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718 CHAPTER 21 Series Representations of Functions

If we apply this method and the limit of the magnitude of successive terms is zero, then the
power series has infinite radius of convergence.
A power series can be differentiated and integrated term by term within its open disk of
convergence.

THEOREM 21.2 Differentiation and Integration of Power Series

Let f be a function defined by
∞

f (z) = c n (z − z 0 ) n
n=0
for z in D :|z − z 0 | < R. Then

1.
∞

n−1
f (z) = nc n (z − z 0 ) for z in D.
n=1
Furthermore, this power series for f (z) has the same radius of convergence as the power

series for f (z).
2. If γ is a path within D, then
∞

n
f (z)dz = c n (z − z 0 ) dz.
γ n=0 γ
We now want to address the possibility of representing a function as a power series about a
point.

THEOREM 21.3 Taylor Expansion

Suppose f is differentiable on an open disk D :|z − z 0 | < R. Then, for z in D,
∞

n
f (z) = c n (z − z 0 ) ,
n=0
where for n = 0,1,2,···,
f (n) (z 0 )
c n = .
n!
Furthermore, this power series converges absolutely in D.

c n is the nth Taylor coefficient of f at z 0 , and this power series is called the Taylor series
or expansion of f about z 0 . In the case z 0 = 0, the Taylor series is also known as the
Maclaurin series.

In Theorem 21.3, R can be ∞, in which case f (z) is differentiable for all z, and the Taylor
series representation of f (z) is valid for all z.
Proof Let z be in D, and choose a number r with
|z − z 0 | <r < R.

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October 14, 2010 15:35 THM/NEIL Page-718 27410_21_ch21_p715-728