Complex Analysis - Maths KU

316 Chapter 6 Series and Residues
D
z0 z
R
s
Figure 6.4 Contour for the proof of
Theorem 6.9
C
This series is called the Taylor series for f centered at z0.A Taylor series
with center z0 =0,
f(z) =
∞�
k=0
f (k) (0)
k!
z k , (7)
is referred to as a Maclaurin series.
We have just seen that a power series with a nonzero radius R of convergence
represents an analytic function.On the other hand we ask:
Question
Ifwe are given a function fthat is analytic in some domain D, can we
represent it by a power series ofthe form (6) or (7)?
Since a power series converges in a circular domain, and a domain D is generally
not circular, the question comes down to: Can we expand f in one or
more power series that are valid—that is, a power series that converges at
z and the number to which the series converges is f(z)—in circular domains
that are all contained in D? The question is answered in the affirmative by
the next theorem.
Theorem 6.9 Taylor’s Theorem
Let f be analytic within a domain D and let z0 beapointinD.Then f
has the series representation
f(z) =
∞�
k=0
f (k) (z0)
(z − z0)
k!
k
valid for the largest circle C with center at z0 and radius R that lies
entirely within D.
Proof Let z be a fixed point within the circle C and let s denote the variable
of integration.The circle C is then described by |s − z0| = R. See Figure 6.4.
To begin, we use the Cauchy integral formula to obtain the value of f at z:
f(z) = 1
�
2πi C
= 1
�
2πi C
�
f(s) 1
ds =
s − z 2πi C
f(s)
(s − z0)
⎧
⎪⎨
f(s)
(s − z0) − (z − z0) ds
(8)
⎫
⎪⎬
1
⎪⎩
z −
ds. (9)
z0
1 −
⎪⎭
s − z0