200 Chapter4
By taking in (4.10-2) a fixed value of n > max(N 1 ,N2), we obtain,
using (4.10-3) to (4.10-5),
lf(z)-Ll<E
provided that z E N 0 (zo) n B. This proves that limz_.zo,zEB f(z) = L ..
Remark We have shown that
lim lim un(z) = lim f(z) = L = lim Ln = lim lim un(z)
z-+-zo n-+oo z-+zo n-+oo n---+-oo Z-tZoThus Theorem 4.16 establishes conditions under which the inversion of the
order of the limits is permissible.
Corollary 4.4 Suppose that the set B in the preceding theorem is open,
that z 0 E B, and that the functions un(z) are continuous at zo. Then it
follows that f ( z) is continuous at z 0 • In other words, the limit function
of an uniformly convergent sequence of continuous functions is again a
continuous function.
Note The conclusion of the corollary still holds if B is not open, z 0 E B,
and the continuity at z 0 of the functions involved is taken in a restricted
sense, i.e., as z -;. z 0 through the set B.
Definidons 4.11 Let
00
L Un(z) = uo(z) + u1(z) + · · · + un(z) + · · · (4.10-6)
n=obe a series of functions un(z ), all of them having a set D "# 0 as a common
domain of definition, and· let
Sn(z) = uo(z) + u1(z) + · · · + un(z) (4.10-7)
be the partial sums of.the series ( 4.10-6). We say that C C Dis the conver-
gence set of the series, or that the series converges (pointwise) on C, if that
set is the convergence set of the sequence {Sn(z)}. Iflimn_. 00 Sn(z) = F(z)
for z EC, the function F(z) is called the sum of the series, and we write
00
F(z) = L un(z), z EC
n=oOn D1 = D - C the sequence {Sn(z)} diverges and we say, accordingly,
that Z::un(z) diverges on D 1. On any nonempty subset B of C where
Sn(z) =+ F(z) the series I: Un(z) is said to converge uniformly to F(z).