1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

286 CHAPTER. 7 • TAYLOR. AND LAURENT SERIES

Theorem 7.13 also allows us to give a simple argument for one version of
L'Hopital's rule.

t Corollary 7.11 (L'Hopital's rule) Suppose that f and g are analytic at a. If

f (a)= 0 and g(a) = 0, but g' (a) =f 0, then

lim f(z) = f'(a).

·~"' g (z) g' (a)

Proof Because g' (a) =f 0, g is not identically zero and, by Theorem 7.13,

there is a punctured disk D; (a) in which g (z) =f O. Thus, the quotient £AA =

~f:l=$f;? is defined for all z ED; (a), and we can write

lim I (z) = lim I (z) - I (a) = lim [f (z) - f (a)]/ (z - a) = f' (a).

z-<> g (z) z-<> g (z) -g (a) •-<> [g (z) -g (a)] / (z - a) g' (a)

•

We can use Theorem 7.14 to get Taylor series for quotients of analytic func-

tions. Its proof involves ideas from Section 7.2, and we leave it as an exercise.

•EXAMPLE 7 .1 5 Find the first few terms of the Maclaurin series for the

function f (z) = secz if lzl <~'and compute f <^4 ) (O).