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21.1 Power Series 721

EXAMPLE 21.4
We will use algebra and the geometric series to write the Taylor expansion of 2i/(4 + iz)
about −3i.
n
Since this expansion about −3i will contain powers (z + 3i) , we attempt to rearrange
2i/(4 + iz) so that we can expand it in a geometric series involving powers of z + 3i. Write
2i 2i
=
4 + iz 4 + i(z + 3i) + 3
2i 2i 1
= =
7 + i(z + 3i) 7 1 + i(z+3i)
7
∞
2i i n
= (−1) n (z + 3i)
7 7
n=0
∞ n+1 n+1
2(−1) i

n
= (z + 3i) .
7 n+1
n=0
This expansion is valid for
i

(z + 3i) < 1

7
or
|z + 3i| < 7.
This expansion has center −3i and a radius of convergence of 7.
We could have predicted the radius of convergence of this power series expansion without
actually writing the series. The function being expanded is f (z) = 2i/(4 + iz), which is differ-
entiable for all z except z = 4i. The radius of convergence of the expansion of f (z) about −3i
will be the distance between the center, −3i, and the closest point to −3i at which f (z) is not
differentiable, in this case 4i. This distance is 7, which we have just seen from the expansion
itself is the radius of convergence.
As a less obvious example, consider g(z) = 1/sin(z). This is differentiable for all z except
integer multiples of π. We can (in theory) expand g(z) in a power series about 3 + i. The radius
of convergence of this series will be the distance between 3 + i and the point nearest 3 + i at
which g(z) is not differentiable. This point is π, so the radius of convergence is the distance
√
2
between 3 + i and π (or (3 − π) + 1).
We will conclude this section with some consequences of power series expansions which
display important properties of complex functions.
Existence of an Antiderivative

If f is differentiable on an open disk D about z 0 , we claim that there must exist a dif-

ferentiable function F such that F (z) = f (z) for all z in D. F is an antiderivative
of f .

To construct F(z), expand f (z) in a power series about z 0 on D as
∞
n
f (z) = c n (z − z 0 )
n=0

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October 14, 2010 15:35 THM/NEIL Page-721 27410_21_ch21_p715-728