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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Unit 7: Complex Variables
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Chapt
er 36: Applications" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 46 "Section 36.5:  the Nyquist stability criterion" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "C
opyright" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 48 "Copyright * 2001 by Addison Wesley Longman, Inc." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All righ
ts reserved.  No part of this publication may be reproduced, stored in
 a retrieval system, or transmitted, in any form or by any means, elec
tronic, mechanical, photocopying, recording, or otherwise, without the
 prior written permission of the publisher. Printed in the United Stat
es of America." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Initializations
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "with(lin
alg):\nwith(plots):\nwith(plottools):\nwith(inttrans):\nwith(student):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(showassumed=0
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 
"" 0 "" {TEXT -1 18 "The Winding Number" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The circle " }{XPPEDIT 18 0 "
z=exp(i*theta),0<=theta" "6$/%\"zG-%$expG6#*&%\"iG\"\"\"%&thetaGF*1\"
\"!F+" }{XPPEDIT 18 0 "``<=2*Pi" "6#1%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT 
-1 80 ", encircles, or wraps around the origin, once.  The closed cont
our described by " }{XPPEDIT 18 0 "z=exp(i*theta),0<=theta" "6$/%\"zG-
%$expG6#*&%\"iG\"\"\"%&thetaGF*1\"\"!F+" }{XPPEDIT 18 0 "``<=4*Pi" "6#
1%!G*&\"\"%\"\"\"%#PiGF'" }{TEXT -1 102 ", is a circle traced twice, a
nd therefore encircles or wraps around the origin twice.  In general, \+
if " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 42 " is a closed contour, \+
the number of times " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 24 " wrap
s around the point " }{XPPEDIT 18 0 "z=zeta" "6#/%\"zG%%zetaG" }{TEXT 
-1 15 " is called the " }{TEXT 260 7 "winding" }{TEXT -1 1 " " }{TEXT 
261 6 "number" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 
-1 17 " with respect to " }{XPPEDIT 18 0 "zeta" "6#%%zetaG" }{TEXT -1 
17 ", and is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "nu(C,zeta)=1/2/Pi/i" "6#/-%#nuG6$%
\"CG%%zetaG**\"\"\"F*\"\"#!\"\"%#PiGF,%\"iGF," }{XPPEDIT 18 0 "Int(1/(
z-zeta),z=C..``)" "6#-%$IntG6$*&\"\"\"F',&%\"zGF'%%zetaG!\"\"F+/F);%\"
CG%!G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "
This integral reflects such a geometric property of the contour " }
{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 139 " because the antiderivative
 is a branch of the logarithm.  The integral actually measures the cha
nge in the argument of the logarithm.  If " }{XPPEDIT 18 0 "C" "6#%\"C
G" }{TEXT -1 14 " wraps around " }{XPPEDIT 18 0 "zeta" "6#%%zetaG" }
{TEXT -1 67 " once, the argument of the logarithm will increase(or dec
rease) by " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 
30 ".  Hence, the number of times " }{XPPEDIT 18 0 "C" "6#%\"CG" }
{TEXT -1 14 " wraps around " }{XPPEDIT 18 0 "zeta" "6#%%zetaG" }{TEXT 
-1 125 " is reflected by the change in the argument of the logarithm g
enerated by the antiderivative for the winding-number integral." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Example 36.24" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " }
{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 15 " be the circle " }{XPPEDIT 
18 0 "z(t)=exp(i*t),0<=t" "6$/-%\"zG6#%\"tG-%$expG6#*&%\"iG\"\"\"F'F-1
\"\"!F'" }{XPPEDIT 18 0 "``<=2*Pi" "6#1%!G*&\"\"#\"\"\"%#PiGF'" }
{TEXT -1 10 ", that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Z := exp(I*t);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "and t
ake " }{XPPEDIT 18 0 "zeta=0" "6#/%%zetaG\"\"!" }{TEXT -1 22 ".  The w
inding number " }{XPPEDIT 18 0 "nu(C,0)" "6#-%#nuG6$%\"CG\"\"!" }
{TEXT -1 8 " is then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "nu(C,0)=1/2/Pi/i" "6#/-%#nuG6$%\"C
G\"\"!**\"\"\"F*\"\"#!\"\"%#PiGF,%\"iGF," }{TEXT -1 1 " " }{XPPEDIT 
18 0 "Int(i,t=0..2*Pi)=1" "6#/-%$IntG6$%\"iG/%\"tG;\"\"!*&\"\"#\"\"\"%
#PiGF.F." }}{PARA 0 "" 0 "" {TEXT -1 6 "since " }{XPPEDIT 18 0 "C" "6#
%\"CG" }{TEXT -1 10 " encloses " }{XPPEDIT 18 0 "z=0" "6#/%\"zG\"\"!" 
}{TEXT -1 6 " once." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 58 "To implement this calculation in Maple, write the integ
ral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "q := (1/2/Pi/I)*Int(diff(Z,t)/Z,t=0..2*Pi);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 25 "and obtain its value with" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(q);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 13 "As expected, " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 10 " encl
oses " }{XPPEDIT 18 0 "z=0" "6#/%\"zG\"\"!" }{TEXT -1 6 " once." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "This woul
d be the result for any other " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 
-1 8 " inside " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 35 ".  For exam
ple, the winding number " }{XPPEDIT 18 0 "nu(C,(1+i)/2)" "6#-%#nuG6$%
\"CG*&,&\"\"\"F)%\"iGF)F)\"\"#!\"\"" }{TEXT -1 25 " is given by the in
tegral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "nu(C,(1+i)/2)=1/2/Pi/i" "6#/-%#nuG6$%\"CG*&,&\"
\"\"F*%\"iGF*F*\"\"#!\"\"**F*F*F,F-%#PiGF-F+F-" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(i*exp(i*t)/(exp(i*t)-(1+i)/2),t=0..2*Pi)=1" "6#/-%$
IntG6$*(%\"iG\"\"\"-%$expG6#*&F(F)%\"tGF)F),&-F+6#*&F(F)F.F)F)*&,&F)F)
F(F)F)\"\"#!\"\"F6F6/F.;\"\"!*&F5F)%#PiGF)F)" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 53 "which we can see if we write the integral in Ma
ple as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "q1 := (1/2/Pi/I)*Int(diff(Z,t)/(Z-(1+I)/2),t=0..2*Pi)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 44 "Maple laborously evaluates this integral to " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "value(q1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "The integral defining the winding
 number is easily computed by Cauchy's residue theorem, and we have" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "nu(C,zeta)=``(1/2/Pi/i)*(2*Pi*i)*Res(1/(z-zeta),z=zeta)
" "6#/-%#nuG6$%\"CG%%zetaG*(-%!G6#**\"\"\"F.\"\"#!\"\"%#PiGF0%\"iGF0F.
*(F/F.F1F.F2F.F.-%$ResG6$*&F.F.,&%\"zGF.F(F0F0/F9F(F." }{TEXT -1 3 " =
 " }{XPPEDIT 18 0 "Res(1/(z-zeta),z=zeta)" "6#-%$ResG6$*&\"\"\"F',&%\"
zGF'%%zetaG!\"\"F+/F)F*" }{TEXT -1 4 " = 1" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "whenever " }{XPPEDIT 18 0 "C" "6
#%\"CG" }{TEXT -1 36 " is a simple closed curve enclosing " }{XPPEDIT 
18 0 "zeta" "6#%%zetaG" }{TEXT -1 11 " just once." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 13 "Example 36.25" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "C" 
"6#%\"CG" }{TEXT -1 45 " be the limacon defined in polar coordinates \+
" }{XPPEDIT 18 0 "``(r,t)" "6#-%!G6$%\"rG%\"tG" }{TEXT -1 4 " by " }
{XPPEDIT 18 0 "r=1+3*cos(t)" "6#/%\"rG,&\"\"\"F&*&\"\"$F&-%$cosG6#%\"t
GF&F&" }{TEXT -1 10 ", that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "R := 1+3*cos(t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 98 "The distinguishing feature of the limacon is the way it loops o
ver itself, as seen in Figure 36.14" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "display(plot([R,t,t=0..2*
Pi],coords=polar, color=black, xtickmarks=5, ytickmarks=5, numpoints=5
00), labels=[x,y], labelfont=[TIMES,ITALIC,12], scaling=constrained);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 9 "We write " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 4 "
 as " }{XPPEDIT 18 0 "z(t)=r(t)*exp(i*t)" "6#/-%\"zG6#%\"tG*&-%\"rG6#F
'\"\"\"-%$expG6#*&%\"iGF,F'F,F," }{TEXT -1 13 ", that is, as" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Z
 := R*exp(I*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "and compute the winding number abo
ut " }{XPPEDIT 18 0 "z=1" "6#/%\"zG\"\"\"" }{TEXT -1 37 ", a point ins
ide the \"inner\" loop, as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "nu(C,1)=1/2/Pi/i" "6#/-%#nu
G6$%\"CG\"\"\"**F(F(\"\"#!\"\"%#PiGF+%\"iGF+" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "int(i*(1+3*exp(i*t))*exp(i*t)/((1+3*cos(t))*exp(i*t)-1)
,t=0..2*Pi)=2" "6#/-%$intG6$**%\"iG\"\"\",&F)F)*&\"\"$F)-%$expG6#*&F(F
)%\"tGF)F)F)F)-F.6#*&F(F)F1F)F),&*&,&F)F)*&F,F)-%$cosG6#F1F)F)F)-F.6#*
&F(F)F1F)F)F)F)!\"\"F?/F1;\"\"!*&\"\"#F)%#PiGF)FD" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 47 "indicating that the limacon wraps twice a
round " }{XPPEDIT 18 0 "z=1" "6#/%\"zG\"\"\"" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "In Maple, this \+
winding number " }{XPPEDIT 18 0 "nu(C,1)" "6#-%#nuG6$%\"CG\"\"\"" }
{TEXT -1 5 " for " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 25 " is give
n by the integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 45 "q := Int(diff(Z,t)/(Z-1),t=0..2*Pi)/(2*Pi*I)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 39 "the value of which Maple computes to be" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value
(q);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 25 "The winding number about " }{XPPEDIT 18 0 "zeta=3
" "6#/%%zetaG\"\"$" }{TEXT -1 63 ", a point inside the limacon, but no
t inside the inner loop, is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "nu(C,3)=1/2/Pi/i" "6#/-%#nu
G6$%\"CG\"\"$**\"\"\"F*\"\"#!\"\"%#PiGF,%\"iGF," }{TEXT -1 1 " " }
{XPPEDIT 18 0 "int(i*(1+3*exp(i*t))*exp(i*t)/((1+3*cos(t))*exp(i*t)-3)
,t=0..2*Pi)=1" "6#/-%$intG6$**%\"iG\"\"\",&F)F)*&\"\"$F)-%$expG6#*&F(F
)%\"tGF)F)F)F)-F.6#*&F(F)F1F)F),&*&,&F)F)*&F,F)-%$cosG6#F1F)F)F)-F.6#*
&F(F)F1F)F)F)F,!\"\"F?/F1;\"\"!*&\"\"#F)%#PiGF)F)" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "indicatin
g that the limacon wraps around " }{XPPEDIT 18 0 "z=1" "6#/%\"zG\"\"\"
" }{TEXT -1 19 " but a single time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 54 "In Maple, this winding number is given b
y the integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 46 "q1 := Int(diff(Z,t)/(Z-3),t=0..2*Pi)/(2*Pi*I);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 18 "which evaluates to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "value(q1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 38 "As expected, the winding number is 1, " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 13 "Example 36.26" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "C" 
"6#%\"CG" }{TEXT -1 83 " be closed contour consisting of four semicirc
les taken from the following circles." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Z1 := 1+3*exp(I*t);\nZ2
 := 2+2*exp(I*t);\nZ3 := 1+exp(I*t);\nZ4 := 2*exp(I*t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 12 "The contour " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 30 " is th
en seen in Figure 36.15." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "p1 := complexplot(Z1,t=-Pi..0, col
or=black):\np2 := complexplot(Z2,t=0..Pi, color=black):\np3 := complex
plot(Z3,t=-Pi..0, color=black):\np4 := complexplot(Z4,t=0..Pi, color=b
lack):\ndisplay([p||(1..4)],scaling=constrained, labels=[`     x`,`y  \+
 `], labelfont=[TIMES,ITALIC,12], xtickmarks=7, ytickmarks=6);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 7 "Inside " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 79 " there
 are points which are enclosed twice by the contour.  The winding numb
er " }{XPPEDIT 18 0 "nu(C,1)" "6#-%#nuG6$%\"CG\"\"\"" }{TEXT -1 45 " i
s computed as the sum of the four integrals" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "q := (Int(diff(Z
1,t)/(Z1-1),t=-Pi..0)+\nInt(diff(Z2,t)/(Z2-1),t=0..Pi)+\nInt(diff(Z3,t
)/(Z3-1),t=-Pi..0)+\nInt(diff(Z4,t)/(Z4-1),t=0..Pi))/(2*Pi*I);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 16 "and evaluates to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(q);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "As \+
anticipated, the winding number with respect to " }{XPPEDIT 18 0 "z=1
" "6#/%\"zG\"\"\"" }{TEXT -1 35 " is 2, indicating that the contour " 
}{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 14 " wraps around " }{XPPEDIT 
18 0 "z=1" "6#/%\"zG\"\"\"" }{TEXT -1 7 " twice." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The point " }{XPPEDIT 18 
0 "z=3" "6#/%\"zG\"\"$" }{TEXT -1 83 " is inside the contour, but not \+
inside the \"inner\" loop.  Thus, the winding number " }{XPPEDIT 18 0 
"nu(C,3)" "6#-%#nuG6$%\"CG\"\"$" }{TEXT -1 42 " is given by the sum of
 the four integrals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 146 "q1 := (Int(diff(Z1,t)/(Z1-3),t=-Pi..0)+
\nInt(diff(Z2,t)/(Z2-3),t=0..Pi)+\nInt(diff(Z3,t)/(Z3-3),t=-Pi..0)+\nI
nt(diff(Z4,t)/(Z4-3),t=0..Pi))/(2*Pi*I);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "whose value i
s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "value(q1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "As expected, the winding n
umber with respect to " }{XPPEDIT 18 0 "z=3" "6#/%\"zG\"\"$" }{TEXT 
-1 35 " is 1, indicating that the contour " }{XPPEDIT 18 0 "C" "6#%\"C
G" }{TEXT -1 14 " wraps around " }{XPPEDIT 18 0 "z=3" "6#/%\"zG\"\"$" 
}{TEXT -1 10 " but once." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 29 "The Principle of the Argument" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "1.  " }{XPPEDIT 18 0 "D" 
"6#%\"DG" }{TEXT -1 30 " is a simply connected domain;" }}{PARA 0 "" 
0 "" {TEXT -1 4 "2.  " }{XPPEDIT 18 0 "G(z)" "6#-%\"GG6#%\"zG" }{TEXT 
-1 16 " is analytic in " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT -1 47 ", \+
except, perhaps for a finite number of poles;" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "3.  " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 51 " is a sim
ple closed positively-oriented contour in " }{XPPEDIT 18 0 "D" "6#%\"D
G" }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 4 "4.  " }{XPPEDIT 18 0 
"G(z)" "6#-%\"GG6#%\"zG" }{TEXT -1 32 " has neither poles nor zeros on
 " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 6 ";\n5.  " }{XPPEDIT 18 0 "
G(z)" "6#-%\"GG6#%\"zG" }{TEXT -1 5 " has " }{XPPEDIT 18 0 "N" "6#%\"N
G" }{TEXT -1 11 " zeros and " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 
14 " poles inside " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 1 ";" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "==" }
{XPPEDIT 18 0 "`>`" "6#%\">G" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "1/2/Pi/i" "6#**\"\"\"F$\"\"#!\"\"%#PiGF&%\"iGF&" }
{XPPEDIT 18 0 "Int(`G'`(z)/G(z),z=C..``)=N-P" "6#/-%$IntG6$*&-%#G'G6#%
\"zG\"\"\"-%\"GG6#F+!\"\"/F+;%\"CG%!G,&%\"NGF,%\"PGF0" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "A pr
oof of this theorem can be found in [1], or in any similar text.  The \+
following example contains the key ideas from which the proof is typic
ally constructed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Exa
mple 36.27" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 30 "Let the following three points" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "P[1] := 2+3*I;\nP
[2] := 4-I;\nP[3] := -5+2*I;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "be a zero, and two simpl
e poles, respectively, for the function" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "G := (z-P[1])/(z-P[2])
/(z-P[3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "`G'`/G" "6#*&%#G'G
\"\"\"%\"GG!\"\"" }{TEXT -1 8 " will be" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "GG := simplify(diff(G,
z)/G);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 12 "Notice that " }{XPPEDIT 18 0 "`G'`/G" "6#*&%#G'
G\"\"\"%\"GG!\"\"" }{TEXT -1 59 " now has simple poles at all three po
ints which lie within " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 13 ", t
he circle " }{XPPEDIT 18 0 "abs(z)=7" "6#/-%$absG6#%\"zG\"\"(" }{TEXT 
-1 22 ".  Hence, parametrize " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 
3 " by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "Z := 7*exp(I*t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "and write the
 integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 42 "q := Int(subs(z=Z,GG),t=0..2*Pi)/(2*Pi*I);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 48 "Unfortunately, Maple evaluates this incorrectly," }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "simplify(value(q));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "even if the real and imagina
ry parts of the integrand are separated, as in" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "simplify(in
t(evalc(integrand(q)),t=0..2*Pi)/(2*Pi*I));" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "  However,
 the integral can be evaluated by the Cauchy residue theorem, and we h
ave" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 258 "" 0 "" {XPPEDIT 18 
0 "1/2/Pi/i" "6#**\"\"\"F$\"\"#!\"\"%#PiGF&%\"iGF&" }{XPPEDIT 18 0 "In
t(`G'`(z)/G(z),z=C..``)=1/2/Pi/i" "6#/-%$IntG6$*&-%#G'G6#%\"zG\"\"\"-%
\"GG6#F+!\"\"/F+;%\"CG%!G**F,F,\"\"#F0%#PiGF0%\"iGF0" }{XPPEDIT 18 0 "
``(2*Pi*i)" "6#-%!G6#*(\"\"#\"\"\"%#PiGF(%\"iGF(" }{XPPEDIT 18 0 "Sum(
Res(`G'`/G,P[k]),k=1..3)=Sum(Res(`G'`/G,P[k]),k=1..3)" "6#/-%$SumG6$-%
$ResG6$*&%#G'G\"\"\"%\"GG!\"\"&%\"PG6#%\"kG/F2;F,\"\"$-F%6$-F(6$*&F+F,
F-F.&F06#F2/F2;F,F5" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "N-P" "6#,&%\"NG
\"\"\"%\"PG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 40 "that is, we have the individual residues" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "for k fr
om 1 to 3 do\nresidue(GG,z=P[k]);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "and for their
 sum" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "add(residue(GG,z=P[k]),k=1..3) = N-P;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 61 "thereby verifying the principle of the argument in this case." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Principle of the Ar
gument - Extended" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 102 "The following theorem combines the constructs of \+
the winding number and the principle of the argument." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "1.  the positively-ori
ented, simple closed contour " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 
17 " is described by " }{XPPEDIT 18 0 "z=f(t),alpha<=t" "6$/%\"zG-%\"f
G6#%\"tG1%&alphaGF(" }{XPPEDIT 18 0 "``<=beta" "6#1%!G%%betaG" }{TEXT 
-1 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 4 "2.  " }{XPPEDIT 18 0 "H(z)" "6#
-%\"HG6#%\"zG" }{TEXT -1 16 " is analytic on " }{XPPEDIT 18 0 "C" "6#%
\"CG" }{TEXT -1 9 "; inside " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 
2 ", " }{XPPEDIT 18 0 "H(z)" "6#-%\"HG6#%\"zG" }{TEXT -1 35 " may have
 a finite number of poles;" }}{PARA 0 "" 0 "" {TEXT -1 4 "3.  " }
{XPPEDIT 18 0 "H(z)<>zeta[0]" "6#0-%\"HG6#%\"zG&%%zetaG6#\"\"!" }
{TEXT -1 4 " on " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 1 ";" }}
{PARA 0 "" 0 "" {TEXT -1 4 "4.  " }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" 
}{TEXT -1 17 " is the image of " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 
-1 7 " under " }{XPPEDIT 18 0 "H" "6#%\"HG" }{TEXT -1 11 "; that is, \+
" }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 13 " is given by " }
{XPPEDIT 18 0 "zeta=F(t)" "6#/%%zetaG-%\"FG6#%\"tG" }{TEXT -1 3 " = " 
}{XPPEDIT 18 0 "H(f(t)),alpha<=t" "6$-%\"HG6#-%\"fG6#%\"tG1%&alphaGF)
" }{XPPEDIT 18 0 "``<=beta" "6#1%!G%%betaG" }{TEXT -1 1 ";" }}{PARA 0 
"" 0 "" {TEXT -1 11 "5.  inside " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 
-1 15 ", the function " }{XPPEDIT 18 0 "H(z)-zeta[0]" "6#,&-%\"HG6#%\"
zG\"\"\"&%%zetaG6#\"\"!!\"\"" }{TEXT -1 5 " has " }{XPPEDIT 18 0 "N" "
6#%\"NG" }{TEXT -1 11 " zeros and " }{XPPEDIT 18 0 "P" "6#%\"PG" }
{TEXT -1 55 " poles, counted with respect to multiplicity and order;" 
}}{PARA 0 "" 0 "" {TEXT -1 4 "6.  " }{XPPEDIT 18 0 "nu(Gamma,zeta[0])
" "6#-%#nuG6$%&GammaG&%%zetaG6#\"\"!" }{TEXT -1 26 " is the winding nu
mber of " }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 17 " about the
 point " }{XPPEDIT 18 0 "zeta[0]" "6#&%%zetaG6#\"\"!" }{TEXT -1 1 ";" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "==" }
{XPPEDIT 18 0 "`>`" "6#%\">G" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "nu(Gamma,zeta[0])=N-P" "6#/-%#nuG6$%&GammaG&%%zetaG6#\"
\"!,&%\"NG\"\"\"%\"PG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 35 "In essence, this theorem says that " }{XPPEDIT 
18 0 "N-P" "6#,&%\"NG\"\"\"%\"PG!\"\"" }{TEXT -1 18 " for the function
 " }{XPPEDIT 18 0 "H(z)" "6#-%\"HG6#%\"zG" }{TEXT -1 8 " inside " }
{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 27 " is the winding number for \+
" }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 15 ", the image of " }
{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 30 " under the mapping induced b
y " }{XPPEDIT 18 0 "H(z)" "6#-%\"HG6#%\"zG" }{TEXT -1 61 ".  To find o
ut something about the number of poles and zeros " }{XPPEDIT 18 0 "H(z
)" "6#-%\"HG6#%\"zG" }{TEXT -1 12 " has inside " }{XPPEDIT 18 0 "C" "6
#%\"CG" }{TEXT -1 32 ", obtain the winding number for " }{XPPEDIT 18 
0 "Gamma" "6#%&GammaG" }{TEXT -1 15 ", the image of " }{XPPEDIT 18 0 "
C" "6#%\"CG" }{TEXT -1 19 " under the mapping " }{XPPEDIT 18 0 "w=H(z)
" "6#/%\"wG-%\"HG6#%\"zG" }{TEXT -1 28 ".  If the winding number of " 
}{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 135 " can be obtained ge
ometrically, without resort to integration, then the theorem gives a g
eometric tool for counting poles and zeros of " }{XPPEDIT 18 0 "H(z)" 
"6#-%\"HG6#%\"zG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "H(z)" "6#-%\"HG6#%\"z
G" }{TEXT -1 99 " is a polynomial, it has no poles in the finite compl
ex plane.  Furthermore, if the winding number " }{XPPEDIT 18 0 "nu(Gam
ma,zeta[0])" "6#-%#nuG6$%&GammaG&%%zetaG6#\"\"!" }{TEXT -1 73 " were k
nown, then this extension of the principle of the argument yields " }
{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 22 ", the number of zeros " }
{XPPEDIT 18 0 "H(z)-zeta[0]" "6#,&-%\"HG6#%\"zG\"\"\"&%%zetaG6#\"\"!!
\"\"" }{TEXT -1 12 " has inside " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
106 "The proof of this theorem is straightforward.  Starting with the \+
definition of the winding number, we have" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "nu(Gamma,zeta[
0])=1/2/Pi/i" "6#/-%#nuG6$%&GammaG&%%zetaG6#\"\"!**\"\"\"F-\"\"#!\"\"%
#PiGF/%\"iGF/" }{XPPEDIT 18 0 "Int(1/(zeta-zeta[0]),zeta=Gamma..``)=1/
2/Pi/i" "6#/-%$IntG6$*&\"\"\"F(,&%%zetaGF(&F*6#\"\"!!\"\"F./F*;%&Gamma
G%!G**F(F(\"\"#F.%#PiGF.%\"iGF." }{XPPEDIT 18 0 "Int(`F'`(t)/(F(t)-zet
a[0]),t=alpha..beta)=1/2/Pi/i" "6#/-%$IntG6$*&-%#F'G6#%\"tG\"\"\",&-%
\"FG6#F+F,&%%zetaG6#\"\"!!\"\"F5/F+;%&alphaG%%betaG**F,F,\"\"#F5%#PiGF
5%\"iGF5" }{XPPEDIT 18 0 "Int(`H'`(f(t))*`f '`(t)/(H(f(t))-zeta[0]),t=
alpha..beta)=1/2/Pi/i" "6#/-%$IntG6$*(-%#H'G6#-%\"fG6#%\"tG\"\"\"-%$f~
'G6#F.F/,&-%\"HG6#-F,6#F.F/&%%zetaG6#\"\"!!\"\"F=/F.;%&alphaG%%betaG**
F/F/\"\"#F=%#PiGF=%\"iGF=" }{XPPEDIT 18 0 "Int(`H'`(z)/(H(z)-zeta[0]),
z=C..``)" "6#-%$IntG6$*&-%#H'G6#%\"zG\"\"\",&-%\"HG6#F*F+&%%zetaG6#\"
\"!!\"\"F4/F*;%\"CG%!G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 31 "Now, the last integral is just " }{XPPEDIT 18 0 "N-P
" "6#,&%\"NG\"\"\"%\"PG!\"\"" }{TEXT -1 36 ", a result which follows b
y setting " }{XPPEDIT 18 0 "G(z)=H(z)-zeta[0]" "6#/-%\"GG6#%\"zG,&-%\"
HG6#F'\"\"\"&%%zetaG6#\"\"!!\"\"" }{TEXT -1 34 " in the principle of t
he argument." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Example
 36.28" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 4 "Let " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 59 " be the c
ontour consisting of the right half of the circle " }{XPPEDIT 18 0 "ab
s(z)=R" "6#/-%$absG6#%\"zG%\"RG" }{TEXT -1 48 " and that part of the i
maginary axis satisfying " }{XPPEDIT 18 0 "-i*R<=z" "6#1,$*&%\"iG\"\"
\"%\"RGF'!\"\"%\"zG" }{XPPEDIT 18 0 "``<=i*R" "6#1%!G*&%\"iG\"\"\"%\"R
GF'" }{TEXT -1 59 ".  This is the letter \"D\" shown on the left in Fi
gure 36.16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 794 "R := 'R': C := 'C': G := 'G':\np1 := plot([[0,-5],
[0,5]],color=cyan,thickness=3):\np2 := plot([5,t,t=-Pi/2..Pi/2],coords
=polar,color=cyan,thickness=3):\np3 := plot([r,Pi/4,r=0..5],coords=pol
ar,color=black):\np4 := textplot([2.5,1.8,`R`]):\np5 := plot([[0,0],[5
.5,0]],color=black):\np6 := plot([[7,-5],[7,5]],color=cyan,thickness=3
):\np7 := plot([7+5*cos(t),5*sin(t),t=-Pi/2..Pi/2], color=cyan,thickne
ss=3):\np8 := plot([[7,0],[7+5/sqrt(2),5/sqrt(2)]],color=black):\np9 :
= plot([[9,-5],[9,5]],color=black):\np10 := plot([[7,0],[12.5,0]],colo
r=black):\np11 := textplot([10,2.2,`R`]):\np12 := textplot([3.8,4.5,`C
`],font=[TIMES,ROMAN,12]):\np13 := textplot(\{[-.3,-.1,`0`],[6.6,-.1,`
-3`],[9.4,-.5,`0`]\}):\np14 := textplot([11.24, 4.646,`G`],font=[SYMBO
L,12]):\ndisplay([p||(1..14)],scaling=constrained,axes=none);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 3 "If " }{XPPEDIT 18 0 "H(z)=z-3" "6#/-%\"HG6#%\"zG,&F'\"\"\"
\"\"$!\"\"" }{TEXT -1 10 ", that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "H := z-3;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 13 "the equation " }{XPPEDIT 18 0 "H(z)=0" "6#/-%\"HG6#%\"zG\"\"!" 
}{TEXT -1 19 " has one solution, " }{XPPEDIT 18 0 "z=3" "6#/%\"zG\"\"$
" }{TEXT -1 34 ", in the right half-plane.  Thus, " }{XPPEDIT 18 0 "H(
z)" "6#-%\"HG6#%\"zG" }{TEXT -1 52 " has one zero, and no poles in the
 right half-plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 13 "The image of " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 15 
" under the map " }{XPPEDIT 18 0 "H(z)" "6#-%\"HG6#%\"zG" }{TEXT -1 
16 " is the contour " }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 
95 ", shown on the right in Figure 36.16, and traced dynamically in th
e following animation.  Both " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 53 " are trace
d in the counterclockwise (positive) sense." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 409 "H1 := subs(z=5*e
xp(I*t),H):\nH2 := subs(z=-I*t,H):\np5 := s -> complexplot(H1,t=-Pi/2.
