{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 258 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis " -1 262 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Em phasis" -1 263 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Da rk Red Emphasis" -1 264 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" 263 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 262 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 257 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 257 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 262 270 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " 263 271 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 263 272 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 263 273 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 264 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 263 279 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 283 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2 " -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 2 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 "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 "Norma l" -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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 62 "Trigonometrical and hyperbolic fu nctions of a complex variable" }}{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 35 "load procedures for comp lex numbers" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 283 9 "complex.m" }{TEXT -1 37 " contains the code for the procedure \+ " }{TEXT 0 13 "conformalplot" }{TEXT -1 25 " used in this worksheet. \+ " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session \+ by a command similar to the one that follows, where the file path give s its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \" K:\\\\Maple/procdrs/complex.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "The sine and cosine functions of \+ a complex variable " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 259 1 "z" }{TEXT -1 23 " is co mplex, we define " }{XPPEDIT 18 0 "sin*z;" "6#*&%$sinG\"\"\"%\"zGF%" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "cos*z" "6#*&%$cosG\"\"\"%\"zGF%" } {TEXT -1 3 " by" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "si n*z = z-z^3/3!+z^5/5!+` . . . `+(-1)^n*z^(2*n+1)/(2*n+1)!+` . . .`;" " 6#/*&%$sinG\"\"\"%\"zGF&,.F'F&*&F'\"\"$-%*factorialG6#F*!\"\"F.*&F'\" \"&-F,6#F0F.F&%(~.~.~.~GF&*(),$F&F.%\"nGF&)F',&*&\"\"#F&F7F&F&F&F&F&-F ,6#,&*&F;F&F7F&F&F&F&F.F&%'~.~.~.GF&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum((-1)^n*z^(2*n+1)/(2*n+1)! ,n = 0 .. infinity);" "6#/%!G-%$SumG6$*(),$\"\"\"!\"\"%\"nGF+)%\"zG,&* &\"\"#F+F-F+F+F+F+F+-%*factorialG6#,&*&F2F+F-F+F+F+F+F,/F-;\"\"!%)infi nityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*z = 1-z^2/2!+z^4/4-z^6/6!+` . \+ . . `+(-1)^n*z^(2*n)/(2*n)!+` . . . `;" "6#/*&%$cosG\"\"\"%\"zGF&,0F&F &*&F'\"\"#-%*factorialG6#F*!\"\"F.*&F'\"\"%F0F.F&*&F'\"\"'-F,6#F2F.F.% (~.~.~.~GF&*(),$F&F.%\"nGF&)F'*&F*F&F9F&F&-F,6#*&F*F&F9F&F.F&F5F&" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Sum((-1)^n*z^(2*n)/(2*n)!,n = 0 .. infinity);" "6#/%!G-%$SumG6$*(),$ \"\"\"!\"\"%\"nGF+)%\"zG*&\"\"#F+F-F+F+-%*factorialG6#*&F1F+F-F+F,/F-; \"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 63 "The p ower series can be differentiated term by term, so we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Diff([sin*z],z)= cos*z,``],[Diff([cos*z],z)=-sin*z,``])" "6#-%*PIECEWISEG6$7$/-%%DiffG6 $7#*&%$sinG\"\"\"%\"zGF.F/*&%$cosGF.F/F.%!G7$/-F)6$7#*&F1F.F/F.F/,$*&F -F.F/F.!