1550251515-Classical_Complex_Analysis__Gonzalez_

Functions. Limits and Continuity. Arcs and Curves 141

Examples

1. f(z) = o {z^2 } as z ~ 0 means that f(z) is an infinitesimal of order
   higher than z^2 (as z ~ 0).


2. f(z) = o(l) as z ~ a may be used to indicate that f(z) is an
   infinitesimal as z ~ a.


3. z^2 + z + 1 = O(z^2 ) as z ~ oo means that (z^2 + z + 1)/z^2 is bounded
   as z ~ oo.


4. f(z) = 0(1), z EA, means that IJ(z)[ is bounded for z in the set A.

Definition 3. 7 If limz_,a f ( z) = oo (a finite or oo) it is said that the

function f(z) is infinitely large as z ~ a. Clearly, the reciprocal of an
infinitesimal function is an infinitely large function, and conversely.
Infinitely large functions can be compared in a manner similar to that
of infinitesimals (see Exercises 3.1, problem 8).

Exercises 3.1

1. Show that:

(a) lim. [3y^2 + i tan(y - l)] = 3 + i

z-+i y -1

(b) z---j.2-i lim. z = 2 + i

(cl) li~ z-+i ~1 Z + = 00

2. Find:

( )^1

. z + i

c 1m--=l

z->oo Z - 1

(e) z-+oo lim (z^3 +1) = oo

(a) lim z

2

+^4 (b) lim z

2

+z

Z-+2i Z - 2i Z->-i 1 + Z

1

. z^2 + 3z - 1 (
1
. z^2 + 3
(c) 1m d) 1m --
z-+oo 2z2 + z + 3 z-+oo z

3. If f(z) and g(z) are two infinitesimals as z ~a, and A and Bare any

two constants, prove the following.

(a) Af(z) + Bg(z) is an infinitesimal as z ~ a.

(b) f(z)g(z) is also an infinitesimal as z ~ a.

(c) if h(z) is a bounded function [i.e., if jh(z)[:::; Mas z ~a], f(z)h(z)

is also an infinitesimal z ~ a.

4. If f(z) and g(z) are infinitely large as z ~ a, then f(z)g(z) is again

an infinitely large as z ~ a. However, Af(z) + Bg(z) need not be

infinitely large (give an example), and f(z)h(z) is infinitely large if h(z)

is bounded away from zero, i.e., if [h(z)I :'.'.'. m > 0.

5. Prove that the sum of a finite number of infinitesimals is an infinitesimal.
   Show by means of an example that this is not necessarily the case if the
   number of terms in the sum increases indefinitely.