1550251515-Classical_Complex_Analysis__Gonzalez_

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138 Chapter^3


  1. limz_,a(f /g)(z) = oo


4. limz-->a (g / !)( Z) = 0


  1. limz-->af(z) = 00

  2. limz->a IJ(z)I = 00


Proofs (1) By Theorem 3.2-8 we have limz-->a lg(z)I = ILi. Hence there
is a 01 > 0 such that z E N6 1 (a) n D implies that lg(z)I < ILi + 1. By

definition 3.3, for any given I< > 0 there is a Oz > 0 such that z E N6 2 (a )nD

implies that lf(z)I > ]{ + ILi + 1. Thus 8 = min(o1, Oz), z E N6(a) n D
implies. that


IJ(z) ± g(z)I ~ IJ(z)l - lg(z)I >I<

(2) If L i- 0, corresponding to E =^1 / 2 ILI there is o 1 > 0 such that
z E N6 1 (a) n D implies that


-^1 /2ILI < lg(z)I - ILi <^1 /2ILI

and hence lg(z)I >^1 / 2 ILI. Also, for any given ]{ > 0 there is Oz > 0

such that z E N6 2 (a) n D implies that IJ(z)I > 2I</ILI. Therefore, for
o = min(o1,0 2 ) and z E N8(a) n D, we have


IJ(z)g(z)I = IJ(z)ljg(z)I > ]{

The proofs of the remaining properties are left as exercises.

Example Consider the function F(z) = 1/[(z - 2)(.i + 1)]. Let J(z) =
1/(z - 2) and g(z) = 1/(z + 1). Since


lim J(z) = oo
z-->2

and limg(z) = 1/s i- 0

z-->2

it follows, by property 2 above, that limz_. 2 F(z) = oo.


3.9 Infinite Limit at Infinity


Definition 3.4 Let f: D --; C be a single-valued complex function, and
suppose that oo is an accumulation point of D. The point oo is said to be
the limit of J(z) as z approaches oo [or the limit of f(z) at oo] if for every


]{ > 0 there is an M > 0 (depending on K) such that

IJ(z)I >I< whenever lzl > M and z ED

or alternatively, if


z E N6(oo) n D implies f(z) E N~(oo)

with 8 = 1/(1 + M^2 )^112 and E = 1/(1 + ]{2)1/2.

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