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724 CHAPTER 21 Series Representations of Functions

EXAMPLE 21.6
3
cos (z)
(z − π/2) 2
has a zero of order 1 at π/2, because the numerator has a zero of order of 3 there, and the
denominator has a zero of order 2.

SECTION 21.1 PROBLEMS

In each of Problems 1 through 6, find the radius of con- (a) First, compute the Taylor coefficients at 0.
vergence and open disk of convergence of the power (b) Find the first seven terms of the product of the
series. Maclaurin series for sin(z) with itself.
2
(c) Write sin (z) in terms of the exponential func-
∞ n + 1
1. (z + 3i) n tion and use the Maclaurin expansion of this
n=0 2 n function.
2
(d) Write sin (z) = (1 − cos(2z))/2, and use the
1
2. (−1) n (z − i) 2n Maclaurin expansion of cos(z).
∞
n=0 (2n + 1) 2
n n 17. Show that
∞
3. (z − 1 + 3i) n
n=0 (n + 1) n ∞
1 1 2π

n e 2z cos(θ) dθ.
2i 2
=
4. (z + 3 − 4i) n n=0 (n!) 2π 0
∞
n=0 5 + i
∞ i n Hint: Show that
5. (z + 8i) n
n=0 2 n+1 z n 2 1 z n
zw
= e dw
∞ (1 − i) n n! 2πi γ n!w n+1
6. (z − 3) n
n=0 n + 2
for n = 0,1,2,··· and γ is the unit circle about the

n
7. Is it possible for ∞ c n (z − 2i) to converge at 0 and
n=0 origin.
diverge i?
n
∞
8. Is it possible for c n (z − 4 + 2i) to converge at i
n=0 In each of Problems 18 through 24, determine the order of
and diverge at 1 + i?
the zero of the function.
In each of Problems 9 through 14, find the Taylor expan-
3
sion of the function about the point. 18. f (z) = z cos(z); z = 0
2
2
19. f (z) = z sin (z); z = 0
2
9. cos(2z); z = 0 20. f (z) = (z − π/2) cos(z); z = π/2
3
−z
10. e ; z =−3i 21. f (z) = cos (z); z = 3π/2
4
2
11. z − 3z + i; z = 2 − i 22. f (z) = cos (z − π/2), z = 0
2
4
z
12. e − i sin(z); z = 0 23. f (z) = sin(z )/z , z = 0
2
13. (z − 9) ;1 + i 24. f (z) = (z−π) 5 , z = π
sin 2 (z)
14. sin(z + i);−i 25. Suppose
15. Suppose f is differentiable in an open disk about 0
∞ ∞
and satisfies f (z) = 2 f (z) + 1. Suppose f (0) = 1

n
f (z) = a n (z − z 0 ) = b n (z − z 0 ) n
and f (0)=i. Find the first six terms of the Maclaurin

n=0 n=0
expansion of f (z).
16. Find the first seven terms of the Maclaurin expansion in some open disk D about z 0 . Show that a n = b n for
2
of f (z) = sin (z) in four ways, as follows. n = 0,1,2,···.
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October 14, 2010 15:35 THM/NEIL Page-724 27410_21_ch21_p715-728