.s,color=black,thickness=3):\np6 := s -> complexplot(H2,t=-5..s,color=
black, thickness=3):\np7 := complexplot(H1,t=-Pi/2..Pi/2,color=black, \+
thickness=3):\np8 := s -> display([p6(s),p7]):\ndisplay([seq(p5(k*Pi/2
0),k=-10..10),seq(p8(k/2),k=-10..10)], insequence=true, scaling=constr
ained, xtickmarks=[-2,0,1], ytickmarks=[-5,0,5]);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The \+
contour " }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 66 " encircles
 the origin once, so the winding number with respect to " }{XPPEDIT 
18 0 "zeta=0" "6#/%%zetaG\"\"!" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "nu(
Gamma,0)=1" "6#/-%#nuG6$%&GammaG\"\"!\"\"\"" }{TEXT -1 10 ".  Hence, \+
" }{XPPEDIT 18 0 "H(z)" "6#-%\"HG6#%\"zG" }{TEXT -1 46 " has exactly o
ne zero in the right half-plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 27 "We verify this by computing" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "nu(Gamma,0)=1/2/Pi/i" "6#/-%#nuG6$%&GammaG\"\"!**\"\"\"F*\"\"#!\"\"
%#PiGF,%\"iGF," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(`H'`(z)/(H(z)-zeta
[0]),z=C..``)" "6#-%$IntG6$*&-%#H'G6#%\"zG\"\"\",&-%\"HG6#F*F+&%%zetaG
6#\"\"!!\"\"F4/F*;%\"CG%!G" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/2/Pi/i
" "6#**\"\"\"F$\"\"#!\"\"%#PiGF&%\"iGF&" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "Int(`H'`(z)/H(z),z=C..``)=1/2/Pi/i" "6#/-%$IntG6$*&-%#H'G6#%\"zG\"
\"\"-%\"HG6#F+!\"\"/F+;%\"CG%!G**F,F,\"\"#F0%#PiGF0%\"iGF0" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "Int(1/(z-3),z=C..``)=1" "6#/-%$IntG6$*&\"\"\"F(,
&%\"zGF(\"\"$!\"\"F,/F*;%\"CG%!GF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "where, since " }{XPPEDIT 
18 0 "zeta[0]=0" "6#/&%%zetaG6#\"\"!F'" }{TEXT -1 24 ", the integrand \+
becomes " }{XPPEDIT 18 0 "`H'`(z)/(H(z)-zeta[0])=`H'`(z)/H(z)" "6#/*&-
%#H'G6#%\"zG\"\"\",&-%\"HG6#F(F)&%%zetaG6#\"\"!!\"\"F2*&-F&6#F(F)-F,6#
F(F2" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 82 "The rightmost contour integral can be evaluated in M
aple as follows.  The contour " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 
-1 37 " is parametrized by the two functions" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Z1 := R*exp(I*t);
\nZ2 := I*t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 47 "The contour integral is, except for the f
actor " }{XPPEDIT 18 0 "1/2/Pi/i" "6#**\"\"\"F$\"\"#!\"\"%#PiGF&%\"iGF
&" }{TEXT -1 30 ", the sum of the two integrals" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "q1 := Int(d
iff(Z1,t)/(Z1-3),t=-Pi/2..Pi/2);\nq2 := Int(diff(Z2,t)/(Z2-3),t=R..-R)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 16 "For the contour " }{XPPEDIT 18 0 "C" "6#%\"CG" }
{TEXT -1 12 " to enclose " }{XPPEDIT 18 0 "z=3" "6#/%\"zG\"\"$" }
{TEXT -1 14 ", the zero of " }{XPPEDIT 18 0 "H(z)" "6#-%\"HG6#%\"zG" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "R" "6#%\"RG" }{TEXT -1 46 " must be la
rger than 3, so make the assumption" }}{PARA 0 "" 0 "" {TEXT -1 1 " " 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(R>3);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 33 "and find the winding number to be" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "simplify(value( map(
evalc,q1) + q2)/(2*Pi*I));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Example 36.29" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " }
{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 50 " be the contour defined in E
xample 36.28, and let " }{XPPEDIT 18 0 "H(z)" "6#-%\"HG6#%\"zG" }
{TEXT -1 18 " be the polynomial" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "H := (z-2)*(z-3);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 9 "Clearly, " }{XPPEDIT 18 0 "H(z)" "6#-%\"HG6#%\"zG" }{TEXT 
-1 16 " has two zeros, " }{XPPEDIT 18 0 "z=2" "6#/%\"zG\"\"#" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "z=3" "6#/%\"zG\"\"$" }{TEXT -1 46 ", both
 in the right half-plane.  The image of " }{XPPEDIT 18 0 "C" "6#%\"CG
" }{TEXT -1 30 " under the mapping induced by " }{XPPEDIT 18 0 "H(z)" 
"6#-%\"HG6#%\"zG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "Gamma" "6#%&Gamma
G" }{TEXT -1 24 ", shown in Figure 36.17." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 327 "H1 := subs(z=5*exp(
I*t),H):\nH2 := subs(z=-I*t,H):\np5 := s -> complexplot(H1,t=-Pi/2..s,
color=black,thickness=1):\np6 := s -> complexplot(H2,t=-5..s,color=bla
ck, thickness=1):\np7:=textplot([-8.519, 23.07,`G`],font=[SYMBOL,12]):
\ndisplay([p5(Pi/2),p6(5),p7], scaling=constrained, xtickmarks=[-15,0,
20], ytickmarks=[-15,-5,0,5,15]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The contour \+
" }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 50 " is traced dynamic
ally in the following animation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 428 "H1 := subs(z=5*exp(I*t),H):
\nH2 := subs(z=-I*t,H):\np5 := s -> complexplot(H1,t=-Pi/2..s,color=bl
ack,thickness=3):\np6 := s -> complexplot(H2,t=-5..s,color=black, thic
kness=3):\np7 := complexplot(H1,t=-Pi/2..Pi/2,color=black, thickness=3
):\np8 := s -> display([p6(s),p7]):\ndisplay([seq(p5(k*Pi/20),k=-10..1
0),seq(p8(k/2),k=-10..10)], insequence=true, scaling=constrained, xtic
