{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 24 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 " " 1 18 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Helvetica" 1 14 128 0 0 1 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "War ning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 38 "Complex Contours and Co ntour Integrals" }}{PARA 257 "" 0 "" {TEXT 259 19 "Mth 611 Spring 2002 " }}{PARA 0 "" 0 "" {TEXT 260 21 "May 31 2002 - Maple 6" }}{PARA 0 "" 0 "" {TEXT 257 20 "Revised June 1, 2002" }}{PARA 0 "" 0 "" {TEXT 262 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 258 39 "Filename: 611s20 02_complex_contours.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 7 " " 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Here's a little routine to plot curves in the complex plane:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "cplot:=(z,R)->plot([Re(z),Im(z),R],args[3..nargs]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 299 "Here z \+ is an expression (not a function) in some real variable, say t, and \+ R is a range of the form t=a..b Note you can pass any number of va riables to a Maple routine, so we make a provision for extra variables by picking them up from the args list and then passing them to the pl ot command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "One can also simply use Maple's built-in command " }{TEXT 261 13 "complexplot()" }{TEXT -1 249 ". It is very flexible and faster tha n cplot(). However, Maple has many built-in commands. Sometimes it is \+ difficult and time consuming to find the one you want. Therefore it is useful to have a little experience at devising rough homebrew solutio ns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Le t's test our routine by plotting a circle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cplot(exp(I*t),t= 0..2*Pi,color=blue,thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6#7S7$$\"\"\"\"\"!$F*F*7$$\"+uBO1**!# 5$\"+#=y_O\"F/7$$\"+*4-Qn*F/$\"+f*)GLDF/7$$\"+zke[#*F/$\"+C.6.QF/7$$\" +>42`')F/$\"+R4U7]F/7$$\"+L?D/zF/$\"+qceDhF/7$$\"+pTd#3(F/$\"++FafqF/7 $$\"+ZdX;hF/$\"+\\!>8\"zF/7$$\"+a:x5]F/$\"+Sh-a')F/7$$\"+Z'R,#QF/$\"+5 bcT#*F/7$$\"+ziQDDF/$\"+M\"oen*F/7$$\"+?o8X8F/$\"+kt64**F/7$$!+'pr5(f! #8$\"+<#)******F/7$$!+eB[i8F/$\"+)GZn!**F/7$$!+\\@=YEF/$\"+]D`V'*F/7$$ !+I*3Jx$F/$\"+Ol'3E*F/7$$!+r_JU]F/$\"+v]oN')F/7$$!+s%*[RgF/$\"+U+BqzF/ 7$$!+WjJ*3(F/$\"+j?x_qF/7$$!+1U?/zF/$\"+%QZc7'F/7$$!+QnIf')F/$\"+8/k,] F/7$$!+si$4B*F/$\"+AHvXQF/7$$!