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}}{PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 207 "" 0 "" {TEXT 214 19 " Stand 8.Januar 2007" }{TEXT 214 0 "" }}{PARA 207 "" 0 "" {TEXT 214 0 " " }}}{EXCHG {PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 217 11 "Aufgabe 1: " }{TEXT 214 15 "Potenzen von i" }{TEXT 203 0 "" }}{PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 15 "Berechnen Sie " }{XPPEDIT 18 0 "i^2, i^3, i^4, i^5" "6&*$%\"iG\" \"#*$F$\"\"$*$F$\"\"%*$F$\"\"&" }{TEXT 214 14 ", ... sowie " } {XPPEDIT 18 0 "1/i, 1/i^2, 1/i^3" "6%*&\"\"\"F$%\"iG!\"\"*&F$F$*$F%\" \"#F&*&F$F$*$F%\"\"$F&" }{TEXT 214 8 ", ... ." }{TEXT 203 0 "" }}} {EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 19 "I^2, I^3, I^4, I^5;" }{MPLTEXT 1 0 0 "" }}{PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 18 "1/I, 1/I^2, 1/I^3;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 217 11 "Auf gabe 2: " }{TEXT 214 21 "Multiplikation mit i" }{TEXT 203 0 "" }} {PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 109 "Mu ltiplikation mit i kann geometrisch als Drehung um 90\260 gedeutet we rden. Verifizieren Sie dies, indem Sie " }{XPPEDIT 18 0 " z" "6#%\"zG " }{TEXT 214 7 " und " }{XPPEDIT 18 0 "i*z;" "6#*&%\"iG\"\"\"%\"zGF% " }{TEXT 214 2 " " }{TEXT 214 0 "" }{TEXT 214 36 "\nsowie den Drehwin kel zeichnen f\374r " }{XPPEDIT 18 0 "z=4+2*i, z=-1+i, z=5-2*i" "6%/% \"zG,&\"\"%\"\"\"*&\"\"#F'%\"iGF'F'/F$,&F'!\"\"F*F'/F$,&\"\"&F'F(F-" } {TEXT 214 2 " ." }{TEXT 203 0 "" }}{PARA 207 "" 0 "" {TEXT 214 0 "" }} }{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 38 "restart: with(plots): wit h(plottools):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 25 "z := 4+2*I; # -1+I, 5-2*I" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 10 "\nz2:= z*I;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 21 "\np hi := argument(z); " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 22 "\nphi2 := ar gument(z2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 34 "A := complexplot([0,z],color=red):" }{MPLTEXT 1 0 0 " " }{MPLTEXT 1 0 37 "\nB := complexplot([0,z2],color=blue):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 40 "\nC := arc([0,0],1,phi..phi2,color=blue):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 36 "\ndisplay(A,B,C,scaling=constrain ed);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 217 11 "Aufgabe 3: " }{TEXT 214 62 "Arithmetische Operationen, R eal- und Imagin\344rteil, Konjugation" }{TEXT 203 0 "" }}{PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 14 "a) Es seien " }{XPPEDIT 18 0 "z[1]=4+3*i, z[2]=2-5*i" "6$/&%\"zG6#\"\"\",&\"\"%F'*& \"\"$F'%\"iGF'F'/&F%6#\"\"#,&F0F'*&\"\"&F'F,F'!\"\"" }{TEXT 214 49 " . Berechnen Sie folgende Ausdr\374cke in der Form " }{XPPEDIT 18 0 "x+ i*y" "6#,&%\"xG\"\"\"*&%\"iGF%%\"yGF%F%" }{TEXT 214 2 " :" }{TEXT 203 0 "" }}{PARA 208 "" 0 "" {TEXT 214 5 " " }{XPPEDIT 18 0 "z[1]*z[2] , (3*z[1]-z[2])^2, 1/z[1], 25*z[2]/z[1], Re(z[1]^3), Re(z[1])^3, (z[1] -z[2])/(z[1]+z[2]), z[1]*conjugate(z[2]), conjugate(z[1])*z[2], 1/z[1] ^2, 1/conjugate(z[1])^2,conjugate(z[1])/conjugate(z[2]), conjugate(z[1 ]/z[2]), z[2]*conjugate(z[2])/(z[1]*conjugate(z[1]))" "60*&&%\"zG6#\" \"\"F'&F%6#\"\"#F'*$,&*&\"\"$F'F$F'F'F(!\"\"F**&F'F'F$F/*(\"#DF'F(F'F$ F/-%#ReG6#*$F$F.*$-F46#F$F.*&,&F$F'F(F/F',&F$F'F(F'F/*&F$F'-%*conjugat eG6#F(F'*&-F?F9F'F(F'*&F'F'*$F$F*F/*&F'F'*$FBF*F/*&FBF'F>F/-F?6#*&F$F' F(F/*(F(F'F>F'*&F$F'FBF'F/" }{TEXT 203 0 "" }}{PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 11 "b) Es sei " }{XPPEDIT 18 0 "z=x+i*y" "6#/%\"zG,&%\"xG\"\"\"*&%\"iGF'%\"yGF'F'" }{TEXT 214 36 " \+ . Berechnen Sie folgende Ausdr\374cke:" }{TEXT 214 0 "" }{TEXT 214 5 " \n " }{XPPEDIT 18 0 "Im(1/z), Im(z^4), Im(z^2)^2, (1+i)^16, Re(z/c onjugate(z)), Re(z^2/conjugate(z))" "6(-%#ImG6#*&\"\"\"F'%\"zG!\"\"-F$ 6#*$F(\"\"%*$-F$6#*$F(\"\"#F2*$,&F'F'%\"iGF'\"#;-%#ReG6#*&F(F'-%*conju gateG6#F(F)-F86#*&F(F2F;F)" }{TEXT 203 0 "" }}{PARA 207 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 210 "" 0 " " {TEXT 212 2 "a)" }{TEXT 212 0 "" }}{PARA 207 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 25 "z1 := 4+3*I; z2 := 2-5*I;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 16 "'z1*z2' = z1*z2;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 28 "'(3*z1-z2)^2' = (3*z1-z2)^2;" }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 14 "'1/z1' = 1/z1;" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 23 "'25*z 2/z1' = 25*z2/z1; " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 38 "'z1^3' = z1^3; 'Re(z1^3)' = Re(z1^3); " }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 42 "\n'Re(z1)' = Re(z1); 'Re(z1)^3' = Re(z1)^3;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 36 "'(z1 -z2)/(z1+z2)' = (z1-z2)/(z1+z2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 38 "'z1*conjugate(z2)' = z1*conjugate(z2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 38 "'co njugate(z1)*z2' = conjugate(z1)*z2;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 18 "'1/z1^2' = 1/z1^2;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 41 "\n'1/conjugate(z1)^2' = 1/conjugate(z1)^2;" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 60 "'conj ugate(z1)/conjugate(z2)' = conjugate(z1)/conjugate(z2);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 39 "\n'conjugate(z1/z2)' = conjugate(z1/z2);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 76 "'z2*c onjugate(z2)/(z1*conjugate(z1))' = z2*conjugate(z2)/(z1*conjugate(z1)) ;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 214 0 "" }} {PARA 211 "" 0 "" {TEXT 209 3 "b) " }{TEXT 209 0 "" }}{PARA 207 "" 0 " " {TEXT 214 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 20 "restar t: z := x+I*y;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 8 "Im(1/z);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 107 "Maple kann diesen Ausdruck nicht weiter vereinfachen, solange wir nicht zu erkennen geb en, da\337 x und y " }{TEXT 217 6 "reelle" }{TEXT 214 7 " Gr\366\3 37en" }{TEXT 203 0 "" }}{PARA 207 "" 0 "" {TEXT 214 50 "sein sollen. D azu ben\366tigt man den Befehl \"evalc\":" }{TEXT 214 0 "" }}{PARA 207 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 44 "Im(1/z) = evalc(Im(1/z)); # evaluate complex" }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 25 "Im(z^4) = evalc(Im(z^4)) ;" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 26 "\nIm(z^2) = evalc(Im(z^2));" } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 30 "\nIm(z^2)^2 = evalc(Im(z^2)^2);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 9 "(1+I)^ 16;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 36 "r := abs(1+I); phi := argument(1+I);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 59 "'r*(cos(phi)+I*sin(phi))' = evalc (r*(cos(phi)+I*sin(phi)));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> \+ " 0 "" {MPLTEXT 1 0 70 "'(r*(cos(phi)+I*sin(phi)))^16' = evalc((r*(cos (phi)+I*sin(phi)))^16); " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 47 "Re(z/conjugate(z)) = evalc(Re(z/conjugate(z)));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 51 "Re( z^2/conjugate(z)) = evalc(Re(z^2/conjugate(z)));" }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 217 12 "Aufgabe 4: " } {TEXT 214 35 "Wie sehen die folgenden Mengen aus?" }{TEXT 203 0 "" }} {PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 208 "" 0 "" {TEXT 214 11 "a) \+ \{ z | " }{XPPEDIT 18 0 "abs(z+i)<1" "6#2-%$absG6#,&%\"zG\"\"\"%\"iG F)F)" }{TEXT 214 16 " \} , b) \{ z | " }{XPPEDIT 18 0 "abs(z+i)<2 " "6#2-%$absG6#,&%\"zG\"\"\"%\"iGF)\"\"#" }{TEXT 214 18 "\} intersect \{ z | " }{XPPEDIT 18 0 "abs(z-i)<2" "6#2-%$absG6#,&%\"zG\"\"\"%\"iG!\" \"\"\"#" }{TEXT 214 28 " \} , c) \{ z | Re(z) > 2 \} " }{TEXT 203 0 "" }}{PARA 207 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 207 "" 0 "" {TEXT 214 2 "a)" }{TEXT 214 0 "" }}{PARA 207 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 11 "z := x+I*y;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 " > " 0 "" {MPLTEXT 1 0 18 "evalc(abs(z+I)<1);" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 21 "\nevalc(abs(z+I)^2<1);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 65 "\nimplicitplot(x^2+y^2+2*y=0,x=-2..2,y=-2..2,scaling =constrained);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 214 0 "" }}{PARA 207 "" 0 "" {TEXT 214 2 "b)" }{TEXT 214 0 "" }}{PARA 207 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 22 "evalc(abs(z+I)^2<2^2);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 23 "\neva lc(abs(z-I)^2<2^2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 59 "implicitplot(\{x^2+y^2+2*y=3,x^2+y^2-2*y=3\},x=-2..2, y=-3..3," }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 24 "\n scaling=constrained );" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 214 0 "" }} {PARA 207 "" 0 "" {TEXT 214 2 "c)" }{TEXT 214 0 "" }}{PARA 207 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 15 "evalc(R e(z)>2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 168 "inequal(\{2 < x\}, x=-1..6, y=-3..4,optionsfeasible=(color=red, thickness=2), optio nsexcluded=(color=blue,thickness=2));" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 212 "" 0 "" {TEXT 218 0 "" }}{PARA 213 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }