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20.3 Consequences of Cauchy’s Theorem 703

theorem and the Cauchy-Riemann equations. If D is the region containing the path and all points
enclosed by γ , then

f (z)dz = udx − v dy + i v dx + udy
γ γ γ
∂(−v) ∂u ∂u ∂v

= − dA + i − dA = 0,
D ∂x ∂y D ∂x ∂y
because by the Cauchy-Riemann equations,
∂u ∂v ∂u ∂v
= and =− .
∂x ∂y ∂y ∂x
Cauchy’s theorem has several important consequences, which are the object of the next
section.

SECTION 20.2 PROBLEMS

In each of Problems 1 through 12, evaluate the integral 5. f (z) = z;γ is the unit circle about the origin.
of the function over the closed path. All curves are ori-
6. f (z)=1/z;γ is the circle of radius 5 about the origin.
ented counterclockwise. In some cases, Cauchy’s theorem
z
applies, while in others it does not, but may still be useful 7. f (z) = ze ;γ is the circle |z − 3i|= 8.
2
(as in Example 20.10). 8. f (z) = z − 4z + i;γ is the rectangle with vertices
1,8,8 + 4i and 1 + 4i.
2
1. f (z) = sin(3z);γ is the circle |z|= 4. 9. f (z)=|z| ;γ is the circle of radius 7 about the origin.
2. f (z) = 2z/(z − i);γ is the circle |z − i|= 3. 10. f (z) = sin(1/z);γ is the circle |z − 1 + 2i|= 1.
3
3. f (z) = 1/(z − 2i) ;γ is given by |z − 2i|= 2. 11. f (z) = Re(z);γ is given by |z|= 2.
2
2
4. f (z) = z sin(z);γ is the square having vertices 0,1,i 12. f (z) = z + Im(z);γ is the square with vertices
and 1 + i. 0,−2i,2and 2 − 2i.

20.3 Consequences of Cauchy’s Theorem

This section develops some consequences of Cauchy’s theorem.

20.3.1 Independence of Path

Independence of path was mentioned briefly in connection with evaluating f (z)dz in terms
γ
of an antiderivative F of f . Independence of path can also be viewed from the perspective of
Cauchy’s theorem.
Suppose f is differentiable on a simply connected domain G, and z 0 and z 1 are points of G.
Let γ 1 and γ 2 be paths in G from z 0 to z 1 (Figure 20.6).

If we reverse the orientation on γ 2 , we form a closed path = γ 1 (−γ 2 ). By Cauchy’s

theorem, f (z)dz = 0, so

f (z)dz + f (z)dz = 0.
γ 1 −γ 2
But

f (z)dz =− f (z)dz,
−γ 2 γ 2

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October 14, 2010 15:32 THM/NEIL Page-703 27410_20_ch20_p695-714