+4Nij'*F/$\"+GQ%=d#F/7$$!+>+.2**F/$\"+4] Ug8F/7$$!+4>&*****F/$\"+='485$!#77$$!+>!Q%3**F/$!+^=8]8F/7$$!+oa1u'*F/ $!+FIGKDF/7$$!+;AEj#*F/$!+HNAnPF/7$$!+Lh=u')F/$!+')3zv\\F/7$$!+vx1YzF/ $!+lUCrgF/7$$!+Sfc5rF/$!+5yMJqF/7$$!+Pke[gF/$!+$eGL'zF/7$$!+@%)o!*\\F/ $!+5Mil')F/7$$!+%[\"ysPF/$!+R***4E*F/7$$!+$el'3EF/$!+&p[Pl*F/7$$!+!ym: H\"F/$!+#*>C;**F/7$$!+@OQdIF`s$!+iK&*****F/7$$\"+]Ks(G\"F/$!+Y=u;**F/7 $$\"+O8/bDF/$!+K'z!o'*F/7$$\"+#Q)zNQF/$!+%>x]B*F/7$$\"+r0I.]F/$!+9wMe' )F/7$$\"+#R]$4hF/$!+)H2o\"zF/7$$\"+&Q+$*4(F/$!+i?sUqF/7$$\"+)Qy,!zF/$! +g&Q38'F/7$$\"+8vLu')F/$!+yp_v\\F/7$$\"+\\4+D#*F/$!+)eo*fQF/7$$\"+o))o a'*F/$!+XQ=0EF/7$$\"+Nb&p!**F/$!+8t'4O\"F/7$F($\"+J_8/#)!#>-%+AXESLABE LSG6$Q!6\"F[[l-%'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%*THICKNESSG6#\"\"$ -%%VIEWG6$%(DEFAULTGF[\\l" 1 2 0 1 10 3 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "Here's the same plot as produced by comp lexplot()." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "complexplot(exp(I*t),t=0..2*Pi,color=blue,thickness =3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVE SG6#7S7$$\"\"\"\"\"!$F*F*7$$\"+uBO1**!#5$\"+#=y_O\"F/7$$\"+*4-Qn*F/$\" +f*)GLDF/7$$\"+zke[#*F/$\"+C.6.QF/7$$\"+>42`')F/$\"+R4U7]F/7$$\"+L?D/z F/$\"+qceDhF/7$$\"+pTd#3(F/$\"++FafqF/7$$\"+ZdX;hF/$\"+\\!>8\"zF/7$$\" +a:x5]F/$\"+Sh-a')F/7$$\"+Z'R,#QF/$\"+5bcT#*F/7$$\"+ziQDDF/$\"+M\"oen* F/7$$\"+?o8X8F/$\"+kt64**F/7$$!+'pr5(f!#8$\"+<#)******F/7$$!+eB[i8F/$ \"+)GZn!**F/7$$!+\\@=YEF/$\"+]D`V'*F/7$$!+I*3Jx$F/$\"+Ol'3E*F/7$$!+r_J U]F/$\"+v]oN')F/7$$!+s%*[RgF/$\"+U+BqzF/7$$!+WjJ*3(F/$\"+j?x_qF/7$$!+1 U?/zF/$\"+%QZc7'F/7$$!+QnIf')F/$\"+8/k,]F/7$$!+si$4B*F/$\"+AHvXQF/7$$! +4Nij'*F/$\"+GQ%=d#F/7$$!+>+.2**F/$\"+4]Ug8F/7$$!+4>&*****F/$\"+='485$ !#77$$!+>!Q%3**F/$!+^=8]8F/7$$!+oa1u'*F/$!+FIGKDF/7$$!+;AEj#*F/$!+HNAn PF/7$$!+Lh=u')F/$!+')3zv\\F/7$$!+vx1YzF/$!+lUCrgF/7$$!+Sfc5rF/$!+5yMJq F/7$$!+Pke[gF/$!+$eGL'zF/7$$!+@%)o!*\\F/$!+5Mil')F/7$$!+%[\"ysPF/$!+R* **4E*F/7$$!+$el'3EF/$!+&p[Pl*F/7$$!+!ym:H\"F/$!+#*>C;**F/7$$!+@OQdIF`s $!+iK&*****F/7$$\"+]Ks(G\"F/$!+Y=u;**F/7$$\"+O8/bDF/$!+K'z!o'*F/7$$\"+ #Q)zNQF/$!+%>x]B*F/7$$\"+r0I.]F/$!+9wMe')F/7$$\"+#R]$4hF/$!+)H2o\"zF/7 $$\"+&Q+$*4(F/$!+i?sUqF/7$$\"+)Qy,!zF/$!+g&Q38'F/7$$\"+8vLu')F/$!+yp_v \\F/7$$\"+\\4+D#*F/$!+)eo*fQF/7$$\"+o))oa'*F/$!+XQ=0EF/7$$\"+Nb&p!**F/ $!+8t'4O\"F/7$F($\"+J_8/#)!#>-%+AXESLABELSG6$Q!6\"F[[l-%'COLOURG6&%$RG BGF+F+$\"*++++\"!