Theorem 8.9 (Weierstrass Double Series Theorem). Suppose that:

1. All the series f m(z) = L~o amkZk (m = O, 1, 2, ... ) are convergent at
   least for lzl < R.


2. The series
   00
   F(z) = L fm(z) = [aoo + ao1z + · · · + aokZk + · · ·]
   m=O

• [a10 + auz + · · · + alkzk + · · ·]
  +···


• [amo + am1Z + · · · + amkZk + · · ·]
  +···

is uniformly convergent for lzl ::; Ri < R for every Ri < R.