658 Chapter^9
provided that 0 < lz -al< min(8i,82).
- Conversely, suppose that limz--+a IJ(z)I = oo. Then there exists a
8 > 0 such that 0 < Jz -al < 8 implies that lf(z)I > 1. Let F(z) = 1/ J(z).
For 0 < lz - al < 8 we have IF(z)i < 1. Since J(z) does not vanish
in 0 < lz - al < 8, F(z), along with f(z), is analytic in some deleted
neighborhood of a. Hence F( z) is locally bounded at a, and therefore
regular at that point. Since
lim z-+a IF(z)i = z-+a lim IJ(l z )I = 0
F(z) becomes analytic at a by letting F(a) = 0. So we must haveF(z) = (z -ar?/J(z)where mis the multiplicity of the zero at a, ?j;(z) is analytic in a neighbor-
hood of a, and ?j;( a) ':/: 0. Hence it follows from Theorem 9.2 that z = a
is a pole of order m of f(z) = 1/F(z).
- If a= oo, consider g(z) = f(l/z) at z = 0. By parts (1) and (2),
z = 0 is a pole of g( z) i:ff
lim lg(z)I = oo
z-+0
Letting z = 1/ z' the condition above is equivalent toz^1 lim --+oo jg(~) Z I= zlim^1 --+oo lf(z')I = oo
9.4 Behavior at an Essential Singularity
Theorem 9.4 (Casorati-Weierstrass). Let a':/: oo be an isolated essential
singularity of f(z), and let e and 8 be two arbitrary positive numbers. Then:
1. If b is any given complex number, the inequality
IJ(z)-bl<e
holds for some points z such that 0 < lz - al < 8.
2. If I< is any given positive real number, the inequality
IJ(z)I >I<
holds for some points z such that 0 < lz - al < 8.
If a = oo, the same properties hold in any given deleted neighborhood
of oo, namely, for R < lzl < oo, where R is arbitrary. In other words, at
some points of any prescribed deleted neighborhood of the isolated essential