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2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 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 "Normal" -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 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 43 "An elementary treatment of Euler' s Formula " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B. C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "The series expansion of " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 58 "In the following we make use of the mathematical constant " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 25 " defined \+ as the value of " }{XPPEDIT 18 0 "Limit((1+t)^(1/t),t = 0);" "6#-%&Lim itG6$),&\"\"\"F(%\"tGF(*&F(F(F)!\"\"/F)\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 47 " has the approximate decimal value 2.718281828." }}{PARA 0 "" 0 " " {TEXT -1 39 "The corresponding exponential function " }{XPPEDIT 18 0 "f(x)=exp(x)" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 27 " has the p roperty that f '(" }{TEXT 268 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "`` =exp(x)" "6#/%!G-%$expG6#%\"xG" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 44 " is a solution \+ of the differential equation " }{XPPEDIT 18 0 "dy/dx=y" "6#/*&%#dyG\" \"\"%#dxG!\"\"%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "Th e gradient of the tangent line to the curve " }{XPPEDIT 18 0 "y=exp(x) " "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 13 " at any point" }{XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 45 " on the curve is numeri cally the same as the " }{TEXT 269 1 "y" }{TEXT -1 26 " coordinate of \+ the point. " }}{PARA 0 "" 0 "" {TEXT -1 59 "For example, the gradient \+ of the tangent line to the curve " }{XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG -%$expG6#%\"xG" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(0,1)" "6#-%!G6$\"\"!\"\"\"" }{TEXT -1 55 " is 1 and the gradient of the tang ent line at the point" }{XPPEDIT 18 0 "``(1,exp(1))" "6#-%!G6$\"\"\"-% $expG6#F&" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\" \"" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 348 471 471 {PLOTDATA 2 "63-%'CURVESG6%7S7$$!33+++++++;!#<$\"36al%*z^' *=?!#=7$$!3!ommm5\\-`\"F*$\"3ighbNt\"[;#F-7$$!3MLLLF#f&p9F*$\"3f-F%R_o -I#F-7$$!3!omm1628S\"F*$\"3WN4))\\%[FY#F-7$$!3wmmmuDgK8F*$\"394e2W$)*y j#F-7$$!3KLLLzXAk7F*$\"3B2m(>N#eCGF-7$$!3(ommYUH3?\"F*$\"3=^Jxv\\W4IF- 7$$!3B+++9u=N6F*$\"3BuStz)=O@$F-7$$!3!pmmEN+t1\"F*$\"3'QzI^qd$RMF-7$$! 3+,++g+J'***F-$\"3G/kD:@:!o$F-7$$!3OLLL`wC+$*F-$\"3c!R%Q#RRa%RF-7$$!3C nmm1b:(o)F-$\"3g'H!Hq,!\\>%F-7$$!3;+++g*ep*zF-$\"3GGVJwil%\\%F-7$$!3a, +++%GRI(F-$\"3.6bUTr><[F-7$$!3(4+++/lgj'F-$\"3s]de`m!*\\^F-7$$!3onmmE6 eHgF-$\"3oD%[C)e!>Z&F-7$$!36NLL$48%3`F-$\"3[E-t!z*4\")eF-7$$!3]LLLt\"* [(p%F-$\"3!R(R@_@f^iF-7$$!3C,++?%Ro)RF-$\"3'*ze\\o!G?r'F-7$$!3=OLLtRzd LF-$\"3E!**)o4x!y9(F-7$$!3k+++?9jnEF-$\"3wf@P([W&ewF-7$$!3n+++gMV5?F-$ \"3Q!o**3zp(y\")F-7$$!3QLLLLjrC8F-$\"3!)f<04yFf()F-7$$!3yVLLL&R,&p!#>$ \"3$eCYtJ)eG$*F-7$$!3mSnmmm[z:!#?$\"3mF\"f1gF -$\"3NT)=$35S<7F*7$$\"3a(*****>6W_EF-$\"3s\\+w(>\\PI\"F*7$$\"3Y)*****R G$GK$F-$\"3QYRKKx9%R\"F*7$$\"3#z******Hr9(RF-$\"3Cd'G3yuv[\"F*7$$\"31) *****>Pn\"p%F-$\"3+\\=rZDm)f\"F*7$$\"3vhmmEw!)Q`F-$\"3o?9\\r#Qbq\"F*7$ $\"3:,+++7wHgF-$\"3'z^B@s\\v#=F*7$$\"3)4LLL$y'el'F-$\"3Ey.z]=jX>F*7$$ \"3Z******fxOStF-$\"33>LMtTZ$3#F*7$$\"3SimmY)GW)zF-$\"3+i-A,#y?A#F*7$$ \"3s+++g#ewl)F-$\"3_bP-Cc#oP#F*7$$\"3IlmmmI'eJ*F-$\"3'y%)*)p(G`QDF*7$$ \"3f******\\U\\+5F*$\"30rDKrci>FF*7$$\"3]LLL*ygo1\"F*$\"3'eXY&f=C1HF*7 $$\"3;LLLh,tM6F*$\"3rY6CTTL5JF*7$$\"3zmmmSv.-7F*$\"3C!3?Dp))oK$F*7$$\" 3#)*****>.')QE\"F*$\"3q/&>F-F- *&&F*6#\"\"&F-*$F'FCF-F-%(~.~.~.~GF-*&&F*6#%\"nGF-)F'FIF-F-FEF-" } {TEXT -1 14 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Putting " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 19 " in (i) shows that " }{XPPEDIT 18 0 "a[0]=1" "6#/&%\"aG6# \"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "Different iating both sides of (i) with respect to " }{TEXT 263 1 "x" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x) = a[1]+2*a[2]*x+3*a[3]*x^2+4*a[4]*x^3+5*a[5]*x^4+` . . . `+n*a[n]*x^( n-1)+` . . . `;" "6#/-%$expG6#%\"xG,2&%\"aG6#\"\"\"F,*(\"\"#F,&F*6#F.F ,F'F,F,*(\"\"$F,&F*6#F2F,F'F.F,*(\"\"%F,&F*6#F6F,F'F2F,*(\"\"&F,&F*6#F :F,F'F6F,%(~.~.~.~GF,*(%\"nGF,&F*6#F?F,)F',&F?F,F,!\"\"F,F,F=F," } {TEXT -1 15 " ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 99 "(Here we \+ are also assuming that it is reasonable to differentiate an infinite s eries term by term)." }}{PARA 0 "" 0 "" {TEXT -1 8 "Putting " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 20 " in (ii) shows that \+ " }{XPPEDIT 18 0 "a[1] = 1;" "6#/&%\"aG6#\"\"\"F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 51 "Differentiating both sides of (ii) with r espect to " }{TEXT 264 1 "x" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x) = 2*a[2]+3*`.`*2*a[3]*x+4*`.`* 3*a[4]*x^2+5*`.`*4*a[5]*x^3+` . . . `+n*(n-1)*a[n]*x^(n-2)+` . . . `; " "6#/-%$expG6#%\"xG,0*&\"\"#\"\"\"&%\"aG6#F*F+F+*,\"\"$F+%\".GF+F*F+& F-6#F0F+F'F+F+*,\"\"%F+F1F+F0F+&F-6#F5F+F'F*F+*,\"\"&F+F1F+F5F+&F-6#F9 F+F'F0F+%(~.~.~.~GF+**%\"nGF+,&F>F+F+!\"\"F+&F-6#F>F+)F',&F>F+F*F@F+F+ F F'F'F'F'*&F'F'-%*factorialG6#\"\"#!\"\"F'*&F'F'-F+6#\"\"$F.F'*&F'F'-F+ 6#\"\"%F.F'*&F'F'-F+6#\"\"&F.F'*&F'F'-F+6#\"\"'F.F'*&F'F'-F+6#\"\"(F.F '*&F'F'-F+6#\"\")F.F'*&F'F'-F+6#\"\"*F.F'*&F'F'-F+6#\"#5F.F'*&F'F'-F+6 #\"#6F.F'*&F'F'-F+6#\"#7F.F'%(~.~.~.~GF'" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1+1+1/2+1/6+1/24+1/120 +1/720+1/5040+1/40320+1/362880+1/3628800+1/39916800+1/479001600+` . . \+ . `" "6#/%!G,>\"\"\"F&F&F&*&F&F&\"\"#!\"\"F&*&F&F&\"\"'F)F&*&F&F&\"#CF )F&*&F&F&\"$?\"F)F&*&F&F&\"$?(F)F&*&F&F&\"%S]F)F&*&F&F&\"&?.%F)F&*&F&F &\"'!)GOF)F&*&F&F&\"(+)GOF)F&*&F&F&\")+o\"*RF)F&*&F&F&\"*+;+z%F)F&%(~. ~.~.~GF&" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 96 "Since adding successively more terms of the series gives progressively more accurate values for " }{XPPEDIT 18 0 "exp(1) " "6#-%$expG6#\"\"\"" }{TEXT -1 25 ", we say that the series " }{TEXT 260 9 "converges" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "exp(1)" "6#-%$exp G6#\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 74 "Indeed, fro m a theoretical point of view, it can be shown that the series " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+x+x^2/2!+x^3/3!+x^4 /4!+x^5/5!+` . . . `+x^n/n!+` . . . `" "6#,4\"\"\"F$%\"xGF$*&F%\"\"#-% *factorialG6#F'!\"\"F$*&F%\"\"$-F)6#F-F+F$*&F%\"\"%-F)6#F1F+F$*&F%\"\" &-F)6#F5F+F$%(~.~.~.~GF$*&)F%%\"nGF$-F)6#F;F+F$F8F$" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 13 "converges to " }{XPPEDIT 18 0 "exp(x)" " 6#-%$expG6#%\"xG" }{TEXT -1 21 " for any real number " }{TEXT 284 1 "x " }{TEXT -1 24 ", that is, the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x)=1+x+x^2/2!+x^3/3!+x^4/4!+x^5/5!+` . . \+ . `+x^n/n!+` . . . `" "6#/-%$expG6#%\"xG,4\"\"\"F)F'F)*&F'\"\"#-%*fact orialG6#F+!\"\"F)*&F'\"\"$-F-6#F1F/F)*&F'\"\"%-F-6#F5F/F)*&F'\"\"&-F-6 #F9F/F)%(~.~.~.~GF)*&)F'%\"nGF)-F-6#F?F/F)FN#e CGF-7$$!3(ommYUH3?\"F*$\"3=^Jxv\\W4IF-7$$!3B+++9u=N6F*$\"3BuStz)=O@$F- 7$$!3!pmmEN+t1\"F*$\"3'QzI^qd$RMF-7$$!3+,++g+J'***F-$\"3G/kD:@:!o$F-7$ $!3OLLL`wC+$*F-$\"3c!R%Q#RRa%RF-7$$!3Cnmm1b:(o)F-$\"3g'H!Hq,!\\>%F-7$$ !3;+++g*ep*zF-$\"3GGVJwil%\\%F-7$$!3a,+++%GRI(F-$\"3.6bUTr><[F-7$$!3(4 +++/lgj'F-$\"3s]de`m!*\\^F-7$$!3onmmE6eHgF-$\"3oD%[C)e!>Z&F-7$$!36NLL$ 48%3`F-$\"3[E-t!z*4\")eF-7$$!3]LLLt\"*[(p%F-$\"3!R(R@_@f^iF-7$$!3C,++? %Ro)RF-$\"3'*ze\\o!G?r'F-7$$!3=OLLtRzdLF-$\"3E!**)o4x!y9(F-7$$!3k+++?9 jnEF-$\"3wf@P([W&ewF-7$$!3n+++gMV5?F-$\"3Q!o**3zp(y\")F-7$$!3QLLLLjrC8 F-$\"3!)f<04yFf()F-7$$!3yVLLL&R,&p!#>$\"3$eCYtJ)eG$*F-7$$!3mSnmmm[z:!# ?$\"3mF\"f1gF-$\"3NT)=$35S<7F*7$$\"3a(*****>6 W_EF-$\"3s\\+w(>\\PI\"F*7$$\"3Y)*****RG$GK$F-$\"3QYRKKx9%R\"F*7$$\"3#z ******Hr9(RF-$\"3Cd'G3yuv[\"F*7$$\"31)*****>Pn\"p%F-$\"3+\\=rZDm)f\"F* 7$$\"3vhmmEw!)Q`F-$\"3o?9\\r#Qbq\"F*7$$\"3:,+++7wHgF-$\"3'z^B@s\\v#=F* 7$$\"3)4LLL$y'el'F-$\"3Ey.z]=jX>F*7$$\"3Z******fxOStF-$\"33>LMtTZ$3#F* 7$$\"3SimmY)GW)zF-$\"3+i-A,#y?A#F*7$$\"3s+++g#ewl)F-$\"3_bP-Cc#oP#F*7$ $\"3IlmmmI'eJ*F-$\"3'y%)*)p(G`QDF*7$$\"3f******\\U\\+5F*$\"30rDKrci>FF *7$$\"3]LLL*ygo1\"F*$\"3'eXY&f=C1HF*7$$\"3;LLLh,tM6F*$\"3rY6CTTL5JF*7$ $\"3zmmmSv.-7F*$\"3C!3?Dp))oK$F*7$$\"3#)*****>.')QE\"F*$\"3q/&>$!3inmmYd-ELF-7$F C$!3?LLL$zXAk#F-7$FH$!3nommYUH3?F-7$FM$!3G-++ST(=N\"F-7$FR$!3h*ommEN+t 'Fir7$FW$\"3Y.!******R**o$!#@7$Ffn$\"3OmmmmM_(*pFir7$F[o$\"3vKLL$\\WGJ \"F-7$F`o$\"3$)******R5/.?F-7$Feo$\"3W)******frgp#F-7$Fjo$\"3/******f \\$RO$F-7$F_p$\"3KKLLt)=/(RF-7$Fdp$\"3!\\mmm!pe\"p%F-7$Fip$\"3^mmmE3^- `F-7$F^q$\"3w)*****z0;8gF-7$Fcq$\"3#Qmmm-1Ak'F-7$Fhq$\"3O******z&oBL(F -7$F]r$\"3K******Rlc*)zF-7$Fbr$\"3immmmOGv')F-7$Fgr$\"3ilmmYg)\\I*F-7$ F]s$\"3gKLLL^?%)**F-7$Fcs$\"3;LLL(Gs*o5F*7$Fhs$\"3))*****R\"yQI6F*7$F] t$\"3CLLLl#=n>\"F*7$Fbt$\"3w*****>6W_E\"F*7$Fgt$\"3%)*****RG$GK8F*7$F \\u$\"3y******Hr9(R\"F*7$Fau$\"3#)*****>Pn\"p9F*7$Ffu$\"3=mmmi2)Q`\"F* 7$F[v$\"37+++?h(Hg\"F*7$F`v$\"35LLL$y'el;F*7$Fev$\"3&******fxOSt\"F*7$ Fjv$\"3Cmmm%)GW)z\"F*7$F_w$\"31+++Eewl=F*7$Fdw$\"3`mmm1jeJ>F*7$Fiw$\"3 f******\\U\\+?F*7$F^x$\"3]LLL*ygo1#F*7$Fcx$\"3;LLLh,tM@F*7$Fhx$\"3zmmm Sv.-AF*7$F]y$\"3#)*****>.')QE#F*7$Fby$\"3tmmm#QrZL#F*7$Fgy$\"3-LLL&3s \")R#F*7$F\\z$\"3=+++e/xlCF*7$Faz$\"3*)*****fsq/`#F*7$Ffz$\"33+++++++E F*-F[[l6&F][lFa[lFa[lF^[l-Fd[l6#\"\"\"-F$6%7S7$F($\"3g,++++++oF-7$F/$ \"3*4*Rgv0#eS'F-7$F4$\"32%Qp)R$HC5'F-7$F9$\"3.U\"e`)pB0eF-7$F>$\"3iyWP jB7`bF-7$FC$\"3'[Q>;9t!\\`F-7$FH$\"3%>sf!*Gi;?&F-7$FM$\"3wO+_%=y84&F-7 $FR$\"32*GXtoYE-&F-7$FW$\"34gGy!o+++&F-7$Ffn$\"3UvKLnE[C]F-7$F[o$\"3e$ y$=L!yh3&F-7$F`o$\"3*R@'Rq'31?&F-7$Feo$\"3=Gjh.,Wj`F-7$Fjo$\"3I4bv?H!e c&F-7$F_p$\"3)p2'[,8@)y&F-7$Fdp$\"3l..9&Q\\05'F-7$Fip$\"3ngeM`5$eS'F-7 $F^q$\"3o&HV!3]!z!oF-7$Fcq$\"3jZV.X]%f?(F-7$Fhq$\"35g&['\\9=)o(F-7$F]r $\"3[yV&[ne;>)F-7$Fbr$\"3Ism&[LFIw)F-7$Fgr$\"3(o1LkEQ\"H$*F-7$F]s$\"3+ $)RA2w@%)**F-7$Fcs$\"3?HLav3Nr5F*7$Fhs$\"31\\)>]I)))Q6F*7$F]t$\"3/67H. t1;7F*7$Fbt$\"3!RLv9L@/I\"F*7$Fgt$\"3?7HTu$*[(Q\"F*7$F\\u$\"33=OV^+,w9 F*7$Fau$\"3!HpuMQE#z:F*7$Ffu$\"3yO%Rq4&Rw;F*7$F[v$\"3OF^k?iw%y\"F*7$F` v$\"3wRtSm'*3()=F*7$Fev$\"3&pBg-xTM+#F*7$Fjv$\"3szSq/%)><@F*7$F_w$\"3? 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f@))36i(F-7$Fgx$!3O4*)\\eDQ*y)F-7$F\\y$!3;7U#f&4Z(f*F-7$Ffy$!3ue)f[h$) *o**F-7$Fjz$!3MNQ;h&fA#**F-7$Fd[l$!3t[^psg'pS*F-7$Fi[l$!3w[M)p(R()[&)F -7$F^\\l$!3G.@/=>vfsF-7$Fc\\l$!3\\3>`fYj1eF-7$Fh\\l$!3qm8Yw$[O(RF-7$Fb ]l$!398'\\tR%e'3#F-F[^l7$Fg^l$\"3)f.\\N`:p)>F-7$Fa_l$\"3H1S#QFbCF-7$F^gl$!3zr*4mmZ2L&F-7$Fhgl$!3 \\qcO'esQU)F-7$F]hl$!31!o&H)*3%e>\"F*7$Fbhl$!3(\\>(*=\\*p8;F*7$Fghl$!3 (G7='*fOn5#F*7$Fail$!3iO='y**[wy#F*7$Fejl$!3i2_>=n&3k$F*7$Fi[m$!347ic' fL8%[F*7$Fc\\m$!3mtAAbvV\"H'F*7$Fh\\m$!3N=Y1\"QS9V)F*7$F]]m$!3bI9E^YK< 6F]hm7$Fb]m$!3#*>BTW<-y9F]hm7$Fg]m$!3A3)3LZtJ(>F]hm7$Fa^m$!3u,/s?jIjDF ]hm7$F[_m$!3B*QzH.`DL$F]hm7$Fe_m$!3Gi#3/=C+U%F]hm7$F_`m$!37]eUm^KncF]h m7$Fd`m$!3yL$H'Q,>bsF]hm7$Fi`m$!3c_J2;*4sD*F]hm7$$\"3M^8NJZ4%H(F*$!3Z' )[q.s#G.\"Fdfn7$F^am$!3#>S7&*3F0:\"Fdfn7$$\"3%o-F5o&4&\\(F*$!3I)p2l@/@ H\"Fdfn7$Fcam$!3E7%)4F9[[9Fdfn-%&COLORG6&FjamF^bm$\"\")!\"\"F^bmF`bm-% %TEXTG6%7$$\"#wF_dp$!\"#F_dpQ\"x6\"-Fham6&FjamF_bmF_bmF_bm-Fadp6%7$$! \"$F_dp$\"#DF_dpQ\"yFidpFjdp-Fadp6%7$$\"#jF_dp$\"\"*F_dpQ*y~=~sin~xFid pFgam-%+AXESLABELSG6%%!GF_fp-%%FONTG6#%(DEFAULTG-%*AXESTICKSG6$\"\"(\" \"'-%%VIEWG6$;$!