{VERSION 4 0 "HP RISC UNIX" "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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 41 "Themenvorschl\344ge f \374r den 23. Oktober 2002" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 301 "a) - e): Hier ist der Integrationsweg der Kreis mit Radius zwei u m den Nullpunkt. F\374r alle Funktionen, die im Innern dieses Kreises \+ holomorph sind, verschwindet das Integral nach dem Cauchyschen Integra lsatz; damit haben die Integrale b) und e) den Wert null. Integral c) \+ verschwindet ebenfalls, denn " }{XPPEDIT 18 0 "1/(z^2);" "6#*&\"\"\"F$ *$%\"zG\"\"#!\"\"" }{TEXT -1 59 " hat die \374berall au\337er im Nullp unkt erkl\344rte Stammfunktion " }{XPPEDIT 18 0 "-1/z;" "6#,$*&\"\"\"F %%\"zG!\"\"F'" }{TEXT -1 105 " ; insbesondere ist diese in einer Umgeb ung des Integrationswegs definiert.\nDas Integral a) hat den Wert " } {XPPEDIT 18 0 "2*Pi*I;" "6#*(\"\"#\"\"\"%#PiGF%%\"IGF%" }{TEXT -1 150 ", denn laut Vorlesung kommt es dabei auf den Radius des Kreises nicht an.\nF\374r d) schlie\337lich ist eine Partialbruchzerlegung des Inte granden notwendig:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "conve rt(1/(z^2+1), parfrac, z, I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&^ ##!\"\"\"\"#\"\"\",&%\"zGF)^#F'F)F'F)*&^##F)F(F),&F+F)^#F)F)F'F)" }}} {EXCHG {PARA 258 "" 0 "" {TEXT -1 83 "Beide Nennernullstellen liegen i m Kreis; das Integral \374ber den ersten Summanden ist" }}{PARA 259 "> " 0 "" {MPLTEXT 1 0 15 "(2*Pi*I) * I/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG!\"\"" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 20 "das \374b er den zweiten" }}{PARA 261 "> " 0 "" {MPLTEXT 1 0 18 "(2*Pi*I) * (-I/ 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#PiG" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 144 "Insgesamt verschwindet also auch dieses Integral.\n f) Der neue Integrationsweg ist ein Kreis mit Radius eins um den Punkt 3. In diesem Kreis ist " }{XPPEDIT 18 0 "1/z;" "6#*&\"\"\"F$%\"zG!\" \"" }{TEXT -1 52 " holomorph, so da\337 das erste Integral verschwinde t; " }{XPPEDIT 18 0 "1/(z-3);" "6#*&\"\"\"F$,&%\"zGF$\"\"$!\"\"F(" } {TEXT -1 72 " ist aber hier nicht mehr holomorph; das Integral b) hat also den Wert " }{XPPEDIT 18 0 "2*Pi*I;" "6#*(\"\"#\"\"\"%#PiGF%%\"IG F%" }{TEXT -1 364 " . Die Integranden c) bis e) sind wieder holomorph, also verschwinden die Integrale.\n\ng) - i) Nun ist der Integrationsw eg nur noch ein Halbkreis mit Radius zwei um den Nullpunkt; er geht vo n -2i \374ber 2 nach 2i. Die Integrale k\366nnen \374ber Stammfunktion en berechnet werden, wobei es f\374r h) wichtig ist, da\337 der Integr ationsweg die negative reelle Achse nicht kreuzt. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "g)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "int(z, z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"zG\"\"#\"\"\"#F(F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "F := unapply(%, z); F(2*I) - F(-2*I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6#%\"zG6\"6$%)oper atorG%&arrowGF(,$*$)9$\"\"#\"\"\"#F1F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 2 "h)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "int(1/z, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ln(I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&^##\"\"\" \"\"#F&%#PiGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand(ln (2*I));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#\"\"#\"\"\"*&^##F (F'F(%#PiGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "expand(ln (-2*I));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#\"\"#\"\"\"*&^## !\"\"F'F(%#PiGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "%% - % ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&^#\"\"\"F%%#PiGF%" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 2 "i)" }}{PARA 265 "> " 0 "" {MPLTEXT 1 0 14 "int(1/z^2, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%%\"z G!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "-1/(2*I) - (-1/ (-2*I));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#\"\"\"" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 2 "j)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "z := x + I*y;\n(1 - I*'z')/(1 + I*'z')=evalc(sim plify(evalc((1 - I*z)/(1 + I*z))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"zG,&%\"xG\"\"\"*&^#F'F'%\"yGF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&\"\"\"F&*&^#!\"\"F&%\"zGF&F&F&,&F&F&*&^#F&F&F*F&F&F),&*&,(F&F& *$)%\"yG\"\"#F&F)*$)%\"xGF4F&F)F&,*F&F&*&F4F&F3F&F)F1F&F5F&F)F&*&*&^#! \"#F&F7F&F&F8F)F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "1-2*y+ y^2+x^2 = x^2 + (1-y)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*\"\"\"F %*&\"\"#F%%\"yGF%!\"\"*$)F(F'F%F%*$)%\"xGF'F%F%,&*$),&F%F%F(F)F'F%F%F, F%" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 83 "k) Wegen h) ist f(z) auf \+ jeden Fall holomorph. Wir m\374ssen in den Tangens einsetzen:" }} {PARA 268 "> " 0 "" {MPLTEXT 1 0 21 "convert(tan(w), exp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&^#!\"\"\"\"\",&*$)-%$expG6#*&^#F'F'%\"wG F'\"\"#F'F'F'F&F'F',&F)F'F'F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "z := 'z': w := I/2*ln((1-I*z)/(1+I*z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG*&^##\"\"\"\"\"#F(-%#lnG6#*&,&F(F(*&^#!\"\"F(%\"z GF(F(F(,&F(F(*&^#F(F(F2F(F(F1F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "I*w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#*&,&\"\"\"F)* &^#!\"\"F)%\"zGF)F)F),&F)F)*&^#F)F)F-F)F)F,#F,\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "u := exp(2*I*w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG*&,&\"\"\"F'*&^#F'F'%\"zGF'F'F',&F'F'*&^#!\"\"F'F *F'F'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(-I*(u-1 )/(u+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"zG" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 69 "Ableitungenen von Umkehrfunktionen kann man w ie im Reellen berechnen:" }}{PARA 270 "> " 0 "" {MPLTEXT 1 0 15 "diff( tan(z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*$)-%$tanG6#% \"zG\"\"#F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(arcta n(z), z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*$)%\"zG\"\"# F$F$F$F$!\"\"" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 148 "l) Der Integr ationswert geht von 1/2 nach -1/2 und die gerade betrachtete Stammfunk tion ist holomorph in einer Umgebung davon. Also ist das Integral" }} {PARA 272 "> " 0 "" {MPLTEXT 1 0 27 "arctan(-1/2) - arctan(1/2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%'arctanG6##\"\"\"\"\"#!\"#" }}} {EXCHG {PARA 273 "" 0 "" {TEXT -1 152 "m) offensichtlich meromorph: Po lstellen bei i und -i, sonst \374berall holomorph.\nn) Wir m\374ssen d ie Caucy-Riemannschen Differentialgleichungen nachrechnen:" }}{PARA 274 "> " 0 "" {MPLTEXT 1 0 48 "z := x + I*y; evalc(exp(z) - exp(conjug ate(z)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG,&%\"xG\"\"\"*&^#F' F'%\"yGF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(^#\"\"#\"\"\"-%$expG6 #%\"xGF&-%$sinG6#%\"yGF&" }}}{EXCHG {PARA 275 "" 0 "" {TEXT -1 210 "Di e Funktion nimmt also nur rein imagin\344re Werte in und ist somit nir gends holomorph und damit auch nicht meromorph.\no) W\344re diese Funk tion meromorph, so auch ihr Inverses; also weder holomorph noch meromo rph. " }{MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }