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 8 RESIDUE THEORY

Section 8.8 The Argument Principle and Rouche's Theorem

We will now derive two results based on Cauchy's residue theorem. They have important practical applications and pertain only to functions all of whose isolated singularities are poles.

Theorem 8.8 (Argument Principle) Let be meromorphic in the simply connected domain .
Let be a simple closed positively oriented contour in along which and .

Then ,

where is the number of zeros of that lie inside and is the number of poles that lie inside .

Theorem 8.9 (Winding Numbers)

Suppose is meromorphic in the simply connected domain . If is a simple closed, positively oriented contour in such that for , and , then

=

known as the winding number of about , counts the number of times the curve winds around the point . If , the integral counts the number of times the curve winds around the origin.

Theorem 8.10 (Rouche's theorem)

Suppose and are meromorphic functions defined in the simply connected domain , that is a simply closed contour in , and that and have no zeros or poles for . If the strict inequality holds for all , then .

Example 8.26, Page 350. Show that all four zeros of the polynomial
lie in the disk .

> g:='g': z:='z':
g := z -> z^4 - 7*z - 1:
`g(z) ` = g(z);

> Zn := sort([fsolve(g(z), z, complex)]):
`The complex roots of g(z) ` = g(z);
z1 := subs(z=Zn[1],z): z[1] = z1;
z2 := subs(z=Zn[2],z): z[2] = z2;
z3 := subs(z=Zn[3],z): z[3] = z3;
z4 := subs(z=Zn[4],z): z[4] = z4;

> print(abs(z[1]),`< 2 `, abs(z1)<2, evalb(evalf(abs(z1))<2));
print(abs(z[2]),`< 2 `, abs(z2)<2, evalb(evalf(abs(z2))<2));
print(abs(z[3]),`< 2 `, abs(z3)<2, evalb(evalf(abs(z3))<2));
print(abs(z[4]),`< 2 `, abs(z4)<2, evalb(evalf(abs(z4))<2));

Here is the set-up for Rouche's theorem. Use the function :

> f:='f': z:='z':
f := z ->z^4:
`f(z) ` = f(z);

We must verify that the inequality holds
for all points on the circle : .

> A:='A': B:='B': s:='s': t:='t': y1:='y1': y1:='y1':
A := z -> abs(f(z)-g(z)): `|f(z)-g(z)| ` = A(z);
B := z -> abs(f(z)): `|f(z)| ` = B(z);
y1 := t -> A(2*exp(I*t)): `|f(z(t))-g(z(t))| ` = y1(t);
y2 := t -> B(2*exp(I*t)): `|f(z(t))| ` = y2(t);

> plot({y1(t),y2(t)}, t=0..2*Pi,
title=`|f(z(t))-g(z(t))| < |f(z(t))| on C`,
labels=[`t`,`s`],
tickmarks=[7,6],
view=[0..6.3,12..17]);

We see that the inequality holds on the circle .
Hence and have the same number of zeros that lie inside the circle .
Since has zeros in this circle, so does .

End of Section 8.8.