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704 CHAPTER 20 Complex Integration

y
z 1
1
x
–
2
z 0
FIGURE 20.6 Independence of path.

so

f (z)dz = f (z)dz.
γ 1 γ 2

This means that f (z)dz is independent of path on G, because the integral over any path in G
γ
depends only on the endpoints of the path. In such a case, we sometimes write
z 1

f (z)dz = f (z)dz.
γ z 0
20.3.2 The Deformation Theorem
The deformation theorem enables us, under certain conditions, to replace one closed path of
integration with another, perhaps more convenient one.

THEOREM 20.2 The Deformation Theorem

Let and γ be closed paths in the plane with γ in the interior of . Suppose f is differentiable
on an open set containing both paths and all points between them. Then

f (z)dz = f (z)dz.
γ
Figure 20.7 suggests the reason for the name of the theorem. Think of γ as made of rubber,
and continuously stretch and deform γ into . In doing this, it is important that the interme-
diate stages of the deformation only cross over points at which f is differentiable, hence the
assumption that f is differentiable at all points between the two curves.

y

x

FIGURE 20.7 The deformation theorem.

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October 14, 2010 15:32 THM/NEIL Page-704 27410_20_ch20_p695-714