\")-%*THICKNESSG6#\"\"$-%%VIEWG6$%(DEFAULTGF[\\l" 1 2 0 1 10 3 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 191 "I t may not look like a circle because the axes are scaled differently. \+ We can force Maple to use the same scale on each axis (but it is hardl y ever worth doing so). Here's that circle again -" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "cplot(exp(I *t),t=0..2*Pi,color=blue,thickness=3,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6#7S7$$\"\" \"\"\"!$F*F*7$$\"+uBO1**!#5$\"+#=y_O\"F/7$$\"+*4-Qn*F/$\"+f*)GLDF/7$$ \"+zke[#*F/$\"+C.6.QF/7$$\"+>42`')F/$\"+R4U7]F/7$$\"+L?D/zF/$\"+qceDhF /7$$\"+pTd#3(F/$\"++FafqF/7$$\"+ZdX;hF/$\"+\\!>8\"zF/7$$\"+a:x5]F/$\"+ Sh-a')F/7$$\"+Z'R,#QF/$\"+5bcT#*F/7$$\"+ziQDDF/$\"+M\"oen*F/7$$\"+?o8X 8F/$\"+kt64**F/7$$!+'pr5(f!#8$\"+<#)******F/7$$!+eB[i8F/$\"+)GZn!**F/7 $$!+\\@=YEF/$\"+]D`V'*F/7$$!+I*3Jx$F/$\"+Ol'3E*F/7$$!+r_JU]F/$\"+v]oN' )F/7$$!+s%*[RgF/$\"+U+BqzF/7$$!+WjJ*3(F/$\"+j?x_qF/7$$!+1U?/zF/$\"+%QZ c7'F/7$$!+QnIf')F/$\"+8/k,]F/7$$!+si$4B*F/$\"+AHvXQF/7$$!+4Nij'*F/$\"+ GQ%=d#F/7$$!+>+.2**F/$\"+4]Ug8F/7$$!+4>&*****F/$\"+='485$!#77$$!+>!Q%3 **F/$!+^=8]8F/7$$!+oa1u'*F/$!+FIGKDF/7$$!+;AEj#*F/$!+HNAnPF/7$$!+Lh=u' )F/$!+')3zv\\F/7$$!+vx1YzF/$!+lUCrgF/7$$!+Sfc5rF/$!+5yMJqF/7$$!+Pke[gF /$!+$eGL'zF/7$$!+@%)o!*\\F/$!+5Mil')F/7$$!+%[\"ysPF/$!+R***4E*F/7$$!+$ el'3EF/$!+&p[Pl*F/7$$!+!ym:H\"F/$!+#*>C;**F/7$$!+@OQdIF`s$!+iK&*****F/ 7$$\"+]Ks(G\"F/$!+Y=u;**F/7$$\"+O8/bDF/$!+K'z!o'*F/7$$\"+#Q)zNQF/$!+%> x]B*F/7$$\"+r0I.]F/$!+9wMe')F/7$$\"+#R]$4hF/$!+)H2o\"zF/7$$\"+&Q+$*4(F /$!+i?sUqF/7$$\"+)Qy,!zF/$!+g&Q38'F/7$$\"+8vLu')F/$!+yp_v\\F/7$$\"+\\4 +D#*F/$!+)eo*fQF/7$$\"+o))oa'*F/$!+XQ=0EF/7$$\"+Nb&p!**F/$!+8t'4O\"F/7 $F($\"+J_8/#)!#>-%+AXESLABELSG6$Q!6\"F[[l-%(SCALINGG6#%,CONSTRAINEDG-% 'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%*THICKNESSG6#\"\"$-%%VIEWG6$%(DEFA ULTGF_\\l" 1 2 0 1 10 3 2 6 1 4 1 1.000000 45.000000 45.000000 0 0 "Cu rve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 197 "Recall the number of solutions of f(w)=z in the disk of radius r (for example) is given by the winding number about z \+ of the image under f of the boundary of the disk. Here's an example :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eqn:=z=w^3+w^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $eqnG/%\"zG,&*$)%\"wG\"\"$\"\"\"F,*$)F*\"\"#F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "h:=subs(w=r*exp(I*t),rhs(eqn)): `h(t)`=simpli fy(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%h(t)G,&*&)%\"rG\"\"$\"\" \"-%$expG6#*&^#F)F*%\"tGF*F*F**&)F(\"\"#F*-F,6#*&^#F3F*F0F*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "cplot(subs(r=2,h),t=0..2*Pi, color=brown,thickness=3,title=\"w^3+w^2, r=2\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6#7]q7$$\"#7\"\"!$F*F*7 $$\"+QEVz6!\")$\"+:D/y@!\"*7$$\"+M\"3&=6F/$\"+KuAxUF27$$\"+n)=W.\"F/$ \"+o/W!)fF27$$\"+_!4!R#*F2$\"+sexF2$\"+'fp]+*F27$$\"+7 <>hfF2$\"+<<6=5F/7$$\"+_Zn7SF2$\"+HxU,6F/7$$\"+A[hb>F2$\"+f_(p9\"F/7$$ !+7u$HE\"!#5$\"+&yLO:\"F/7$$!+*o\"Hp@F2$\"+Y(o>7\"F/7$$!+X1cmRF2$\"+Z% )zf5F/7$$!+?07;cF2$\"+!=.Wo*F27$$!+3Li()o@5F/$!+_O*>`#FY7$$!+c39%z*F2$!+/%zU5#F27$$! +SSLN!*F2$!+#[&>6QF27$$!+Cd9>\")F2$!+C@hY^F27$$!+CFp0qF2$!+!y?sG'F27$$ !+#z*)\\c&F2$!+?0Z-tF27$$!+#emc)RF2$!+?Sw/!)F27$$!+LDuEBF2$!+_(o$y$)F2 7$$!+[&)pCmFY$!+?Us;%)F27$$\"+%)>%>%))FY$!+K(Rj9)F27$$\"+$p\"3=BF2$!+k YX&f(F27$$\"+*3n,[$F2$!+ORCyoF27$$\"+7)zaZ%F2$!+'fj^)fF27$$\"+)[$=1aF2 $!+7cnUZF27$$\"+gN8LgF2$!+#4ygO$F27$$\"+C!oRJ'F2$!+56mY@F27$$\"+sNVjjF 2$!+kq5T#*FY7$$\"+SA$*QhF2$\"+/NbfWFY7$$\"+CS]LcF2$\"+O/f+FY$\"+ g(>(pXF27$$!+JzXo7F2$\"+_j&z@%F27$$!+a\\k9AF2$\"+==!)ROF27$$!+uoDOHF2$ \"+>:5QHF27$$!++'[p[$F2$\"+G#Gi5#F27$$!+cNAkQF2$\"+;&Rh5\"F27$$!+$pI(* *RF2$\"+o=,i\\!#67$$!+ba*y(QF2$!+/4u\\5F27$$!+5fg%\\$F2$!+]bM\"4#F27$$ !+lIoiHF2$!+u$Gf!HF27$$!+QR9nAF2$!+,01)f$F27$$!+Rn&GO\"F2$!+gK1tTF27$$ !+_j?\"R$FY$!++atQXF27$$\"+s1s\\zFY$!+O%>Jn%F27$$\"+!*[G_>F2$!+KU%oa%F 27$$\"+R*)>_IF2$!+WpRpTF27$$\"+[G*o1%F2$!+$QH/b$F27$$\"+3!)G?\\F2$!+&z W2u#F27$$\"+!e4eg&F2$!+;VO^`[FY7$$\"+c6;ijF2$\" +o$R3,*FY7$$\"++M:2jF2$\"+5*pg>#F27$$\"+!3;D*fF2$\"+u5f&[$F27$$\"+o3ss `F2$\"+cUb)z%F27$$\"+_P%\\Z%F2$\"+k%Rd)fF27$$\"+8LPWMF2$\"+g]&[!pF27$$ \"+1R8PAF2$\"+W(Gaj(F27$$\"+kH[vvFY$\"++**G!=)F27$$!+#*zKd$)FY$\"+GuDG %)F27$$!+/MX%Q#F2$\"+oZ3r$)F27$$!+F*[l#RF2$\"+WC7C!)F27$$!+#zL@Z&F2$\" +oeTatF27$$!+Oja*)oF2$\"+Sfd&Q'F27$$!++$zf4)F2$\"+_eru^F27$$!+wI3w!*F2 $\"+7+eQPF27$$!+Wfq9)*F2$\"+SC%=/#F27$$!+1]GA5F/$\"+W\"\\-1#FY7$$!+i%= !G5F/$!+(4!*=j\"F27$$!+WMN(***F2$!+gIxuMF27$$!+;^1e$*F2$!+!y_xH&F27$$! +WOmz$)F2$!+O0]\"*pF27$$!+'z5=5(F2$!+;i#3[)F27$$!+/!\\lb&F2$!+78@C(*F2 7$$!+$H\"zURF2$!+l]&31\"F/7$$!+Qe3*=#F2$!+))zY@6F/7$$!*C#*\\2(FY$!+*** eR:\"F/7$$\"+uhH(3#F2$!+,lCX6F/7$$\"+cyK%)RF2$!+iXL-6F/7$$\"+w8w*y&F2$ !+1GAF5F/7$$\"+GWR^vF2$!+;o=R\"*F27$$\"+7Q&R3*F2$!+kv\"))p(F27$$\"+Lmi G5F/$!+#R$*e2'F27$$\"+]'=!>6F/$!+#\\)HkUF27$$\"+=Fcz6F/$!+Se?r@F27$$\" +++++7F/$\"+uKKDE!#<-%+AXESLABELSG6$Q!6\"F^jl-%'COLOURG6&%$RGBG$\")#)e qkF/$\"))eqk\"F/Ffjl-%&TITLEG6#Q-w^3+w^2,~r=26\"-%*THICKNESSG6#\"\"$-% %VIEWG6$%(DEFAULTGFd[m" 1 2 0 1 10 3 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 156 "We see if z lies inside the small loop then f(w)=z has 3 solutions (the winding number) in the disk wit h radius 2 and center at the origin, D(0,2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "It is not necessary to s pecify thickness or color. The default is thickness=1 and color=red (a t least in an unmodified Maple 6)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "cplot(subs(r=1,h),t=0..2*Pi ,title=\"w^3+w^2, r=1\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7aq7$$\"\"#\"\"!$F*F*7$$\"+;*3'p>!\"*$\"+NA >0M!#57$$\"+B`\\z=F/$\"+K$G!*p'F27$$\"+48*[v\"F/$\"+_gq!R*F27$$\"+-1q! f\"F/$\"+Xt3&=\"F/7$$\"+V\"eTO\"F/$\"+$4LnU\"F/7$$\"+$[705\"F/$\"+$ftV i\"F/7$$\"+'oPT1)F2$\"+@m6tF/7$$!+\"f%*QY\"F2$\"+iEk')=F/7$$!+D*p7K%F2$\"+cYn?=F/7$$!+j\\ 'Q+(F2$\"+Xxa56>\"F/$!+ f#y%e%)F27$$!+i'z?)**F2$!+5E>,5F/7$$!+`TRhyF2$!+pfr:6F/7$$!+,:XUcF2$!+ 7#oq=\"F/7$$!+B/H\"\\$F2$!+rM1:7F/7$$!+!*)p@S\"F2$!+qsh/7F/7$$\"*V@0%Q F2$!+%oSU;\"F/7$$\"+m[)z,#F2$!+K!ov4\"F/7$$\"+dKH5PF2$!+uRK#*)*F27$$\" +z7\"R3&F2$!+*G#)>c)F27$$\"+,2ptfF2$!+wM]+tF27$$\"+p_$=g'F2$!+E/vofF27 $$\"+!4ZQ+(F2$!+[q[(R%F27$$\"+]1snqF2$!+!=&>uGF27$$\"+M\\gmoF2$!+R.;@; F27$$\"+Y&Q[X'F2$!*&Gu4]F27$$\"+`s&\\!eF2$\"*9T=N&F27$$\"+.:S-]F2$\"+! o]yL\"F27$$\"+9n(=:%F2$\"+)**3s'=F27$$\"+3W0sKF2$\"+&of@;#F27$$\"+K%3, Q#F2$\"+,PAFAF27$$\"+SdDq:F2$\"+B#>W1#F27$$\"*)GH;%*F2$\"+Y%G8u\"F27$$ \"*J'QiXF2$\"+X\"3]G\"F27$$\"*S]\"GEF2$\"*V;s'**F27$$\"*&>-67F2$\"*)y- roF27$$\")sM:LF2$\"+]d#*GO!#67$$\"'_/CF2$\"+J,A,J!#77$$\")yy#[#F2$!+^& \\J9$F][l7$$\"*a%>*3\"F2$!*3!yClF27$$\"*VP?^#F2$!*+Bqv*F27$$\"*7qX\\%F 2$!+h@Tw7F27$$\"*tUW=*F2$!+%z)zCF27$$ \"+>m]k\\F2$!+f;Yn8F27$$\"+D,2^dF2$!*gCp,'F27$$\"+p`h(R'F2$\"*6^@'QF27 $$\"+]/'4%oF2$\"+_k9E:F27$$\"+rnFjqF2$\"+G3[5GF27$$\"+)=B0,(F2$\"+k`-^ VF27$$\"+J7C6mF2$\"+ZI/VfF27$$\"+P>\"H%fF2$\"+'e]HN(F27$$\"+#>U8)\\F2$ \"+$[JU@\"F/ 7$$!+C:&y(eF2$\"+mqg\"=\"F/7$$!+p,pOzF2$\"+@&zC6\"F/7$$!+$o#43**F2$\"+ vE215F/7$$!+_$[+=\"F/$\"+j3Kk&)F27$$!+!)4hW8F/$\"+A!o\\q'F27$$!+Jd#\\Z \"F/$\"+1+-zXF27$$!+A!G#p:F/$\"+.2*H?#F27$$!+NCEC;F/$!*ctt\"[F27$$!+mF sI;F/$!+t#e[G$F27$$!+et'))e\"F/$!+i(R2,'F27$$!+9!R$*\\\"F/$!+\")y\\v') F27$$!+0flf8F/$!+FVvC6F/7$$!+Lj@u6F/$!+lIgd8F/7$$!+\"[^k\\*F2$!+EXDc:F /7$$!+e'zc!pF2$!+3/^:F/7$$\"+ZL`L^F2$!+H@:j=F/7$$\"+D&z6-)F2$!+ex #[x\"F/7$$\"+(fEZ2\"F/$!+Y16S;F/7$$\"+Ns'*Q8F/$!+Zk*)[9F/7$$\"+)RCwc\" F/$!+!fuQ@\"F/7$$\"+2$)HYF/$!+e4\\%R$F27$$\"+++++?F/$\"+:w1-T!#=-%'COLOURG6&%$RGBG$\"#5! \"\"F+F+-%+AXESLABELSG6$Q!6\"Fi[m-%&TITLEG6#Q-w^3+w^2,~r=16\"-%%VIEWG6 $%(DEFAULTGFc\\m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 303 "Note f(w) has a critical point at 0 and at -2/3. The second derivative is nonzero at these points so it follows that \+ f(w) is 2-to-1 in a punctured neighborhood of each critical point. The next plot shows clearly that for any z we have at most 2 solu tions of f(w)=z in the disk D(0,3/5)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "cplot(subs(r=3/5,h), t=0..2*Pi,title=\"w^3+w^2, r=3/5\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7`p7$$\"++++gd!#5$\"\"!F,7$$\"+ T9(3o&F*$\"+5e@@$*!#67$$\"+Vr-YaF*$\"+/%3l$=F*7$$\"+vUu?^F*$\"+\"Gd*zD F*7$$\"+vx3\"p%F*$\"+.pdlKF*7$$\"+'3+i4%F*$\"+L(4*[RF*7$$\"+'*Qb+MF*$ \"+I2k@XF*7$$\"+$[i(>EF*$\"+WzOr\\F*7$$\"+t![nuV#F*7$$!+3)[Ia%F*$\"+S8'4r\"F*7$$!+COpaYF*$\"+O:Z9&*F27$$!+n .6QYF*$\"+Id;D?F27$$!+#)QsAXF*$!+AW%)4VF27$$!+DOM?VF*$!+J?wD5F*7$$!+yn %)**RF*$!+aO=M;F*7$$!+!=Ghf$F*$!+0dHk@F*7$$!+U`^CJF*$!+Md*pg#F*7$$!+4k I0EF*$!+mI!G&HF*7$$!+&G?z2#F*$!+)eP7>$F*7$$!+MI?T:F*$!+gt!pL$F*7$$!+sy Hf5F*$!+K_g\"R$F*7$$!++O3TfF2$!+&)R3xLF*7$$\"+G?j.RF2$!+g4X.JF*7$$\"+ \"*))[O5F*$!+g.dvEF*7$$\"+g(pS`\"F*$!+Gp!31#F*7$$\"+&Fhn7$$\"+PU- S9F*$!+qK4LA!#87$$\"+ox$4X\"F*$\"+T4y0\\Fhn7$$\"+hWe%[\"F*$\"+'oEd4\"F 27$$\"+*G$o%e\"F*$\"+>C'>j#F27$$\"+$)>;8I:HF*7$$\"+kn!H`$F2$\"+\"))z27$F*7$$!+&GwP%fF2 $\"+-?6xLF*7$$!+:v'\\2\"F*$\"+P`$4R$F*7$$!+%)\\*Hd\"F*$\"+?#43L$F*7$$! +\\\\#H7#F*$\"+06wuJF*7$$!+gLdhEF*$\"+Qp7@HF*7$$!+?')oTJF*$\"+siI$f#F* 7$$!+)4@,e$F*$\"+,A#>=#F*7$$!+GbZxRF*$\"+-!G)o;F*7$$!+gEg'H%F*$\"+Wl\\ !3\"F*7$$!+PJ2>XF*$\"+pPG\\WF27$$!+D$>:k%F*$!+%*[\"eN#F27$$!+K&))Hl%F* $!+k!*ey(*F27$$!+;nhQXF*$!+RvtHW#F*7$$!+$R@w&RF* $!+JE`@JF*7$$!+>Bw$[$F*$!+33dhPF*7$$!+39P,HF*$!+`#z`K%F*7$$!+f'f&HAF*$ !+v`U!z%F*7$$!+1\\5![\"F*$!+'Qca9&F*7$$!+=Pt#R(F2$!+*)pAj`F*7$$\"+?zGb MFhn$!+rL2paF*7$$\"+C)*=\"R*F2$!+R)*RYaF*7$$\"+%=ue$=F*$!+G$*Rn_F*7$$ \"+`kJ3EF*$!+![Fn(\\F*7$$\"+`zIKLF*$!+cb:oXF*7$$\"+oG*)HSF*$!+U3L7SF*7 $$\"+$f.1j%F*$!+6ONYLF*7$$\"+*e(G)4&F*$!+i*4?i#F*7$$\"+gy*zW&F*$!+Vf!4 $=F*7$$\"+M@P\"o&F*$!++M*=H*F27$F($\"++dKA6!#=-%'COLOURG6&%$RGBG$\"#5! \"\"F+F+-%+AXESLABELSG6$Q!6\"Fbfl-%&TITLEG6#Q/w^3+w^2,~r=3/56\"-%%VIEW G6$%(DEFAULTGF\\gl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 173 "Now let's create a little routine to com pute contour integrals - actually we will define two routines, one ine rt. Note we assume we are dealing with differentiable contours." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "cint:=proc(beta,R,f,z)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " l ocal t,g; t:=lhs(R);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " g:=subs( z=beta,f)*diff('beta',t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " int( g,R,args[5..nargs]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "Cint:=proc(beta,R,f,z)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " local t,g; t:=lhs(R);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " g:=subs(z=beta,f)*diff('beta',t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " Int(g,R,args[5..nargs]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "Here beta is the contour (an expression in a r eal variable, not a function), R is a range for the parameter in be ta, f is an expression in a complex variable and z is that variabl e. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 197 " Note we have used deferred evaluation of beta in the differentiation command. This prevents the int() command from chocking on booleans \+ (as would occur in dealing with piecewise smooth curves)." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "1/(2 *Pi*I)*cint(subs(r=2,h),t=0..2*Pi,1/z,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "That is certainly the winding number of the co ntour \"w^3+w^2, r=2\" about about the origin. Let try the winding num ber about 10. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 468 "If you try cint() it will take a long time and you will \+ get quite a mess. We use the inert Cint() instead and then evalf() ( evaluate as floating point). The evalf() call after an inert integral \+ call alerts Maple that we want to compute the integral numerically rat her than symbolically. Maple uses its default numerical method though \+ we could specify any of a number of methods. Skipping the attempt to d o a symbolic evaluation can save a substantial amount of time." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "1/(2*Pi*I)*Cint(subs(r=2,h),t=0..2*Pi,1/(z-10),z): evalf(%,8); e valf(%%,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\")+++5!\"($\")#[Yx %!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+&*********!#5$!+&e'H\\&* !#A" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "It is pretty clear the imaginary part here is due to roundoff \+ and the winding number is 1 (as expected)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Note cint() can also be used for nonclosed contours:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "h2:=1-t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h2G,&\"\"\"F&%\"tG !