266 CHAPTER 7 • TAYLOR AND LAURENT SERIES
( c) Use the partial sum involving terms up to z^9 to find approximations
to C (1.0) and S (LO).- Let f be defined in a domain that cont ains the origin. T he function f is said to
be even if f (-z) = f (z), and it is called odd if f (- z) = -f (z).
(a} Show that the derivative of an odd function is an even function.
(b) Show that the derivative of an even funct ion is an odd function. Hint:
Use limits.
(c) If f (z) is even, show that all the coefficients of the odd powers of z in
the Maclaurin series are zero.
(d) U f (z) is odd, show that all the coefficients of the even powers of z in
the Maclaurin series are zero.- Verify Identity (7-18) by using mathematical induction.
19. Consider the funct ion
f (z) = { 1 ~, when z # t•
0 whenz= 2.(a) Use Theorem 7.4, Taylor's theorem, to show that the Maclaurin series
00
for f (z) equals L: z".
~=O
(b) Obviously, the radius of convergence of this series equals 1 (ratio test).
However, the distance between 0 and the nearest singularity of f equals!. Explain why this condit ion doesn' t contradict Corollary 7.3.
20. Consider the real-valued function f defined on the real numbers asf (x) = { e-.;.r when x # 0,
0 when x = 0.(a) Show that , for all n > 0, f(n) (0) = 0, where f(n) is the nth derivative
of f. Hint: Use the limit definition for t he derivative to est ablish t he
case for n = 1 and then use mathematical induction to complete your
argument.
(b) Explain why the function f gives an example of a function that, a.J.
though differentiable everywhere on t he real line, is not expressible as
a Taylor series about 0. Hint: Evaluate the Taylor series representation
for f (x) when x # 0, and show that the series does not equal f (x).
( c) Explain why a similar argument could not be made for t he complex-
valued function g defined on t he complex numbers asg(z) = { e-1' when z # 0,
0 when z = 0.Hint: Show that g (z) is not even continuous at z = 0 by taking limits
along the real and imaginary axes.