kmarks=[-20,-10,0,5], ytickmarks=[-15,-10,-5,0,5,10,15]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 12 "The contour " }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 11 
" encircles " }{XPPEDIT 18 0 "zeta[0]=0" "6#/&%%zetaG6#\"\"!F'" }
{TEXT -1 37 " twice, so the the winding number is " }{XPPEDIT 18 0 "nu
(Gamma,0)=2" "6#/-%#nuG6$%&GammaG\"\"!\"\"#" }{TEXT -1 19 ", and the f
unction " }{XPPEDIT 18 0 "H(z)-zeta[0]=H(z)" "6#/,&-%\"HG6#%\"zG\"\"\"
&%%zetaG6#\"\"!!\"\"-F&6#F(" }{TEXT -1 46 " has two zeros in the right
 half-plane.  Since" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 
"" {TEXT -1 1 " " }{XPPEDIT 18 0 "`H'`(z)/H(z)=(2*z-5)/(z-2)/(z-3)" "6
#/*&-%#H'G6#%\"zG\"\"\"-%\"HG6#F(!\"\"*(,&*&\"\"#F)F(F)F)\"\"&F-F),&F(
F)F1F-F-,&F(F)\"\"$F-F-" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(z-2)+1/(
z-3)" "6#,&*&\"\"\"F%,&%\"zGF%\"\"#!\"\"F)F%*&F%F%,&F'F%\"\"$F)F)F%" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "the winding number is" }
}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "nu(Gamma,0)=2*Pi*i/2
/Pi/i" "6#/-%#nuG6$%&GammaG\"\"!*.\"\"#\"\"\"%#PiGF+%\"iGF+F*!\"\"F,F.
F-F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "[Res(`H'`/H,2)+Res(`H'`/H,3)]=2
" "6#/7#,&-%$ResG6$*&%#H'G\"\"\"%\"HG!\"\"\"\"#F+-F'6$*&F*F+F,F-\"\"$F
+F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 75 "Working in Maple, we can obtain a direct evaluation of \+
the contour integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "nu(Gamma,0)=1/2/Pi/i" "6#/-%#nuG6$
%&GammaG\"\"!**\"\"\"F*\"\"#!\"\"%#PiGF,%\"iGF," }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(`H'`(z)/H(z),z=C..``)" "6#-%$IntG6$*&-%#H'G6#%\"zG
\"\"\"-%\"HG6#F*!\"\"/F*;%\"CG%!G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 13 "if we define " }{XPPEDIT 18 0 "Q(z)=`H'`(z)/H(z)" "6#/-%
\"QG6#%\"zG*&-%#H'G6#F'\"\"\"-%\"HG6#F'!\"\"" }{TEXT -1 3 " by" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "Q := diff(H,z)/H;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and compute the winding numbe
r from the integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 93 "q := Int((subs(z=R*exp(I*t),Q)*I*R*exp(I*
t)),t=-Pi/2..Pi/2) + Int((subs(z=I*t,Q)*I),t=R..-R);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "T
he factor " }{XPPEDIT 18 0 "1/2/Pi/i" "6#**\"\"\"F$\"\"#!\"\"%#PiGF&%
\"iGF&" }{TEXT -1 42 " has been held in abeyance so that we can " }
{TEXT 256 3 "map" }{TEXT -1 4 " an " }{TEXT 257 5 "evalc" }{TEXT -1 
66 " onto the integrands before evaluating the integrals.  We then get
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "simplify(value(map(evalc,op(1,q))+map(evalc,op(2,q)))
/(2*Pi*I));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 26 "for the winding number of " }{XPPEDIT 18 
0 "Gamma" "6#%&GammaG" }{TEXT -1 17 " with respect to " }{XPPEDIT 18 
0 "zeta[0]=0" "6#/&%%zetaG6#\"\"!F'" }{TEXT -1 57 ".  This is consiste
nt with the animation of the trace of " }{XPPEDIT 18 0 "Gamma" "6#%&Ga
mmaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 
"Example 36.30" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 5 "With " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 32 " agai
n as in Example 36.28, let " }{XPPEDIT 18 0 "H(z)" "6#-%\"HG6#%\"zG" }
{TEXT -1 18 " be the polynomial" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "H := z^3/10+z^2+9*z+4;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "a function with the three zeros" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fsolve(H=0,z,comp
lex);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 92 "are all in the left half-plane.  There are no z
eros in the right half-plane.  Consequently, " }{XPPEDIT 18 0 "Gamma" 
"6#%&GammaG" }{TEXT -1 15 ", the image of " }{XPPEDIT 18 0 "C" "6#%\"C
G" }{TEXT -1 30 " under the mapping induced by " }{XPPEDIT 18 0 "H(z)
" "6#-%\"HG6#%\"zG" }{TEXT -1 21 " should not encircle " }{XPPEDIT 18 
0 "zeta[0]=0" "6#/&%%zetaG6#\"\"!F'" }{TEXT -1 22 ".  Figure 36.18 sho
ws " }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 6 " when " }
{XPPEDIT 18 0 "R=12" "6#/%\"RG\"#7" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 295 "H1 := subs
(z=12*exp(I*t),H):\nH2 := subs(z=-I*t,H):\np5 := complexplot(H1,t=-Pi/
2..Pi/2,color=black):\np6 := complexplot(H2,t=-12..12,color=red,linest
yle=2):\np7 := textplot([150,200,`G`],font=[SYMBOL,12]):\ndisplay([p5,
p6,p7], scaling=constrained, xtickmarks=[-150,0,100,200,300,400], ytic
kmarks=6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 217 "Careful observation shows the large loop
 (solid black curve) is traced in the counterclockwise sense, whereas \+
the small loop (dotted red curve) is traced in the clockwise sense.  T
hus, there is no net encirclement of " }{XPPEDIT 18 0 "zeta[0] = 0;" "
6#/&%%zetaG6#\"\"!F'" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "Gamma" "6#%&G
ammaG" }{TEXT -1 55 ", and the winding number is correctly determined \+
to be " }{XPPEDIT 18 0 "nu(Gamma,0)=0" "6#/-%#nuG6$%&GammaG\"\"!F(" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "For " }{XPPEDIT 18 0 "R=5" "6#/%\"RG\"\"&" }{TEXT -1 44 ",
 the following animation of the tracing of " }{XPPEDIT 18 0 "Gamma" "6
#%&GammaG" }{TEXT -1 20 " clearly shows that " }{XPPEDIT 18 0 "Gamma" 
