1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/ Applications

v

y

Fig. 9.3

and there is also a point z 2 in N^1 (a) such· that

lf(z2)I > k

661

u

Let C: z = z(t), a:::; t:::; /3, be any arc joining z 1 and z 2 , and contained
in N'(a). Then C': w = f(z(t)), a :::; t :::; /3, is an arc in the w-plane

joining f(z 1 ) and f(z 2 ) (Fig. 9.3). Consider the real continuous function

F(t) = lf(z(t))I over the interval [a, /3]. Fort= a we have

F(a) = lf(z(a))I = lf(z1)I < k

and for t = /3 we have

F(/3) = lf(z(/3))1 = Jf(z2)I > k

By the intermediate value theorem, there exists a point t 0 E (a, /3) such
that F(to) = k. Letting z 0 = z(to) we obtain

F(to) = lf(z(to))I = lf(zo)I = k

Thus there is z 0 E N'(a) C NHa) for which lf(zo)I = k. By starting
all over with a neighborhood N"(a) C N'(a) leaving z 0 without, we can
determine another point Zb #-zo such that lf(zb)I = k, and so on.

9.5 Nonisolated Singularities. Cluster Points

Definition 9.9 A nonisolated singularity a is said to be a cluster point

of the function f: D--+ C if a ED and if in every neighborhood of a there

are infinitely many isolated singular points off.

Examples 1. The function f(z) = csc(7r/z) has poles at z = ±1, ±^1 / 2 ,

±1/ 3 , •••• Hence z = 0 is a cluster point of csc(7r/z).