Complex Analysis

“ca: ch_09” — 2012/3/27 — 21:35 — page 419 — #3

Special Topics

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419

for k = 1, 2, ..., n. Form a finite sequence of neighbourhoods N

0

, N

1

, N

2

, ..., N

n−1

, N

n

(refer Figure

9.4) such that each N

k

has radius d and centre z

k

. Here, all these neighbourhoods are contained in D

and the centre z

k

of any neighbourhood N

k

(

k = 1, 2, ..., n

)

lies in the preceding neighbourhood N

k−1

.

Now, by Theorem 6.6, if f is analytic throughout a neighbourhood N

0

of z

0

and f (z) = 0 at each point

z of a domain D or line segment L containing z

0

, then f

(

z

)

≡ 0inN

0

. But z

1

lies in N

0

then again by

applying Theorem 6.6 we get f

(

z

)

≡ 0inN

1

and by proceeding in this manner, we arrive at the ...