"6#%&GammaG" }{TEXT -1 30 " does not encircle the origin." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 404 "H1 \+
:= subs(z=5*exp(I*t),H):\nH2 := subs(z=-I*t,H):\np5 := s -> complexplo
t(H1,t=-Pi/2..s,color=black,thickness=3):\np6 := s -> complexplot(H2,t
=-5..s,color=black, thickness=3):\np7 := complexplot(H1,t=-Pi/2..Pi/2,
color=black, thickness=3):\np8 := s -> display([p6(s),p7]):\ndisplay([
seq(p5(k*Pi/20),k=-10..10),seq(p8(k/2),k=-10..10)], insequence=true, s
caling=constrained, xtickmarks=6, ytickmarks=[-40,0,40]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 51 "The following animation shows a dynamic tracing of " }{XPPEDIT 
18 0 "Gamma" "6#%&GammaG" }{TEXT -1 13 " in the case " }{XPPEDIT 18 0 
"R=12" "6#/%\"RG\"#7" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 378 "H1 := subs(z=12*exp(I*t)
,H):\nH2 := subs(z=-I*t,H):\np5 := s -> complexplot(H1,t=-Pi/2..s,colo
r=black):\np6 := s -> complexplot(H2,t=-12..s,color=red):\np7 := compl
explot(H1,t=-Pi/2..Pi/2,color=black):\np8 := s -> display([p6(s),p7]):
\ndisplay([seq(p5(k*Pi/20),k=-10..10),seq(p8(k),k=-12..12)], insequenc
e=true, scaling=constrained, xtickmarks=[-150,0,100,200,300,400], ytic
kmarks=6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 47 "Careful observation shows the black porti
on of " }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 120 " is traced \+
in the counterclockwise sense, whereas, the red portion is traced in t
he clockwise sense.  Thus, there is no " }{TEXT 258 3 "net" }{TEXT -1 
17 " encirclement of " }{XPPEDIT 18 0 "zeta[0]=0" "6#/&%%zetaG6#\"\"!F
'" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 
55 ", and the winding number is correctly determined to be " }
{XPPEDIT 18 0 "nu(Gamma,0)=0" "6#/-%#nuG6$%&GammaG\"\"!F(" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "The Nyquis
t Stability Criterion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 276 "In 1932, Harry Nyquist (1889-1976), an enginee
r at Bell Telephone Laboratories, articulated the use of the extended \+
principle of the argument for determining where the poles of a transfe
r function in a linear feedback system are located.  This usage has si
nce been called the " }{TEXT 259 27 "Nyquist stability criterion" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 622 "The transfer function of a feedback system (see Section \+
36.4) arises from a Laplace transform of the system.  The poles of the
 transfer function are actually the eigenvalues or characteristic root
s of the differential equation governing the system.  If any of these \+
poles (or eigenvalues) have postive real part, that is, if any pole is
 located in the right half-plane, the solution may contain exponential
 terms which grow without bound as time increases.  Thus, determining \+
whether or not there are poles of the transfer function in the right h
alf-plane is tantamount to determining the stability of the feedback s
ystem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
102 "If the transfer function of the feedback system is a completely r
educed rational function of the form " }{XPPEDIT 18 0 "T(z)=A(z)/B(z)
" "6#/-%\"TG6#%\"zG*&-%\"AG6#F'\"\"\"-%\"BG6#F'!\"\"" }{TEXT -1 15 ", \+
the poles of " }{XPPEDIT 18 0 "T(z)" "6#-%\"TG6#%\"zG" }{TEXT -1 18 " \+
are the zeros of " }{XPPEDIT 18 0 "B(z)" "6#-%\"BG6#%\"zG" }{TEXT -1 
44 ".  Determining the location of the zeros of " }{XPPEDIT 18 0 "B(z)
" "6#-%\"BG6#%\"zG" }{TEXT -1 43 " is equivalent to determining the po
les of " }{XPPEDIT 18 0 "T(z)" "6#-%\"TG6#%\"zG" }{TEXT -1 19 ", and t
he poles of " }{XPPEDIT 18 0 "T(z)" "6#-%\"TG6#%\"zG" }{TEXT -1 387 " \+
are the eigenvalues, or characteristic roots, of the system.  The solu
tion of the differential equation underlying the feedback system will \+
contain multiples of exponential terms whose exponents contain the eig
envalues.  If any of the eigenvalues have positive real parts (because
 they fall in the right half-plane), the solution might grow unbounded
 as time increases.  Thus, poles of " }{XPPEDIT 18 0 "T(z)" "6#-%\"TG6
#%\"zG" }{TEXT -1 77 " falling in the right half-plane means the feedb
ack system could be unstable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 280 "Engineering texts on feedback control (f
or example, [2] and [3]), contain additional detail on the use of the \+
Nyquist stability criterion.  In addition, we even find complex variab
les texts such as [4] discussing the notion of feedback systems and th
e Nyquist stability criterion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 92 "[1] Louis L. Pennisi, Elements of Complex Variable
s, Holt, Rinehart and Winston, Inc., 1963." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "[2] Charles L. Phillips and Ro
yce D. Harbor, Feedback Control Systems, Alternate Second Edition, Pre
ntice Hall, Inc., 1991." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 139 "[3] Gene F. Franklin, J. David Powell and Abbas Eme
mi-Naeini, Feedback Control of Dynamic Systems, Addison-Wesley Publish
ing Company, 1986." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 114 "[4] A. David Wunsch, Complex Variables with Applications
, Second Edition, Addison-Wesley Publishing Company, 1994." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }