**for**

**6.6 The Theorems of Morera and Liouville
and Extensions**

In this section we investigate some of the qualitative properties of analytic and harmonic functions. Our first result shows that the existence of an antiderivative for a continuous function is equivalent to the statement that the integral of f(z) is independent of the path of integration. This result stated in a form that will serve as a converse to the Cauchy-Goursat theorem.

**Theorem 6.13 (**__Morera's
Theorem__**).** Let
f(z) be a continuous function in a
simply connected domain D. If for
every closed contour in D, then
f(z) is analytic in D.

**Proof of Theorem 6.13 is in the book.
**

Cauchy's integral formula show how the value can be represented by a certain contour integral. If we choose the contour of integration C to be a circle with center , then we can show that the value is the integral average of the values of f(z) at points z on the circle C.

**Theorem 6.14 (Gauss's Mean Value
Theorem).** If f(z)
is analytic in a simply connected domain D that
contains the circle , then

.

**Proof of Theorem 6.14 is in the book.
**

We now prove an important result concerning the modulus of an analytic function.

**Theorem 6.15 (Maximum Modulus
Principle).** Let f(z)
be analytic and nonconstant in the bounded domain D. Then does
not attain a maximum value at any point
in D.

**Proof of Theorem 6.15 is in the book.
**

We sometimes state the maximum modulus principle in the following form.

**Theorem 6.16 (**__Maximum
Modulus
Principle__**).** Let
f(z) be analytic and nonconstant in
the bounded domain D. If f(z)
is continuous on the closed region R
that consists of D and all of its
boundary points B, then
assumes its maximum value, and does so only at point(s)
on the boundary B.

**Proof of Theorem 6.16 is in the book.
**

**Example
6.26.** Let . If
we set our domain D to
be , then
f(z) is continuous on the closed
region . Prove
that

,

and this value is assumed by f(z) at
a point on
the boundary of D.

Solution. From the triangle inequality and the fact
that
in D, it follows that

(6-58) .

If we choose , where , then

so the vectors
and
lie on the same ray through the origin. This is the
requirement for the Inequality (6-58) to
be an equality (see Exercise 19 in Section
1.3).

Hence , and
the result is established.

**Extra Example
1.** Let . If
we set our domain D to
be , then
f(z) is continuous on the closed
region . Show
that , and
this value is assumed by
at a point on
the boundary of D.

**Explore
Solution for Extra Example 1.**

**Extra Example
2.** Let . If
we set our domain D to
be , then
f(z) is continuous on the closed
region . Show
that , and
this value is assumed by
at a point on
the boundary of D.

**Explore
Solution for Extra Example 2.**

**Theorem 6.17 (Cauchy's
Inequalities).** Let f(z)
be analytic in the simply connected domain D
that contains the circle . If holds
for all points , then

for .

**Proof of Theorem 6.17 is in the book.
**

Theorem 6.18 shows that a nonconstant entire function cannot be a bounded function.

**Theorem 6.18 (**__Liouville's
Theorem__**).** If
f(z) is an entire function and is
bounded for all values of z in the
complex plane, then f(z) is
constant.

**Proof of Theorem 6.18 is in the book.
**

**Example 6.27.** Show
that the function sin(z) is
not a bounded function.

Solution. We established this characteristic with a somewhat tedious argument in Section 5.4. All we need do now is observe that f(z) is not constant, and hence it is not bounded.

We can use __Liouville's__
__theorem__
to establish an important theorem of algebra.

**Theorem 6.19 (**__Fundamental
Theorem of
Algebra__**).** If
P(z) is a polynomial of degree
,
then P(z) has at least one zero.

**Proof of Theorem 6.19 is in the book.
**

**Corollary 6.4.** Let
P(z) be a polynomial of degree
. Then
P(z) can be expressed as the product
of linear factors. That is,

where are
the zeros of P(z) counted according
to multiplicity an A is a
constant.

**Extra Example
3.** Find the n
zeros of the equation .

**Explore
Solution for Extra Example 3.**

**Extra Example
4.** Find the n
zeros of the equation .

**Explore
Solution for Extra Example 4.**

**Extra Example
5.** Find the roots of the
Chebyshev polynomial.

**Explore
Solution for Extra Example 5.**

**Exercises
for Section 6.6. The Theorems of Morera and Liouville and
Extensions**** **

__Fundamental
Theorem of Algebra__

**The Next Module
is**

**The
Fundamental Theorem of Algebra**

**Return to the Complex
Analysis Modules**

__Return
to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell