Page 739 - Advanced_Engineering_Mathematics o'neil

21.1 Power Series 719

D

z 0

FIGURE 21.1 γ in the proof of Theorem 21.3.

Let γ be the circle of radius r about z 0 (Figure 21.1), so γ has a center at z 0 and encloses z.By
the Cauchy integral formula,
1 f (w)
f (z) = dw.
2πi γ w − z
An algebraic manipulation allows us to write
1 1 1 1
= = .
w − z w − z 0 − (z − z 0 ) w − z 0 1 − z−z 0
w−z 0
Now

z − z 0
< 1,

w − z 0
so we can write the convergent geometric series
n
1 z − z 0
∞
=
w − z w − z 0
n=0
∞
1
n
= (z − z 0 ) .
(w − z 0 ) n+1
n=0
Then
∞
f (w) f (w)
n
= (z − z 0 ) .
w − z (w − z 0 ) n+1
n=0
It can be shown that this series can be integrated term by term, so
1 f (w)
f (z) = dw
2πi γ w − z

∞

1 f (w) n
= (z − z 0 ) dw
2πi C (w − z 0 ) n+1
n=0
∞
1 f (w) n
= dw (z − z 0 )
2πi γ (w − z 0 ) n+1
n=0
∞ n
f (z 0 ) n
= (z − z 0 ) .
n!
n=0
At the last line, we used the Cauchy representation formula for higher derivatives.
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October 14, 2010 15:35 THM/NEIL Page-719 27410_21_ch21_p715-728