{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "" -1 256 "symbol" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "symbol" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "with(plots):setoptio ns(scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=z->exp(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "This function c onverts complex numbers to cartesian coordinates" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "cart:=z->op([Re(z),Im(z)]);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 50 "Aspect ratio is the ratio of an ellipse's semiaxes" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "aspect:=5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "loop:=r->r*(cos(t)+I*aspect*sin(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Here is an ellipse with " }{TEXT 260 1 "x " }{TEXT -1 11 "-semiaxis 1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot ([cart(loop(1)),t=-Pi..Pi]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "E volution of the image of an ellipse as the size of the ellipse grows. " }}{PARA 0 "" 0 "" {TEXT -1 78 "Notice how initially only one point i nside the ellipse goes to 1, namely zero." }}{PARA 0 "" 0 "" {TEXT -1 86 "In accordance with the argument principle the image of the ellipse goes around 1 once." }}{PARA 0 "" 0 "" {TEXT -1 46 "As the ellipse gr ows and eventually encloses 2" }{TEXT 256 1 "p" }{TEXT 258 1 "i" } {TEXT -1 7 " and -2" }{TEXT 257 1 "p" }{TEXT 259 2 "i " }{TEXT -1 46 " the image of the ellipse goes around 1 thrice." }}{PARA 0 "" 0 "" {TEXT -1 120 "The exponential function is never 0, so as the ellipse g rows, the winding number of its image with respect to 0 stays 0." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "animate([cart(f(loop(r))),t=-Pi..Pi ],r=0.2..1.4,numpoints=400);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "8" 0 }{VIEWOPTS 1 1 0 1 1 1803 }