660 Chapter^9
for some points z such that z E NHO) n Dg, where Dg = {z: l/z En,}.
Also,
lg(z)I >I<
for some points z such that z E NHO) n D 9. Letting z = l/z' we obtain
lf(z') - bl < €
for some points z' such that z' E Nk( oo) n D fl and also
lf(z')I >I<
for some points z' such that z' E Nk( oo) n D f.
Example The function f(z) = e^1 fz has z = 0 as an isolated essential
singularity. In fact, this function has the Laurent expansion
e l/z =!+-+--+···^1 1 1
z 2! z^2
valid for z =I-0. Ifwe let z = x we have e^1 /x - 00 if x - o+ while e^1 /x - 0
if x - o-. On the other hand, if we let z = iy we have
e^1 fiy = e-i/y =cos .!_ - i sin .!_
y y
so that in this case e^1 fz moves around the circle lwl = 1 as z = iy - 0.
For any given b, let e =/:- 0 be such that le - bl < €. Then e^1 /z = e for
l/z =loge= In lei+ i(O + 2br). For lkl large enough there are values of
z, given by the preceding equation, which lie in NHO).
It can be shown that in the vicinity of an isolated essential singularity
f(z) actually assumes every complex value with one exception at most.
This is Picard's theorem, which is rather deep and will be proved elsewhere
(Selected Topics, Theorem 3.41).
Example In any deleted neighborhood of z = 0 the function e^1 /z takes
on every value, excepting 0.
A simpler result is contained in the following theorem.
Theorem 9.5 In any given deleted neighborhood of an isolated essential
singularity a of f there are infinitely many points where IJ(z)I = k, k
being a given positive real number.
Proof We shall consider together the cases a =/:-oo and a = oo, denoting
by N'(a) a deleted circular neighborhood contained in the given deleted
neighborhood NH a) and such that f be analytic in N' (a). By the Casorati-
Weierstrass theorem there is a point z 1 E N' (a) such that
lf(zi)I = lf(z1) - OI < k