\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "cint(h2,t=0..1,1/( w-z),w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#*&%\"zG\"\"\",&!\" \"F(F'F(F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Now let's consider a piecewise smooth contour - a square: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "h3a:=(1-t)*(-1-I)+t*(1-I):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "h3b:=(1-t)*(1-I)+t*(1+I):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "h3c:=(1-t)*(1+I)+t*(-1+I):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "h3d:=(1-t)*(-1+I)+t*(-1-I):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "h3:=piecewise(t<1,h3a,t<2,subs(t=t-1,h3b) ,t<3,subs(t=t-2,h3c),t<=4,subs(t=t-3,h3d));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h3G-%*PIECEWISEG6&7$,&*&^$!\"\"F,\"\"\",&F-F-%\"tGF, F-F-*&^$F-F,F-F/F-F-2F/F-7$,&*&F1F-,&\"\"#F-F/F,F-F-*&^$F-F-F-,&F/F-F- F,F-F-2F/F77$,&*&F9F-,&\"\"$F-F/F,F-F-*&^$F,F-F-,&F/F-F7F,F-F-2F/F@7$, &*&FBF-,&\"\"%F-F/F,F-F-*&F+F-,&F/F-F@F,F-F-1F/FI" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "Note we were care ful at t=4. Otherwise Maple will consider h3(4) to be 0 and cplo t() will produce a line segment joining the origin to one vertex of o ur square." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "cplot(h3,t=0..4,thickness=2,color=brown);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6#7Y7$$!\"\" \"\"!F(7$$!+mwAc#)!#5$!+++++5!\"*7$$!+%o!)*QnF.F/7$$!+mxnK]F.F/7$$!+mV 1:LF.F/7$$!+%[9cg\"F.F/7$$!)=ct?F.F/7$$\"+]YJ?;F.F/7$$\"+#=\"\\qM$F1FT7$$!*+.W2&F1FT7$$!*ep'RmF1FT7$$!*S>4N)F1FT7$$!*w&*f :*F1FT7$$!*7s5'**F1FT7$F/$\"*7\"R(>*F17$F/$\"*OaeN)F17$F/$\"*MU.r'F17$ F/$\"*wVw)\\F17$F/$\"*E![GLF17$F/$\"*'fuJ;F17$F/$!(_Q4&F17$F/$!*!3:(f \"F17$F/$!*c%GpLF17$F/$!*9-V&\\F17$F/$!*WhUk'F17$F/$!*9o " 0 "" {MPLTEXT 1 0 12 "g:=z/sin(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*&%\"zG\"\"\"-%$sinG6#F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Cint(h3,t=0..4,g,z): evalf(%); eval f(%%,16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!\"\"!#7$\"\"!F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"\"!F%F$" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 212 "We expected to get 0 \+ by Cauchy's theorem since z/sin(z) has a removeable singularity at \+ the origin. The default precision in Maple is 10 decimal digits. The nonzero value we get a first is due to round-off." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "41 2 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }