542 Chapter^8
and since L: rm converges for 0 :::; r < 1, the uniform convergence of (8.4-7)
on lzl :::; r follows from the Weierstrass M-test. Now we have
m oo
f mZ=---= ( ) Z L z km =z m + z 2m + z. 3m + + ··· z km + ···
l-zm
k=lwhich converges for lzl < 1 (m = 1, 2, ... ). By applying Weierstrass double
series theorem we get
00
for lzl < 1
where 'Tn denotes the number of divisors of n. Note that zn appears in
L: zkm with coefficient 1 iff m is a divisor of n.
Remark The conclusion of the Weierstrass double series theorem, namely
00 00
F(z) = L fm(z) = L bkzk
m=O k=O
may be written as follows:
00 00 oo .. 00
L L amkZk = L L amkZk
m=O k=O k=O m=O
so that it amounts to the feasibility of reversing the order of the summation
symbols under the conditions set forth in the statement of the theorem.
Theorem 8.10 Let w = J(z) = L::;=O anzn for lzl < R1 and g(w) =
r::=O An(w - wor for lw - wol < R2, where Wo = f(O) = ao. Then
F(z) = g(f(z)) is analytic in some neighborhood of the origin, and
F(z)~ ~A. (t,•;z;) • ~ ~V (8.4-8)
for lzl < r :::; R1, r being such that lzl < r will imply that lf(z) - aol <
R < R2. The coefficients bn in (8.4-8) are obtained by raising the series
L: ajzi to the successive powers n = 1, 2, ... and then collecting like terms.
Proof Such an r > 0 exists by the continuity of f(z) at z = 0. The series
expansion
00 00
F(z) = g(f(z)) = L An [f(z) - aor = L Fn(z) (8.4-9)
n=O n=O
converges uniformly whenever lf(z)-a 0 j < R < R 2 , or lzl < r, and so also
converges uniformly on every compact subset of lz I < r. Since the functions