Complex Analysis

“ca: ch_02” — 2012/3/27 — 19:14 — page 58 — #18

58

|

Complex Analysis

Let there is a point z in the neighbourhood

|

z − z

0

|

<δat which f (z) = 0. Then, equation (2.22)

gives

|

f (z

0

)

|

<ε. Choosing ε =

|

f (z

0

)

|

2

, we have

|

f (z

0

)

|

<

|

f (z

0

)

|

2

, which is a contradiction.

Thus, f (z) is non-zero at some point in the neighbourhood of z

0

.

Theorem 2.10:

If a function f (z) is continuous on a closed and bounded set S ⊂ C, then the minimum

and maximum value of

|

f (z)

|

exist on S.

Proof: We have, f (z) = u(x, y) + iv(x, y) is continuous on set S.

⇒ u(x , y) and v(x, y) are also continuous on set S.

⇒

|

f (z)

|

=



[

u(x , y)

]

2

+

[

v(x, y)

]

2

is real-valued continuous function on set S.