142 Chapter^36. If J(z) and g(z) are infinitesimals, f(z)-:/= g(z), and if J(z),...., g(z), show
that J(z) -g(z) is of an order higher than either of them.- Prove that the order of a finite sum of infinitesimals of different orders is
that of the term of the smallest order. This term is called the principal
term or the principal part of the infinitesimal sum.
8. If F( z) and G( z) are infinitely large as z ---+ a, it is said that they are of
the same order if there are two positive constants K1 and K2 such thatK1 < IF(z)/G(z)I < K2 and, in particular, if limz->a F(z)/G(z) = L
with L -:/= 0, oo. If L = 1, we say that F(z) and G(z) are equivalent,
and write F(z) ,...., G(z). If limz->a F(z)/G(z) = 0, F(z) is said to be
of smaller order than G(z), or that G(z) is of higher order than F(z).
Prove the following.
(a) The limit as z ---+ a of an expression containing F(z) as a factor
(or divisor) does not change if F(z) is replaced by G(z) whenever
F(z) ,...., G(z).
(b) I:f F(z),...., G(z) and F(z)-:/= G(z), the difference is of smaller order
than either function.9. Let f and g be defined on a set E, and let z 0 be an accumulation point
of E. Suppose that
lim f(z) = L and lim g(z) = L'
z->zo ,zEE z->zo ,zEEwhere Land L' are finite. If h(z) = J(z)g(z), z EE, prove that
lim h(z) =LL'
z->zo,zEE3.11 Continuity of Complex Functions
Definition 3.8 Let f: D ---+ C be a single-valued complex function, and
suppose that:1. a ED [i.e., f(a) is defined] and that a is an accumulation point of D.
- limz->a f(z) exists.
- limz->a f(z) = f(a).
Then the function f is said to be continuous at a.
If f is continuous at every point of a set A C D, then f is said to be
continuous on A. A function that is continuous on C is called a continuous
function. For example, J(z) = z^2 is a continuous function.In the foregoing definition concerning continuity of f at a point a, the
domain D of definition of f could be open, closed, or neither (provided
that a i.s an accumulation point of D); see Fig. 3.9.
However, in many cases of interest D is taken to be open (Fig. 3.9a).