#wF_dp$\"#xF_dp;$!#DF_dpFaep" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "In fact the two s eries above converge to " }{XPPEDIT 18 0 "sin*x" "6#*&%$sinG\"\"\"%\"x GF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xG F%" }{TEXT -1 44 " respectively for all real number values of " } {TEXT 286 1 "x" }{TEXT -1 44 ". The convergence is quite rapid as long as " }{TEXT 287 1 "x" }{TEXT -1 28 " is not too far away from 0." }} {PARA 0 "" 0 "" {TEXT -1 29 "For example, we can estimate " }{XPPEDIT 18 0 "sin*1" "6#*&%$sinG\"\"\"F%F%" }{TEXT -1 60 " (the sine of one ra dian) by means of the truncated series: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1-1/3!+1/5!-1/7!+1/9!-1/11!" "6#,.\"\"\"F$*&F $F$-%*factorialG6#\"\"$!\"\"F**&F$F$-F'6#\"\"&F*F$*&F$F$-F'6#\"\"(F*F* *&F$F$-F'6#\"\"*F*F$*&F$F$-F'6#\"#6F*F*" }{TEXT -1 1 " " }{TEXT 281 1 "~" }{TEXT -1 16 " 0.84147098465. " }}{PARA 0 "" 0 "" {TEXT -1 10 "Not e that " }{XPPEDIT 18 0 "sin*1" "6#*&%$sinG\"\"\"F%F%" }{TEXT -1 1 " \+ " }{TEXT 282 1 "~" }{TEXT -1 89 " 0.841470985 correct to 9 digits, so \+ the value given by the truncated series agrees with " }{XPPEDIT 18 0 " sin*1" "6#*&%$sinG\"\"\"F%F%" }{TEXT -1 14 " to 9 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 87 "The expo nential function of a complex variable defined by a series and Euler's formula " }}{PARA 0 "" 0 "" {TEXT -1 21 "For a complex number " } {TEXT 288 1 "z" }{TEXT -1 16 ", we can define " }{XPPEDIT 18 0 "exp(z) " "6#-%$expG6#%\"zG" }{TEXT -1 33 " by means of the infinite series " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(z) = 1+z+z^2/2! +z^3/3!+z^4/4!+z^5/5!+z^6/6!+z^7/7!+z^8/8!+` . . . `;" "6#/-%$expG6#% \"zG,6\"\"\"F)F'F)*&F'\"\"#-%*factorialG6#F+!\"\"F)*&F'\"\"$-F-6#F1F/F )*&F'\"\"%-F-6#F5F/F)*&F'\"\"&-F-6#F9F/F)*&F'\"\"'-F-6#F=F/F)*&F'\"\"( -F-6#FAF/F)*&F'\"\")-F-6#FEF/F)%(~.~.~.~GF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "z=i" "6#/%\"zG%\"iG" } {TEXT -1 8 ", where " }{TEXT 289 1 "i" }{TEXT -1 32 " is the imaginary unit, we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ex p(i) = 1+i+i^2/2!+i^3/3!+i^4/4!+i^5/5!+i^6/6!+i^7/7!+i^8/8!+` . . . ` " "6#/-%$expG6#%\"iG,6\"\"\"F)F'F)*&F'\"\"#-%*factorialG6#F+!\"\"F)*&F '\"\"$-F-6#F1F/F)*&F'\"\"%-F-6#F5F/F)*&F'\"\"&-F-6#F9F/F)*&F'\"\"'-F-6 #F=F/F)*&F'\"\"(-F-6#FAF/F)*&F'\"\")-F-6#FEF/F)%(~.~.~.~GF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "i^2=-1" "6#/*$%\"iG\"\"#,$\"\"\"!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "i^3=-i" "6#/*$%\"iG\"\"$,$F%!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "i^4=1" "6#/*$%\"iG\"\"%\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "i^5=i" "6#/*$%\"iG\"\"&F%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "i^6=i^2" "6#/*$%\"iG\"\"'*$F%\"\"#" }{XPPEDIT 18 0 "`` =-1" "6#/%!G,$\"\"\"!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "i^7=i^3" " 6#/*$%\"iG\"\"(*$F%\"\"$" }{XPPEDIT 18 0 "``=-i" "6#/%!G,$%\"iG!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "i^8=1" "6#/*$%\"iG\"\")\"\"\"" } {TEXT -1 17 ", etc., we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "exp(i) = 1+i-1/2!-i/3!+1/4!+i/5!-1/6!-i/7!+1/8!+` . . . `;" "6#/-%$expG6#%\"iG,6\"\"\"F)F'F)*&F)F)-%*factorialG6#\"\"#!\"\"F/ *&F'F)-F,6#\"\"$F/F/*&F)F)-F,6#\"\"%F/F)*&F'F)-F,6#\"\"&F/F)*&F)F)-F,6 #\"\"'F/F/*&F'F)-F,6#\"\"(F/F/*&F)F)-F,6#\"\")F/F)%(~.~.~.~GF)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(1-1/2!+1/4!-1/6!+1/8!+`. . . `) +``(i-i/3!+i/5!-i/7!+` . . . `);" "6#/%!G,&-F$6#,.\"\"\"F)*&F)F)-%*fac torialG6#\"\"#!\"\"F/*&F)F)-F,6#\"\"%F/F)*&F)F)-F,6#\"\"'F/F/*&F)F)-F, 6#\"\")F/F)%'.~.~.~GF)F)-F$6#,,%\"iGF)*&F@F)-F,6#\"\"$F/F/*&F@F)-F,6# \"\"&F/F)*&F@F)-F,6#\"\"(F/F/%(~.~.~.~GF)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(1-1/2!+1/4!-1/6!+1/8!+`. . . `)+i*``(1-1/3!+1/5!-1/7!+` \+ . . . `);" "6#/%!G,&-F$6#,.\"\"\"F)*&F)F)-%*factorialG6#\"\"#!\"\"F/*& F)F)-F,6#\"\"%F/F)*&F)F)-F,6#\"\"'F/F/*&F)F)-F,6#\"\")F/F)%'.~.~.~GF)F )*&%\"iGF)-F$6#,,F)F)*&F)F)-F,6#\"\"$F/F/*&F)F)-F,6#\"\"&F/F)*&F)F)-F, 6#\"\"(F/F/%(~.~.~.~GF)F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = cos*1+ i*sin*1;" "6#/%!G,&*&%$cosG\"\"\"F(F(F(*(%\"iGF(%$sinGF(F(F(F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "More generally, if " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 30 " is any real number, we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(i*theta) = 1+i*theta+i^2*theta^2/2! +i^3*theta^3/3!+i^4*theta^4/4!+i^5*theta^5/5!+i^6*theta^6/6!+i^7*theta ^7/7!+i^8*theta^8/8!+` . . . `;" "6#/-%$expG6#*&%\"iG\"\"\"%&thetaGF), 6F)F)*&F(F)F*F)F)*(F(\"\"#F*F.-%*factorialG6#F.!\"\"F)*(F(\"\"$F*F4-F0 6#F4F2F)*(F(\"\"%F*F8-F06#F8F2F)*(F(\"\"&F*F<-F06#F7$F<$\"35+++++++SF?F0F4-F$6%7$7$$! \"\"F)F=7$FGFAF0F4-F$6%7$7$F=F<7$FAFjubk6'F-$\"3!\\@@&\\!>8\"zF-7$$\"3VGuIc:x5]F-$\"3-$ >p(Qh-a')F-7$$\"37C3nW'R,#QF-$\"3aK+16bcT#*F-7$$\"3$oY@iF'QDDF-$\"3s&Q \"[M\"oen*F-7$$\"3OB^hAo8X8F-$\"33)fVPO<\"4**F-7$$!3+qB/u(p5(f!#@$\"2% HhJ<#)******!#<7$$!33)\\T#fB[i8F-$\"3ap%>wGZn!**F-7$$!3)e:d.:#=YEF-$\" 3PA>=\\D`V'*F-7$$!3%*yV1J*3Jx$F-$\"3IzF#e`m3E*F-7$$!3V(\\AOF:B/&F-$\"3 vU'yI2&oN')F-7$$!3[igNw%*[RgF-$\"3!G;)RQ+BqzF-7$$!3/XUwYjJ*3(F-$\"3Afv Og?x_qF-7$$!3&=e_b?/U!zF-$\"3u3HvXQF-7$$!3!p!RS4Nij'*F-$\"3o$p9m#Q%= d#F-7$$!3#p(pT>+.2**F-$\"31V?00]Ug8F-7$$!3/gKG4>&*****F-$\"3vCA\\O%485 $!#?7$$!3__Is=!Q%3**F-$!3q#4*>c=8]8F-7$$!3)>b`sYlSn*F-$!3UNbFGIGKDF-7$ $!37_J.8AEj#*F-$!3q4&=j`Bsw$F-7$$!3%\\F)4Kh=u')F-$!3kENX')3zv\\F-7$$!3 EB*z7xng%zF-$!37W(H!pUCrgF-7$$!3XD84Pfc5rF-$!3Y?#o?\"yMJqF-7$$!3!Q%y6I ke[gF-$!3)4j2yeGL'zF-7$$!3g@UD=%)o!*\\F-$!3I_Mh6Mil')F-7$$!3o+1s\"[\"y sPF-$!3#\\IN,%***4E*F-7$$!30yCH\"el'3EF-$!3=V>(fp[Pl*F-7$$!3g67Cvnc\"H \"F-$!3;$RoI*>C;**F-7$$!3EzyOHMQdIF\\v$!3W1O#>E`*****F-7$$\"3GIAj`Ks(G \"F-$!3lYZ3X=u;**F-7$$\"3EKRqX8/bDF-$!3U$[o%H'z!o'*F-7$$\"3@%)yb&Q)zNQ F-$!36.2<#>x]B*F-7$$\"3O7w4w0I.]F-$!3wHgb5wMe')F-7$$\"3@\\#)[*R]$4hF-$ !3/a*[=H2o\"zF-7$$\"33/wY'Q+$*4(F-$!3)4d)eg?sUqF-7$$\"3[f=s$Ry,!zF-$!3 D1$H)zD(p_v\\F-7$$\"3!G$e@`4+D#*F-$!3i &>2ndo*fQF-7$$\"31mGAp))oa'*F-$!3%)f]nRQ=0EF-7$$\"3@zb'f`bp!**F-$!3/up 91t'4O\"F-7$F<$\"36YKhSr8/#)!#F-F16&F3$\"*++++\"!\")F(F(F4-F$6'7#F*-%' SYMBOLG6#%'CIRCLEGF0-%&STYLEG6#%&POINTGF4-F$6'F[^l-F]^l6#%(DIAMONDGF0F `^lF4-F$6'F[^l-F]^l6#%&CROSSGF0F`^lF4-F$6$7S7$$\"3++++++++DF-F(7$$\"33 M&G'\\([$*\\#F-$\"39Go\"Q4')fq&F\\v7$$\"3=#**p*eEs(\\#F-$\"3Wc?&e,Uo1 \"F?7$$\"3'eh,0-\"f!QaA\\[#F-$\"3$G;K(H:`TFF?7$$\"3G%Q([L1qyCF-$\"3+]c \"G-FkD$F?7$$\"394x;sS8rCF-$\"3GkjDL=4)y$F?7$$\"3%HDxXN4@Y#F-$\"3Q/+ux ()4OVF?7$$\"37!o$z)G/>X#F-$\"3C&eeJ$)=-)[F?7$$\"3uT17DF:SCF-$\"3x$*f&e q8uV&F?7$$\"3W5K6ZIvGCF-$\"3!zyb]&R'e#fF?7$$\"3(Q%4/h,v9CF-$\"3UL:pk#z GZ'F?7$$\"33EfGW.X*R#F-$\"3_b<)p)y\")=qF?7$$\"3w8r^P$QNQ#F-$\"31YbAs$= :a(F?7$$\"3%Ru)=I@5oBF-$\"3,\\a&3@mI,)F?7$$\"3m,$piYL&[BF-$\"3iq:jB!f' p&)F?7$$\"3rM$*[LC$4L#F-$\"3\"=\"[\\wUWP!*F?7$$\"3-fc)Ge'G4BF-$\"3iKx0 2B.x&*F?7$$\"3?$*H(Q$R3*G#F-$\"3hY:v(pM]+\"F-7$$\"3]()yDBG!eE#F-$\"3UJ k%*>TZc5F-7$$\"3XfjTa#fDC#F-$\"3GgdR.h&\\5\"F-7$$\"3O$e*)GZ,s@#F-$\"3[ !Q?*3o*\\:\"F-7$$\"3ymKeM9$H>#F-$\"3%4_@!Q\"Q/?\"F-7$$\"3gfweF$4d;#F-$ \"3'=xkAu!))[7F-7$$\"3\\9_9;0IO@F-$\"37+uI!QX&)H\"F-7$$\"36C:7=Cx4@F-$ \"3t4qV%*f@T8F-7$$\"3-%e*zhW;!3#F-$\"3c#=t)*R#p'Q\"F-7$$\"3#HBY9VZ&[?F -$\"3d#G?ee()HV\"F-7$$\"3r`q%zO>m,#F-$\"3yJ1OT7ex9F-7$$\"3m!=$zML![)>F -$\"31pFaB70?:F-7$$\"3%\\i8z)*H%[>F-$\"37)[R)4PSm:F-7$$\"3Li$ya>B[\">F -$\"3X3oM#37tg\"F-7$$\"3_J[7JK*z(=F-$\"3o#G;%fV>];F-7$$\"3=%Q#*e4!zV=F -$\"3K[RSXSK)o\"F-7$$\"3og+$**)=^0=F-$\"3[ory:$)>HF-7$$\"3-`=5$GF-7$$\"3y(p.r u!e?:F-$\"31&*R')*o(R%)>F-7$$\"3#G!p&443,[\"F-$\"3-L%QZ]lZ,#F-7$$\"3j& )[+*>xHV\"F-$\"3=*oHrpa&[?F-7$$\"3!)e\\Bj(o,R\"F-$\"3%R(*Rg#G%y2#F-7$$ \"3k0YoXt'QM\"F-$\"3xo]pqX33@F-7$$\"3W/:D]Z$*)H\"F-$\"3%*3d\"**yjg8#F- 7$$\"3uYcU+++]7F-$\"3Y\\`@4N1l@F-F0-%%TEXTG6%7$$\"#&)!\"#$\"#**Fj^mQ6e ~~~~~=~cos~~~+~i~sin6\"F0-Fe^m6%7$$\"#bFj^m$\"$0\"Fj^mQ\"iF^_mF0-Fe^m6 %7$Fd_m$Fh]lFj^mQ\"1F^_mF0-Fe^m6%7$$!$0\"Fj^mFj_mQ#-1F^_mF0-Fe^m6%7$$! \"'Fj^m$\"$2\"Fj^mFf_mF0-Fe^m6%7$Fe`m$!$2\"Fj^mQ#-iF^_mF0-Fe^m6%7$$\"# 8FHFe`mQ%RealF^_mF0-Fe^m6%7$$!#:Fj^mFbamQ%ImagF^_mF0-Fe^m6&7$$\"#:Fj^m $F7FHQ\"qF^_mF0-%%FONTG6$F]^l\"#6-Fe^m6&7$$\"#$*Fj^m$\"$$**!\"$FabmF0F bbm-Fe^m6&7$FbamF[cmFabmF0Fbbm-Fe^m6&7$$\"\"'FHFd_mFabmF0Fbbm-%+AXESLA BELSG6%Q!F^_mFicm-Fcbm6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%*AXESTI CKSG6$F)F)-%%VIEWG6$F\\dmF\\dm" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" }}{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "For examp le, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(Pi/4*i)=c os(Pi/4)+i*sin(Pi/4)" "6#/-%$expG6#*(%#PiG\"\"\"\"\"%!\"\"%\"iGF),&-%$ cosG6#*&F(F)F*F+F)*&F,F)-%$sinG6#*&F(F)F*F+F)F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``=sqrt(2)/2+sqrt(2)/2" "6#/%!G,&*&-%%sqrtG6#\"\"#\"\" \"F*!\"\"F+*&-F(6#F*F+F*F,F+" }{TEXT -1 1 " " }{TEXT 290 1 "i" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(Pi/3* i) = cos(Pi/3)+i*sin(Pi/3);" "6#/-%$expG6#*(%#PiG\"\"\"\"\"$!\"\"%\"iG F),&-%$cosG6#*&F(F)F*F+F)*&F,F)-%$sinG6#*&F(F)F*F+F)F)" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`` = 1/2+sqrt(3)/2;" "6#/%!G,&*&\"\"\"F'\"\"#!\"\"F' *&-%%sqrtG6#\"\"$F'F(F)F'" }{TEXT -1 1 " " }{TEXT 291 1 "i" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(Pi/2*i) \+ = cos(Pi/2)+i*sin(Pi/2);" "6#/-%$expG6#*(%#PiG\"\"\"\"\"#!\"\"%\"iGF), &-%$cosG6#*&F(F)F*F+F)*&F,F)-%$sinG6#*&F(F)F*F+F)F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "`` = i;" "6#/%!G%\"iG" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(3*Pi/4*i) = cos(3*Pi/4)+i*sin( 3*Pi/4);" "6#/-%$expG6#**\"\"$\"\"\"%#PiGF)\"\"%!\"\"%\"iGF),&-%$cosG6 #*(F(F)F*F)F+F,F)*&F-F)-%$sinG6#*(F(F)F*F)F+F,F)F)" }{TEXT -1 1 " " } {XPPEDIT 18 0 "`` = -sqrt(2)/2+sqrt(2)/2;" "6#/%!G,&*&-%%sqrtG6#\"\"# \"\"\"F*!\"\"F,*&-F(6#F*F+F*F,F+" }{TEXT -1 1 " " }{TEXT 292 1 "i" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp (Pi*i) = cos*Pi+i*sin*Pi;" "6#/-%$expG6#*&%#PiG\"\"\"%\"iGF),&*&%$cosG F)F(F)F)*(F*F)%$sinGF)F(F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1; " "6#/%!G,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "The exponential form of a g eneral complex number " }}{PARA 0 "" 0 "" {TEXT -1 25 "A general compl ex number " }{XPPEDIT 18 0 "z=a+b*i" "6#/%\"zG,&%\"aG\"\"\"*&%\"bGF'% \"iGF'F'" }{TEXT -1 55 " can be expressed in the polar (modulus-argume nt) form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z=r*(cos *theta+i*sin*theta)" "6#/%\"zG*&%\"rG\"\"\",&*&%$cosGF'%&thetaGF'F'*(% \"iGF'%$sinGF'F+F'F'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 " where " }{XPPEDIT 18 0 "r=abs(z)" "6#/%\"rG-%$absG6#%\"zG" }{XPPEDIT 18 0 "``=sqrt(a^2+b^2)" "6#/%!G-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*$%\"bGF+ F," }{TEXT -1 19 " is the modulus of " }{TEXT 293 1 "z" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "theta = arg(z);" "6#/%&thetaG-%$argG6#%\"zG" } {TEXT -1 44 " is (the principal form of) the argument of " }{TEXT 295 2 "z." }}{PARA 0 "" 0 "" {TEXT -1 22 "Using Euler's formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(i*theta)=cos*theta+i*si n*theta" "6#/-%$expG6#*&%\"iG\"\"\"%&thetaGF),&*&%$cosGF)F*F)F)*(F(F)% $sinGF)F*F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "we see t hat " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z=r*exp(i*the ta)" "6#/%\"zG*&%\"rG\"\"\"-%$expG6#*&%\"iGF'%&thetaGF'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "This is the " }{TEXT 260 16 "exp onential form" }{TEXT -1 23 " of the complex number " }{TEXT 296 1 "z " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 83 "Incorporating the st andard notation for the polar form of a complex number we have " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z = r*exp(i*theta) " "6#/%\"zG*&%\"rG\"\"\"-%$expG6#*&%\"iGF'%&thetaGF'F'" }{XPPEDIT 18 0 " `` = r*`/_`*theta;" "6#/%!G*(%\"rG\"\"\"%#/_GF'%&thetaGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "z[1] = r[1 ]*(cos*theta[1]+i*sin*theta[1]);" "6#/&%\"zG6#\"\"\"*&&%\"rG6#F'F',&*& %$cosGF'&%&thetaG6#F'F'F'*(%\"iGF'%$sinGF'&F06#F'F'F'F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z[2] = r[2]*(cos*theta[2]+i*sin*theta[2]);" "6# /&%\"zG6#\"\"#*&&%\"rG6#F'\"\"\",&*&%$cosGF,&%&thetaG6#F'F,F,*(%\"iGF, %$sinGF,&F16#F'F,F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 " The product of " }{XPPEDIT 18 0 "z[1];" "6#&%\"zG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z[2];" "6#&%\"zG6#\"\"#" }{TEXT -1 3 " is" } }{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "z[1]*z[2] = r[1]* r[2]*(cos*theta[1]+i*sin*theta[1])*(cos*theta[2]+i*sin*theta[2]);" "6# /*&&%\"zG6#\"\"\"F(&F&6#\"\"#F(**&%\"rG6#F(F(&F.6#F+F(,&*&%$cosGF(&%&t hetaG6#F(F(F(*(%\"iGF(%$sinGF(&F66#F(F(F(F(,&*&F4F(&F66#F+F(F(*(F9F(F: F(&F66#F+F(F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " = \+ " }{XPPEDIT 18 0 "r[1]*r[2]*``(cos*theta[1]*cos*theta[2]-sin*theta[1]* sin*theta[2]+i*``(sin*theta[1]*cos*theta[2]+cos*theta[1]*sin*theta[2]) );" "6#*(&%\"rG6#\"\"\"F'&F%6#\"\"#F'-%!G6#,(**%$cosGF'&%&thetaG6#F'F' F0F'&F26#F*F'F'**%$sinGF'&F26#F'F'F7F'&F26#F*F'!\"\"*&%\"iGF'-F,6#,&** F7F'&F26#F'F'F0F'&F26#F*F'F'**F0F'&F26#F'F'F7F'&F26#F*F'F'F'F'F'" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "r[1 ]*r[2]*``(cos(theta[1]+theta[2])+i*sin(theta[1]+theta[2]));" "6#*(&%\" rG6#\"\"\"F'&F%6#\"\"#F'-%!G6#,&-%$cosG6#,&&%&thetaG6#F'F'&F46#F*F'F'* &%\"iGF'-%$sinG6#,&&F46#F'F'&F46#F*F'F'F'F'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 56 "that is, complex numbers in polar form are mult iplied by" }{TEXT 260 48 " multiplying the moduli and adding the argum ents" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Using the exponential notation we have " }{XPPEDIT 18 0 "z[1]=r[1]*exp(i*theta[1])" "6#/&%\"zG6#\"\"\"*&&%\"rG6#F'F'-%$expG6 #*&%\"iGF'&%&thetaG6#F'F'F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z[2]= r[2]*exp(i*theta[2])" "6#/&%\"zG6#\"\"#*&&%\"rG6#F'\"\"\"-%$expG6#*&% \"iGF,&%&thetaG6#F'F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Also " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z[1]*z[2] =r[1]*r[2]*exp(i*(theta[1]+theta[2]))" "6#/*&&%\"zG6#\"\"\"F(&F&6#\"\" #F(*(&%\"rG6#F(F(&F.6#F+F(-%$expG6#*&%\"iGF(,&&%&thetaG6#F(F(&F96#F+F( F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "From another point of view, it turns out that the " } {TEXT 260 18 "rule for exponents" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(u)*`.`*exp(v) = exp(u+v);" "6#/*(-% $expG6#%\"uG\"\"\"%\".GF)-F&6#%\"vGF)-F&6#,&F(F)F-F)" }{TEXT -1 2 ", \+ " }{TEXT 297 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "which we know holds w hen " }{TEXT 298 1 "u" }{TEXT -1 5 " and " }{TEXT 299 1 "v" }{TEXT -1 67 " are real numbers, still applies in the more general context where " }{TEXT 300 1 "u" }{TEXT -1 5 " and " }{TEXT 301 1 "v" }{TEXT -1 26 " are complex numbers (and " }{XPPEDIT 18 0 "exp(u)" "6#-%$expG6#%\"uG " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "exp(v)" "6#-%$expG6#%\"vG" } {TEXT -1 54 " are defined by the corresponding series expansions). " } }{PARA 0 "" 0 "" {TEXT -1 176 "The multiplication rule for complex num bers in polar form (and correspondingly also in exponential form) can \+ be interpreted as being an example of the rule of exponents above. " } }{PARA 0 "" 0 "" {TEXT -1 7 "Indeed " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "z[1]*`.`*z[2] = r[1]*exp(i*theta[1])*`.`*r[2]*exp( i*theta[2]);" "6#/*(&%\"zG6#\"\"\"F(%\".GF(&F&6#\"\"#F(*,&%\"rG6#F(F(- %$expG6#*&%\"iGF(&%&thetaG6#F(F(F(F)F(&F/6#F,F(-F26#*&F5F(&F76#F,F(F( " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "``= r[1]*r[2]*exp (i*theta[1]+i*theta[2])" "6#/%!G*(&%\"rG6#\"\"\"F)&F'6#\"\"#F)-%$expG6 #,&*&%\"iGF)&%&thetaG6#F)F)F)*&F2F)&F46#F,F)F)F)" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= r[1]*r[2]*exp(i*( theta[1]+theta[2]))" "6#/%!G*(&%\"rG6#\"\"\"F)&F'6#\"\"#F)-%$expG6#*&% \"iGF),&&%&thetaG6#F)F)&F46#F,F)F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "Similarly, " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "z[1]/z[2] = r[1]*exp(i*theta[1])/(r[2]*exp(i*theta[2])) ;" "6#/*&&%\"zG6#\"\"\"F(&F&6#\"\"#!\"\"*(&%\"rG6#F(F(-%$expG6#*&%\"iG F(&%&thetaG6#F(F(F(*&&F/6#F+F(-F26#*&F5F(&F76#F+F(F(F," }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=r[1]/r[2]" "6# /%!G*&&%\"rG6#\"\"\"F)&F'6#\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(i*(theta[1]-theta[2]))" "6#-%$expG6#*&%\"iG\"\"\",&&%&thetaG6#F(F (&F+6#\"\"#!\"\"F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 15 "us ing the rule " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "exp(u)/exp(v)=exp(u- v)" "6#/*&-%$expG6#%\"uG\"\"\"-F&6#%\"vG!\"\"-F&6#,&F(F)F,F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 22 "which also holds when " } {TEXT 302 1 "u" }{TEXT -1 5 " and " }{TEXT 303 1 "v" }{TEXT -1 30 " ar e general complex numbers. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Note that the value of the exponential function at a complex number " }{XPPEDIT 18 0 "z=a+b*i" "6#/%\"zG,&%\"aG\"\"\" *&%\"bGF'%\"iGF'F'" }{TEXT -1 34 " given in rectangular form (where " }{TEXT 324 1 "a" }{TEXT -1 5 " and " }{TEXT 325 1 "b" }{TEXT -1 100 " \+ are real numbers) can be obtained by a method which does not require u sing the series expansion of " }{XPPEDIT 18 0 "exp(z)" "6#-%$expG6#%\" zG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(z)=exp(a+b*i)" "6#/-%$expG6#%\"zG-F%6#,&%\"aG\"\"\"*&%\"bGF,%\"iG F,F," }{XPPEDIT 18 0 "``=exp(a)*exp(i*b)" "6#/%!G*&-%$expG6#%\"aG\"\" \"-F'6#*&%\"iGF*%\"bGF*F*" }{XPPEDIT 18 0 "``=exp(a)*(cos*b+i*sin*b)" "6#/%!G*&-%$expG6#%\"aG\"\"\",&*&%$cosGF*%\"bGF*F**(%\"iGF*%$sinGF*F.F *F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Examples: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(2+Pi/2*i)=exp(2)* (cos(Pi/2)+i*sin(Pi/2))" "6#/-%$expG6#,&\"\"#\"\"\"*(%#PiGF)F(!\"\"%\" iGF)F)*&-F%6#F(F),&-%$cosG6#*&F+F)F(F,F)*&F-F)-%$sinG6#*&F+F)F(F,F)F)F )" }{XPPEDIT 18 0 "``=exp(2)*i" "6#/%!G*&-%$expG6#\"\"#\"\"\"%\"iGF*" }{TEXT -1 1 " " }{TEXT 327 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "7.389 056099*i" "6#*&-%&FloatG6$\"+*4c!*Q(!\"*\"\"\"%\"iGF)" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-1+Pi/4*i)=exp(-1 )*(cos(Pi/4)+i*sin(Pi/4))" "6#/-%$expG6#,&\"\"\"!\"\"*(%#PiGF(\"\"%F)% \"iGF(F(*&-F%6#,$F(F)F(,&-%$cosG6#*&F+F(F,F)F(*&F-F(-%$sinG6#*&F+F(F,F )F(F(F(" }{XPPEDIT 18 0 "``=1/exp(1)" "6#/%!G*&\"\"\"F&-%$expG6#F&!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(sqrt(2)/2+sqrt(2)/2*i);" "6#-%! G6#,&*&-%%sqrtG6#\"\"#\"\"\"F+!\"\"F,*(-F)6#F+F,F+F-%\"iGF,F," } {XPPEDIT 18 0 "``=(sqrt(2)/(2*exp(1))*(1+i)" "6#/%!G*(-%%sqrtG6#\"\"# \"\"\"*&F)F*-%$expG6#F*F*!\"\",&F*F*%\"iGF*F*" }{TEXT -1 1 " " }{TEXT 326 1 "~" }{TEXT -1 3 " 0." }{XPPEDIT 18 0 "2601300475+0" "6#,&\"+v/I, E\"\"\"\"\"!F%" }{TEXT -1 1 "." }{XPPEDIT 18 0 "2601300475*i" "6#*&\"+ v/I,E\"\"\"%\"iGF%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 61 "Trigonometric and hyperbolic fun ctions of a complex variable " }}{PARA 0 "" 0 "" {TEXT -1 41 "The hype rbolic sine and cosine functions " }{XPPEDIT 18 0 "sinh*x" "6#*&%%sinh G\"\"\"%\"xGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cosh*x" "6#*&%%cos hG\"\"\"%\"xGF%" }{TEXT -1 18 " of a real number " }{TEXT 305 1 "x" } {TEXT -1 16 " are defined by " }{XPPEDIT 18 0 "sinh*x=(exp(x)-exp(-x)) /2" "6#/*&%%sinhG\"\"\"%\"xGF&*&,&-%$expG6#F'F&-F+6#,$F'!\"\"F0F&\"\"# F0" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cosh*x=(exp(x)+exp(-x))/2" "6# /*&%%coshG\"\"\"%\"xGF&*&,&-%$expG6#F'F&-F+6#,$F'!\"\"F&F&\"\"#F0" } {TEXT -1 14 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 20 "Using the \+ relations " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[sinh*x] = cosh*x;" "6#/7#*&%%sinhG\"\"\"%\"xGF'*&%%coshGF'F(F'" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[cosh*x] = sinh*x;" "6#/7#*&%%coshG\"\"\"%\"xGF'*&%%sin hGF'F(F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 29 "together wit h the facts that " }{XPPEDIT 18 0 "sinh*0=0" "6#/*&%%sinhG\"\"\"\"\"!F &F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cosh*0=1" "6#/*&%%coshG\"\"\" \"\"!F&F&" }{TEXT -1 49 ", it is possible to obtain the series expansi ons:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*x = x+x^ 3/3!+x^5/5!+x^7/7!+x^9/9!+x^11/11!+` . . . `;" "6#/*&%%sinhG\"\"\"%\"x GF&,0F'F&*&F'\"\"$-%*factorialG6#F*!\"\"F&*&F'\"\"&-F,6#F0F.F&*&F'\"\" (-F,6#F4F.F&*&F'\"\"*-F,6#F8F.F&*&F'\"#6-F,6#F " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Question 1 " }}{PARA 0 "" 0 "" {TEXT -1 30 "Obt ain the series expansions: " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "cos*x=1-x^2/2!+x^4/4!-x^6/6!+x^8/8!-x^10/10!+` . . . ` " "6#/*&%$cosG\"\"\"%\"xGF&,0F&F&*&F'\"\"#-%*factorialG6#F*!\"\"F.*&F' \"\"%-F,6#F0F.F&*&F'\"\"'-F,6#F4F.F.*&F'\"\")-F,6#F8F.F&*&F'\"#5-F,6#F " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 32 "________________________________" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 11 "Question 2 " }}{PARA 0 "" 0 "" {TEXT -1 27 "Use th e series expansion: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x = 1-x^2/2!+x^4/4!-x^6/6!+x ^8/8!-x^10/10!+x^12/12!-` . . . `;" "6#/*&%$cosG\"\"\"%\"xGF&,2F&F&*&F '\"\"#-%*factorialG6#F*!\"\"F.*&F'\"\"%-F,6#F0F.F&*&F'\"\"'-F,6#F4F.F. *&F'\"\")-F,6#F8F.F&*&F'\"#5-F,6#F " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "____________ ____________________" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Question 3 " }}{PARA 0 "" 0 "" {TEXT -1 78 "Use Maple, or a graphics calculato r, to compare the graphs of the polynomials " }{XPPEDIT 18 0 "1-x^2/2 " "6#,&\"\"\"F$*&%\"xG\"\"#F'!\"\"F(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "1-x^2/2+x^4/24" "6#,(\"\"\"F$*&%\"xG\"\"#F'!\"\"F(*&F&\"\"%\"#CF(F$" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "1-x^2/2+x^4/24-x^6/720" "6#,*\"\"\"F$ *&%\"xG\"\"#F'!\"\"F(*&F&\"\"%\"#CF(F$*&F&\"\"'\"$?(F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 36 "obtained by truncating the series \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "1-x^2/2!+x^4/4!-x^6/6!+x^8/8!-x^10/10!+x^12/12!-` . \+ . . `;" "6#,2\"\"\"F$*&%\"xG\"\"#-%*factorialG6#F'!\"\"F+*&F&\"\"%-F)6 #F-F+F$*&F&\"\"'-F)6#F1F+F+*&F&\"\")-F)6#F5F+F$*&F&\"#5-F)6#F9F+F+*&F& \"#7-F)6#F=F+F$%(~.~.~.~GF+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 20 ", with the graph of " }{XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"% \"xGF%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "_____________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Question 4 " }}{PARA 0 "" 0 "" {TEXT -1 92 "Express the following complex numbers (given in exponential form) \+ in the (rectangular) form " }{XPPEDIT 18 0 "a+b*i" "6#,&%\"aG\"\"\"*&% \"bGF%%\"iGF%F%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) \+ " }{XPPEDIT 18 0 "exp(Pi/6*i)" "6#-%$expG6#*(%#PiG\"\"\"\"\"'!\"\"%\"i GF(" }{TEXT -1 6 " (b) " }{XPPEDIT 18 0 "exp(2*Pi/3*i)" "6#-%$expG6#* *\"\"#\"\"\"%#PiGF(\"\"$!\"\"%\"iGF(" }{TEXT -1 6 " (c) " }{XPPEDIT 18 0 "exp(Pi*i)" "6#-%$expG6#*&%#PiG\"\"\"%\"iGF(" }{TEXT -1 6 " (d) \+ " }{XPPEDIT 18 0 "exp(2*Pi*i)" "6#-%$expG6#*(\"\"#\"\"\"%#PiGF(%\"iGF( " }{TEXT -1 6 " (e) " }{XPPEDIT 18 0 "sqrt(2)*exp(-3*Pi/4*i);" "6#*&- %%sqrtG6#\"\"#\"\"\"-%$expG6#,$**\"\"$F(%#PiGF(\"\"%!\"\"%\"iGF(F1F(" }{TEXT -1 5 " (f) " }{XPPEDIT 18 0 "exp(1+Pi/3*i)" "6#-%$expG6#,&\"\" \"F'*(%#PiGF'\"\"$!\"\"%\"iGF'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "sqrt(3)/2+i/2;" "6#,&*&-%%sqr tG6#\"\"$\"\"\"\"\"#!\"\"F)*&%\"iGF)F*F+F)" }{TEXT -1 1 " " }{TEXT 329 1 "~" }{TEXT -1 3 " 0." }{XPPEDIT 18 0 "8660254038+0" "6#,&\"+QSDg ')\"\"\"\"\"!F%" }{TEXT -1 1 "." }{XPPEDIT 18 0 "5*i" "6#*&\"\"&\"\"\" %\"iGF%" }{TEXT -1 6 " (b) " }{XPPEDIT 18 0 "-1/2 +sqrt(3)/2*i" "6#,& *&\"\"\"F%\"\"#!\"\"F'*(-%%sqrtG6#\"\"$F%F&F'%\"iGF%F%" }{TEXT -1 1 " \+ " }{TEXT 330 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-0" "6#,$\"\"!!\"\" " }{TEXT -1 1 "." }{XPPEDIT 18 0 "5+0" "6#,&\"\"&\"\"\"\"\"!F%" } {TEXT -1 1 "." }{XPPEDIT 18 0 "8660254038*i" "6#*&\"+QSDg')\"\"\"%\"iG F%" }{TEXT -1 21 " (c) -1 (d) 1 (e) " }{XPPEDIT 18 0 "-1-i" "6#,&\" \"\"!\"\"%\"iGF%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(f) " }{XPPEDIT 18 0 "exp(1)/2" "6#*&-%$expG6#\"\"\"F'\"\"#!\"\"" }{XPPEDIT 18 0 "``(1+sqrt(3)*i)" "6#-%!G6#,&\"\"\"F'*&-%%sqrtG6#\"\"$F'%\"iGF'F' " }{TEXT -1 1 " " }{TEXT 328 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1.3 59140914+2.354101118*i" "6#,&-%&FloatG6$\"+949f8!\"*\"\"\"*&-F%6$\"+=6 5aBF(F)%\"iGF)F)" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 32 "____ ____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "_______________________________ _" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Question 5 " }}{PARA 0 "" 0 "" {TEXT -1 59 "Convert the following complex numbers to exponential form. " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "cos(2*Pi/ 3) + i*sin(2*Pi/3)" "6#,&-%$cosG6#*(\"\"#\"\"\"%#PiGF)\"\"$!\"\"F)*&% \"iGF)-%$sinG6#*(F(F)F*F)F+F,F)F)" }{TEXT -1 6 " (b) " }{XPPEDIT 18 0 "-i" "6#,$%\"iG!\"\"" }{TEXT -1 7 " (c) " }{XPPEDIT 18 0 "2*i" "6# *&\"\"#\"\"\"%\"iGF%" }{TEXT -1 6 " (d) " }{XPPEDIT 18 0 "1/sqrt(2)+i /sqrt(2);" "6#,&*&\"\"\"F%-%%sqrtG6#\"\"#!\"\"F%*&%\"iGF%-F'6#F)F*F%" }{TEXT -1 6 " (e) " }{XPPEDIT 18 0 "1+i" "6#,&\"\"\"F$%\"iGF$" } {TEXT -1 6 " (f) " }{XPPEDIT 18 0 "cos*1-i*sin*1" "6#,&*&%$cosG\"\"\" F&F&F&*(%\"iGF&%$sinGF&F&F&!\"\"" }{TEXT -1 6 " (g) " }{XPPEDIT 18 0 "sqrt(3)+i" "6#,&-%%sqrtG6#\"\"$\"\"\"%\"iGF(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "exp(2*Pi/3*i) " "6#-%$expG6#**\"\"#\"\"\"%#PiGF(\"\"$!\"\"%\"iGF(" }{TEXT -1 6 " (b ) " }{XPPEDIT 18 0 "exp(-Pi/2*i)" "6#-%$expG6#,$*(%#PiG\"\"\"\"\"#!\" \"%\"iGF)F+" }{TEXT -1 6 " (c) " }{XPPEDIT 18 0 "2*exp(Pi/2*i)" "6#*& \"\"#\"\"\"-%$expG6#*(%#PiGF%F$!\"\"%\"iGF%F%" }{TEXT -1 5 " (d) " } {XPPEDIT 18 0 "exp(Pi/4*i)" "6#-%$expG6#*(%#PiG\"\"\"\"\"%!\"\"%\"iGF( " }{TEXT -1 6 " (e) " }{XPPEDIT 18 0 "sqrt(2)*exp(Pi/4*i)" "6#*&-%%sq rtG6#\"\"#\"\"\"-%$expG6#*(%#PiGF(\"\"%!\"\"%\"iGF(F(" }{TEXT -1 6 " \+ (f) " }{XPPEDIT 18 0 "exp(-i)" "6#-%$expG6#,$%\"iG!\"\"" }{TEXT -1 6 " (g) " }{XPPEDIT 18 0 "2*exp(Pi/3*i)" "6#*&\"\"#\"\"\"-%$expG6#*(%#Pi GF%\"\"$!\"\"%\"iGF%F%" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "____________ ____________________" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Question 6 " }}{PARA 0 "" 0 "" {TEXT -1 11 "Given that " }{XPPEDIT 18 0 "theta " "6#%&thetaG" }{TEXT -1 81 " is a real number, establish the followin g formulas by using appropriate series. " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "cos*i*theta = cosh*theta" "6#/*(%$cosG\"\"\"% \"iGF&%&thetaGF&*&%%coshGF&F(F&" }{TEXT -1 7 " (b) " }{XPPEDIT 18 0 "sin*i*theta=i*sinh*theta" "6#/*(%$sinG\"\"\"%\"iGF&%&thetaGF&*(F'F&%% sinhGF&F(F&" }{TEXT -1 7 " (c) " }{XPPEDIT 18 0 "cosh*i*theta=cos*th eta" "6#/*(%%coshG\"\"\"%\"iGF&%&thetaGF&*&%$cosGF&F(F&" }{TEXT -1 7 " (d) " }{XPPEDIT 18 0 "sinh*i*theta=i*sin*theta" "6#/*(%%sinhG\"\"\" %\"iGF&%&thetaGF&*(F'F&%$sinGF&F(F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 32 "________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "____________ ____________________" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Question 7 " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(theta)=cos* theta+i*sin*theta" "6#/-%\"fG6#%&thetaG,&*&%$cosG\"\"\"F'F+F+*(%\"iGF+ %$sinGF+F'F+F+" }{TEXT -1 17 ". Show that f '(" }{XPPEDIT 18 0 "theta " "6#%&thetaG" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "``=i*f(theta)" "6#/%!G* &%\"iG\"\"\"-%\"fG6#%&thetaGF'" }{TEXT -1 15 " in two ways: " }} {PARA 0 "" 0 "" {TEXT -1 53 "(a) by direct differentiation, using the \+ facts that " }{XPPEDIT 18 0 "d/(d*theta)" "6#*&%\"dG\"\"\"*&F$F%%&the taGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[sin*theta]=cos*theta" "6# /7#*&%$sinG\"\"\"%&thetaGF'*&%$cosGF'F(F'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "d/(d*theta)" "6#*&%\"dG\"\"\"*&F$F%%&thetaGF%!