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20.3 Consequences of Cauchy’s Theorem 711

One important consequence of this bound on higher derivatives is Liouville’s theorem, which
states that a bounded function that is differentiable for all z must be constant. This means that, if
f is nonconstant and differentiable for all z, then f cannot be a bounded function. We saw this
with cos(z) and sin(z), which are differentiable for all z and are not bounded functions (over the
entire complex plane).
To prove Liouville’s theorem, suppose | f (z)|≤ M for all z. By Theorem 20.7 with n =1for
any number z 0 ,
M

| f (z 0 )|≤
r
in which r is the radius of a circle about z 0 . Since r can be as large as we want, M/r can be made

arbitrarily small, so | f (z 0 )| must be zero. Then f (z 0 ) = 0. But z 0 is any number so f (z) =0for

all z, and from this it is routine to check, using Theorem 19.5, that f (z) must be constant.
Liouville’s theorem provides a simple proof of the fundamental theorem of algebra, which
states that if p(z) is a complex polynomial of degree n ≥ 1, then for some z 0 , p(z 0 ) = 0. If this
were not true, then we would have p(z) = 0 for all z. Then 1/p(z) would differentiable for all z.
But routine estimates enable us to conclude that 1/p(z) is bounded on the entire plane. By Liou-
ville’s theorem, 1/p(z) would be constant, so p(z) would be constant, which is a contradiction.
Therefore, p(z) must be zero for some complex number.

20.3.6 An Extended Deformation Theorem
The deformation theorem enables us, under certain conditions, to deform one closed path to

another γ without changing the value of f (z)dz. This requires that the deformation of one
γ
path into the other not pass over any points at which f is not differentiable. If γ is enclosed by
, this requires that f be differentiable at all points between these curves.
We will extend this result to the case that encloses any finite number of disjoint closed
paths. As usual, unless otherwise stated, all closed paths are oriented counterclockwise.

THEOREM 20.8 Extended Deformation Theorem

Let be a closed path, and let γ 1 ,··· ,γ n be closed paths enclosed by . Assume that no two
of ,γ 1 ,···γ n intersect and no point interior to any γ j is interior to any other γ k .Let f be
differentiable on an open set containing and each γ j and all points that are both interior to
and exterior to each γ j . Then

n

f (z)dz = f (z)dz.
γ j
j=1
If n = 1, this is the deformation theorem. Here is an example of the theorem in evaluating an
integral.

EXAMPLE 20.15
We will evaluate
z

dz
(z + 2)(z − 4i)
where is a closed path enclosing both −2 and 4i.
As in Figure 20.12, enclose each of −2 and 4i by closed paths γ 1 and γ 2 small enough that
they do not intersect each other or . Then

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October 14, 2010 15:32 THM/NEIL Page-711 27410_20_ch20_p695-714