COMPLEX ANALYSIS: Maple Worksheets, 2001
(c) John H. Mathews Russell W. Howell
mathews@fullerton.edu howell@westmont.edu

Complimentary software to accompany the textbook:

COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9
Jones and Bartlett Publishers, Inc., 40 Tall Pine Drive, Sudbury, MA 01776
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CHAPTER 6 COMPLEX INTEGRATION

Section 6.6 The Theorems of Morera and Liouville and Some Applications

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 is independent of the path of integration. This result is stated in a form that will serve as a converse to the Cauchy-Goursat theorem.

Theorem 6.13 (Morera Theorem) Let be a continuous function in a simply connected domain D.

If for every closed contour in D, then is analytic in D.

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

Theorem 6.14 (Gauss's Mean Value Theorem)

If is analytic in a simply connected domain D that contains the circle C: , then

= .

Theorem 6.15 (Maximum Modulus Principle)

Let be analytic and nonconstant in the domain D. Then does not attain a maximum value at any point in D.

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

Theorem 6.16 (Maximum Modulus Principle)

Let be analytic and nonconstant in the bounded domain D. If is continuous on the closed region that consists of D and all of its boundary points , then assumes its maximum value, and does so only at point(s) on the boundary .

Theorem 6.17 (Cauchy's Inequalities)

Let be analytic and nonconstant in the domain D that contains the circle C: .

If holds for all points z on C, then

for .

The next result shows that a nonconstant entire function cannot be a bounded function.

Theorem 6.18, (Liouville's Theorem)

If is an entire function and is bounded for all values of in the complex plane, then is constant.

Example 6.27, Page 256. Show that the function is NOT a bounded function.

> i:='i': x:='x': y:='y': z:='z':
f := z -> sin(z):
`f(z) ` = f(z);
`f(x + iy) ` = f(x + I*y);
`f(x + iy) ` = evalc(f(x + I*y));
`|f(x + iy)| ` = evalc(abs(f(x + I*y)));
f(pi/2 + i*y) = evalc(f(Pi/2 + I*y));
lim := limit(f(Pi/2 + I*y), y=infinity):
print(`limit(f(pi/2 + I*y), y=infinity)`,` = `,lim);

We can use Liouville's theorem can be used to establish an important theorem of elementary algebra.

Theorem 6.19 (The Fundamental Theorem of Algebra)

If is a polynomial of degree n, then has at least one zero.

Corollary 6.4 Let be a polynomial of degree n. Then can be expressed as the product of linear factors. That is, where are the zeros of counted according to multiplicity an A is a constant.

End of Section 6.6.