Functions. Limits and Continuity. Arcs and Curves 139
Notation We write
lim f(z) = oo
z-+ooor f(z) -too as z -too
Example For f(z) = ez, D = {z: Rez > O}, we have limz-+oo f(z) = oo.
In fact, for any given J( > 0 we have
lf(z)I = lezl =ex > K
whenever x > lnK. Hence it suffices to take M > max (O,lnK), 8 =
1/(1 + M^2 )^112 , and z E N6(oo) n D.
It is worth noting that for f ( z) = ez defined on D 1 = { z: Re z = 0},
i.e., for f(iy) = eiY, there is no limit as z = iy -too, and for f(z) = ez
defined in D2 = {z: Rez < O} we have limz-+oo f(z) = 0.
3.10 Infinitely Small and Infinitely Large Functions
Definition 3.5 If limz-+a f(z) = 0 (a finite or oo) it is said that the
function f(z) is an infinitesimal as z -ta, or that f(z) is infinitely small as
z -ta. Clearly, limz-+a g(z) = L (L finite) iff the function f(z) = g(z) -L
is an infinitesimal as z -t a.
Examples
- f ( z) = z^2 is an infinitesimal as z -t 0.
- F(z) = (z -2)(z^2 + 1) is an infinitesimal z -t 2.
- h(z) = 1/z is an infinitesimal as z -t oo.
Definition 3.6 Two infinitesimals f(z) and g(z) as z -ta are said to be
of the same order if limz-+a f(z)/ g(z) = L, where L /= 0 and L /= oo. More
generally, those two infinitesimals are of the same order as z -t a if there
are two positive constants K 1 and K2 such that K1 < lf(z)/g(z)I < K2.'
If limz-+a f(z)/g(z) = 0 then f(z) is called an infinitesimal of higher
order than g(z) (as z -ta). On the other hand, if limz-+a f(z)/g(z) = oo,
then f(z) is said to be of lower order than g(z) (as z -ta). Obviously, if
f(z) is of higher order than g(z), then g(z) is of lower order than f(z).If limz-+a f(z)/g(z) = 1, the infinitesimals f(z) and g(z) are said to be
equivalent, and we write f(z) "" g(z) as z -t a.
Examples- The infinitesimals f(z) = 2z^2 and g(z) = z^2 + z^4 as z -t 0 are of the
same order since limz-+O f(z)/g(z) = 2.
- The infinitesimals f(z) = z^3 +2z^5 and g(z) = z^3 as z -t 0 are equivalent
since limz-+O f (z) / g(z) = 1.
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