Complex Analysis - Maths KU

320 Chapter 6 Series and Residues
y
z*
|z – 2i| = √5
|z| = 1
1
1 − z =
x
EXAMPLE 3 Taylor Series
Expand f(z) = 1
1 − z in a Taylor series with center z0 =2i.
Solution In this solution we again use the geometric series (15).By adding
and subtracting 2i in the denominator of 1/(1 − z), we can write
1
1 − z =
1
1 − z +2i − 2i =
1
1
=
1 − 2i − (z − 2i)
1
1 − 2i z − 2i
1 −
1 − 2i
1
We now write
as a power series by using (15) with the symbol z
z − 2i
1 −
1 − 2i
z − 2i
replaced by the expression
1 − 2i :
�
1 1 z − 2i
= 1+
1 − z 1 − 2i 1 − 2i +
� �2 � � �
3
z − 2i z − 2i
+
+ ···
1 − 2i 1 − 2i
or
1
1 − 2i +
1
(z − 2i)+
(1 − 2i) 2
1
(1 − 2i) 3 (z − 2i)2 +
1
(1 − 2i) 4 (z − 2i)3 + ··· . (17)
Because the distance from the center z0 =2i to the nearest singularity z =1
is √ 5, we conclude that the circle of convergence for (17) is |z − 2i| = √ 5.
This can be verified by the ratio test of the preceding section.
In (15) and (17) we represented the same function f(z) =1/(1 − z) by
two different power series.The first series (15) has center z0 = 0 and radius
of convergence R = 1.The second series (17) has center z0 =2i and radius of
convergence R = √ 5.The two different circles of convergence are illustrated
in Figure 6.5. The interior of the intersection of the two circles, shown in
color, is the region where both series converge; in other words, at a specified
point z* in this region, both series converge to same value f(z ∗ )=1/(1 − z ∗ ).
Outside the colored region at least one of the two series must diverge.
Figure 6.5 Series (15) and (17) both
converge in the shaded region. Remarks Comparison with Real Analysis
(i) As a consequence of Theorem 5.11, we know that an analytic function
f is infinitely differentiable.As a consequence of Theorem 6.9, we
know that an analytic function f can always be expanded in a power
series with a nonzero radius R of convergence.In real analysis, a
function f can be infinitely differentiable, but it may be impossible
to represent it by a power series.See Problem 51 in Exercises 6.2.