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[cos*theta]=-sin*theta" "6#/7#*&%$cosG \"\"\"%&thetaGF',$*&%$sinGF'F(F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "(b) by using Euler's formula to write " }{XPPEDIT 18 0 "f(theta)=exp(i*theta)" "6#/-%\"fG6#%&thetaG-%$expG6#*&%\"iG\"\" \"F'F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 34 ": Assume that the imaginary unit " }{TEXT 319 1 "i" }{TEXT -1 95 " can be treated in the same manner \+ as an unknown real constant in the differentiation process. " }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "__ ______________________________" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Question 8 " }}{PARA 0 "" 0 "" {TEXT -1 18 "Use the formulas " } {XPPEDIT 18 0 "PIECEWISE([sin(a+i*b) = sin*a*cosh*b+i*cos*a*sinh*b, `` ],[cos(a+b*i) = cos*a*cosh*b-i*sin*a*sinh*b, ``],[sinh(a+b*i) = sinh*a *cos*b+i*cosh*a*sin*b, ``],[cosh(a+b*i) = cosh*a*cos*b+i*sinh*a*sin*b, ``]);" "6#-%*PIECEWISEG6&7$/-%$sinG6#,&%\"aG\"\"\"*&%\"iGF-%\"bGF-F-, &**F)F-F,F-%%coshGF-F0F-F-*,F/F-%$cosGF-F,F-%%sinhGF-F0F-F-%!G7$/-F56# ,&F,F-*&F0F-F/F-F-,&**F5F-F,F-F3F-F0F-F-*,F/F-F)F-F,F-F6F-F0F-!\"\"F77 $/-F66#,&F,F-*&F0F-F/F-F-,&**F6F-F,F-F5F-F0F-F-*,F/F-F3F-F,F-F)F-F0F-F -F77$/-F36#,&F,F-*&F0F-F/F-F-,&**F3F-F,F-F5F-F0F-F-*,F/F-F6F-F,F-F)F-F 0F-F-F7" }{TEXT -1 38 " to find the values of the following. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " } {XPPEDIT 18 0 "sinh(2+Pi*i)" "6#-%%sinhG6#,&\"\"#\"\"\"*&%#PiGF(%\"iGF (F(" }{TEXT -1 6 " (b) " }{XPPEDIT 18 0 "cosh(2+Pi*i)" "6#-%%coshG6#, &\"\"#\"\"\"*&%#PiGF(%\"iGF(F(" }{TEXT -1 6 " (c) " }{XPPEDIT 18 0 "s in(Pi/3+i);" "6#-%$sinG6#,&*&%#PiG\"\"\"\"\"$!\"\"F)%\"iGF)" }{TEXT -1 6 " (d) " }{XPPEDIT 18 0 "cos(Pi/3+i)" "6#-%$cosG6#,&*&%#PiG\"\"\" \"\"$!\"\"F)%\"iGF)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "-sinh*2" "6#,$*&%%sinhG\"\"\"\"\"#F&!\"\" " }{TEXT -1 1 " " }{TEXT 320 1 "~" }{TEXT -1 19 " -3.626860408 (b) " }{XPPEDIT 18 0 "-cosh*2" "6#,$*&%%coshG\"\"\"\"\"#F&!\"\"" }{TEXT -1 1 " " }{TEXT 323 1 "~" }{TEXT -1 19 " -3.762195691 (c) " }{XPPEDIT 18 0 "sqrt(3)/2" "6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "cosh*1+i/2" "6#,&*&%%coshG\"\"\"F&F&F&*&%\"iGF&\"\"# !\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*1" "6#*&%%sinhG\"\"\"F%F %" }{TEXT -1 1 " " }{TEXT 321 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1. 336347030+0" "6#,&-%&FloatG6$\"+IqMO8!\"*\"\"\"\"\"!F)" }{TEXT -1 1 ". " }{XPPEDIT 18 0 "5876005968*i" "6#*&\"+of+we\"\"\"%\"iGF%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }{XPPEDIT 18 0 "1/2" "6#*& \"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cosh*1-sqrt(3)/2" "6#,&*&%%coshG\"\"\"F&F&F&*&-%%sqrtG6#\"\"$F&\"\"#!\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "i*sinh*1" "6#*(%\"iG\"\"\"%%sinhGF%F%F%" }{TEXT -1 1 " " }{TEXT 322 1 "~" }{TEXT -1 3 " 0." }{XPPEDIT 18 0 "7715403174 -1.017754088*i" "6#,&\"+uJS:x\"\"\"*&-%&FloatG6$\"+)3ax,\"!\"*F%%\"iGF %!\"\"" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 32 "______________ __________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "code for p ictures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 9 "graph of " }{XPPEDIT 18 0 "y=exp(x)" "6#/% \"yG-%$expG6#%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 663 "p1 := plot(exp(x),x=-1.6..1 .6,thickness=2):\np2 := plot([[[0,1],[1,exp(1)]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=COLOR(RGB,0,0,.8)):\np3 : = plot([[-.4,.6],[.4,1.4]],color=COLOR(RGB,0,0,.8)):\np4 := plot([[.6, .6*exp(1)],[1.4,1.4*exp(1)]],color=COLOR(RGB,0,0,.8)):\nt1 := plots[te xtplot]([[1.6,-.2,`x`],[-.2,5.2,`y`]],color=black):\nt2 := plots[textp lot]([[.2,.9,`(0,1)`],[1.2,2.6,`(1,e)`],[.7,1.2,`gradient = 1`],\n \+ [1.7,3.3,`gradient = e`]],color=COLOR(RGB,0,0,.8)):\nt3 := plots[textp lot]([[1.1,4.22,`y = e`],[1.31,4.35,`x`]],color=red):\nplots[display]( [p1,p2,p3,p4,t1,t2,t3],tickmarks=[3,6],labels=[``,``],\n view=[-1.6. .1.7,-.2..5.2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 9 "gra ph of " }{XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 41 " together with polynomial approximations " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 341 "p1 := plot( [exp(x),1+x,1+x+x^2/2,1+x+x^2/2+x^3/6],x=-1.6..1.6,\n thickness=[2,1$ 3],color=[red,blue,magenta,coral]):\nt1 := plots[textplot]([[1.6,-.2,` x`],[-.2,5.2,`y`]],color=black):\nt2 := plots[textplot]([[1.1,4.22,`y \+ = e`],[1.31,4.35,`x`]],color=red):\nplots[display]([p1,t1,t2],tickmark s=[3,6],labels=[``,``],\n view=[-1.6..1.7,-.2..5.2]);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 0 "" 0 "" {TEXT -1 9 "graph of " }{XPPEDIT 18 0 "y = si n*x;" "6#/%\"yG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 41 " together with pol ynomial approximations " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 380 "p1 := plot([sin(x),x,x-x^3/6,x-x^3 /6+x^5/120,x-x^3/6+x^5/120-x^7/5040],x=-7.6..7.6,\n numpoints=75,thic kness=[2,1$3],color=[red,blue,magenta,coral,COLOR(RGB,0,.8,0)]):\nt1 : = plots[textplot]([[7.6,-.2,`x`],[-.3,2.5,`y`]],color=black):\nt2 := p lots[textplot]([[6.3,.9,`y = sin x`]],color=red):\nplots[display]([p1, t1,t2],tickmarks=[7,6],labels=[``,``],\n view=[-7.6..7.7,-2.5..2.5]) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 16 "Euler's fo rmula " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 783 "cs := evalf(cos(Pi/3)):\nsn := evalf(sin(Pi/3)):\np1 := plot([cos(t),sin(t),t=0..2*Pi],color=red):\np2 := plot([[[0,0],[cs ,sn]],[[1,-.04],[1,.04]],[[-1,-.04],[-1,.04]],\n [[-.04,1],[.04,1]], [[-.04,-1],[.04,-1]]],color=black):\np3 := plot([[[cs,sn]]$3],style=po int,symbol=[circle,diamond,cross],\n color=black):\np4 := \+ plot([0.25*cos(t),0.25*sin(t),t=0..Pi/3],color=black):\nt1 := plots[te xtplot]([[.85,.99,`e = cos + i sin`],[.55,1.05,`i`],\n [1.0 5,-.08,`1`],[-1.05,-.08,`-1`],\n [-.06,1.07,`i`],[-.06,-1.07,`-i`],[ 1.3,-.06,`Real`],[-.15,1.3,`Imag`]],color=black):\nt2 := plots[textplo t]([[.15,.1,`q`],[.93,.993,`q`],\n [1.3,.993,`q`],[.6,1 .05,`q`]],font=[SYMBOL,11],color=black):\nplots[display]([p2,p1,p3,p4, t1,t2],tickmarks=[0,0],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }