1550251515-Classical_Complex_Analysis__Gonzalez_

132 Chapter 3

Example If f(z) = z^2 + i, D = {z: Rez > O}, then

1

F(z) = z2 + i

G(z)=

1

z^2 +i

z2

M(z)= l+iz2

{

on D- 1-i} ./2

on D

{

1 +.}

on D - ./2

2

By making use of the extended complex plane (or the Riemann sphere)
it is possible to assign the point oo to those points where a denominator
vanishes. This convention will be used often in what follows.

3.5 LIMIT OF A COMPLEX FUNCTION AT A POINT

Definition 3.1 Let f: D ___, C be a single-valued complex function of

a complex variable, and let a (a -:f; oo) be an accumulation point of its
domain D. The number L ( L # oo) is said to be the limit of f ( z) as z

approaches a [or, the limit of f(z) at a] if for every E > 0 there is a 8 > 0

(depending in general on E) such that

lf(z) -LI < E whenever^0 < lz - al <^8 and z E D

Alternatively, if for every E there is a 8 such that

zEN6(a)nD implies f(z) E N€(L)

(Fig. 3.5).

This is expressed briefly by writing

lim J(z) = L

z-->a

or f(z) ___, L asz-?a

y v

NE(l)

/ .....

( D

z

a

N3(a)nD

0 x 0 LI

Fig. 3.5