Complex Analysis - Maths KU

314 Chapter 6 Series and Residues
Important
☞
Theorem 6.6 Continuity
A power series � ∞
k=0 ak(z − z0) k represents a continuous function f
within its circle of convergence | z − z0| = R.
Theorem 6.7 Term-by-Term Differentiation
A power series � ∞
k=0 ak(z − z0) k can be differentiated term by term
within its circle of convergence |z − z0| = R.
Differentiating a power series term-by-term gives,
∞� d
ak(z − z0)
dz
k ∞�
=
k=0
ak
k=0
d
dz (z − z0) k ∞�
= akk(z − z0) k−1 .
Note that the summation index in the last series starts with k = 1 because
the term corresponding to k = 0 is zero.It is readily proved by the ratio test
that the original series and the differentiated series,
∞�
ak(z − z0) k
k=0
and
k=1
∞�
akk(z − z0) k−1
have the same circle of convergence | z − z0| = R.Since the derivative of
a power series is another power series, the first series �∞ k=1 ak(z − z0) k can
be differentiated as many times as we wish.In other words, it follows as a
corollary to Theorem 6.7 that a power series defines an infinitely differentiable
function within its circle of convergence and each differentiated series has the
same radius of convergence R as the original power series.
k=1
Theorem 6.8 Term-by-Term Integration
A power series � ∞
k=0 ak(z − z0) k can be integrated term-by-term within
its circle of convergence | z − z0| = R, for every contour C lying entirely
within the circle of convergence.
The theorem states that
� ∞�
ak(z − z0) k ∞�
dz =
C
k=0
�
ak
k=0
C
(z − z0) k dz
whenever C lies in the interior of | z − z0| = R.Indefinite integration can also