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20.3 Consequences of Cauchy’s Theorem 707

thereby evaluating the integral on the right in terms of the value of f (z) at a point z 0 enclosed by

γ . Keep in mind, however, that we have evaluated ( f (z)/(z − z 0 ))dz, not f (z)dz.
γ γ
EXAMPLE 20.12
We will evaluate

z 2
e
dz
γ z − i
2
z
for any closed path that does not pass through i.Here f (z) = e is differentiable for all z. There
are two cases.
2
z
Case 1 If γ does not enclose i, then e /(z − i) is differentiable on and in the region enclosed
by γ , so by Cauchy’s theorem,
z 2
e
dz = 0.
γ z − i
Case 2 If γ encloses i, then by Cauchy’s integral formula with z 0 = i,
z 2
e
dz = 2πif (i) = 2πie .
−1
γ z − i
EXAMPLE 20.13

Evaluate
e sin(z )
2z 2
dz
z − 2
γ
2
2z
over any closed path that does not pass through 2. Let f (z)=e sin(z ). Then f is differentiable
for all z. There are two cases.
Case 1 If γ does not enclose 2, then f (z)/(z − 2) is differentiable on and within γ ,so
2z 2
e sin(z )
,dz = 0.
z − 2
γ
Case 2 If γ encloses 2, then by Cauchy’s integral formula,
e sin(z )
2z 2
dz = 2πif (2) = 2πie sin(4).
4
z − 2
γ
There is a version of the integral formula for derivatives.

THEOREM 20.4 Cauchy’s Integral Formula for Higher Derivatives

With f , G, γ , and z 0 as in Cauchy’s integral formula (Theorem 20.3), then

n! f (z)
f (n) (z 0 ) = dz (20.3)
2πi γ (z − z 0 ) n+1
in which n is any nonnegative integer.

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October 14, 2010 15:32 THM/NEIL Page-707 27410_20_ch20_p695-714