\"\"F2" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 265 12 "____________" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "The Euler formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "cos*z+i*sin*z = exp(i*z);" "6#/,&*&%$cosG\"\"\"%\"zGF'F '*(%\"iGF'%$sinGF'F(F'F'-%$expG6#*&F*F'F(F'" }{TEXT -1 13 " ------- (i ) " }}{PARA 0 "" 0 "" {TEXT -1 15 "now holds when " }{TEXT 278 1 "z" } {TEXT -1 12 " is complex." }}{PARA 0 "" 0 "" {TEXT -1 24 "Furthermore, since also " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*z -i*sin*z = exp(-i*z);" "6#/,&*&%$cosG\"\"\"%\"zGF'F'*(%\"iGF'%$sinGF'F (F'!\"\"-%$expG6#,$*&F*F'F(F'F," }{TEXT -1 13 " ------- (ii)" }}{PARA 0 "" 0 "" {TEXT -1 30 "by adding (i) and (ii) we have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "cos*z = (exp(i*z)+exp(-i*z))/2;" "6#/*&%$cosG\"\"\"%\"zGF&*&,&-%$expG6#*&%\"iGF&F'F&F&-F+6#,$*&F.F&F'F& !\"\"F&F&\"\"#F3" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 266 13 "_____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "and by subtracting (ii) from (i) we have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "sin*z = (exp(i*z)-exp(-i*z))/(2*i);" " 6#/*&%$sinG\"\"\"%\"zGF&*&,&-%$expG6#*&%\"iGF&F'F&F&-F+6#,$*&F.F&F'F&! \"\"F3F&*&\"\"#F&F.F&F3" }{TEXT -1 3 " . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 267 13 "_____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "By squaring and adding these equations we see that the standard formula" }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "cos^2*z+sin^2*z = 1;" "6#/,&*&%$cosG \"\"#%\"zG\"\"\"F)*&%$sinGF'F(F)F)F)" }{TEXT -1 1 " " }{TEXT 260 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 268 10 "__________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "holds when " }{TEXT 275 1 "z" }{TEXT -1 12 " is complex." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "cz := (exp(I*z)+exp(-I*z))/2;\nsz := (exp(I*z)-exp(-I*z))/(2*I);\n cz^2+sz^2;\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#czG,&-% $expG6#*&%\"IG\"\"\"%\"zGF+#F+\"\"#-F'6#,$F)!\"\"F-" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#szG,$*&%\"IG\"\"\",&-%$expG6#*&F'F(%\"zGF(F(-F+6#, $F-!\"\"F2F(#F2\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$),&-%$expG 6#*&%\"IG\"\"\"%\"zGF,#F,\"\"#-F(6#,$F*!\"\"F.F/F,F,*$),&F'F,F0F3F/F,# F3\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "The addition formulas" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 56 "The sta ndard addition formulas hold for complex numbers." }}{PARA 256 "" 0 " " {TEXT -1 8 " " }{XPPEDIT 18 0 "PIECEWISE([sin(z+w) = sin*z*co s*w+cos*z*sin*w, ``],[sin(z-w) = sin*z*cos*w-cos*z*sin*w, ``],[cos(z+w ) = cos*z*cos*w-sin*z*sin*w, ``],[cos(z-w) = cos*z*cos*w+sin*z*sin*w, \+ ``]);" "6#-%*PIECEWISEG6&7$/-%$sinG6#,&%\"zG\"\"\"%\"wGF-,&**F)F-F,F-% $cosGF-F.F-F-**F1F-F,F-F)F-F.F-F-%!G7$/-F)6#,&F,F-F.!\"\",&**F)F-F,F-F 1F-F.F-F-**F1F-F,F-F)F-F.F-F9F37$/-F16#,&F,F-F.F-,&**F1F-F,F-F1F-F.F-F -**F)F-F,F-F)F-F.F-F9F37$/-F16#,&F,F-F.F9,&**F1F-F,F-F1F-F.F-F-**F)F-F ,F-F)F-F.F-F-F3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 273 21 "_____________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "They can be checked by \+ using the formulas:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*z = (exp(i*z)+exp(-i*z))/2;" "6#/*&%$cosG\"\"\"%\"zGF&*&,&-%$ex pG6#*&%\"iGF&F'F&F&-F+6#,$*&F.F&F'F&!\"\"F&F&\"\"#F3" }{TEXT -1 8 " \+ and " }{XPPEDIT 18 0 "sin*z = (exp(i*z)-exp(-i*z))/(2*i);" "6#/*&%$si nG\"\"\"%\"zGF&*&,&-%$expG6#*&%\"iGF&F'F&F&-F+6#,$*&F.F&F'F&!\"\"F3F&* &\"\"#F&F.F&F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "For example, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*z*cos*w+cos*z*sin*w" "6#,&**%$sinG\"\"\"% \"zGF&%$cosGF&%\"wGF&F&**F(F&F'F&F%F&F)F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(exp(i*z)-exp(-i*z))/(2*i)" "6#/%!G*&,&-%$expG6#*&%\"iG\"\"\" %\"zGF,F,-F(6#,$*&F+F,F-F,!\"\"F2F,*&\"\"#F,F+F,F2" }{TEXT -1 1 " " } {TEXT 276 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "(exp(i*w)+exp(-i*w))/2 + (exp(i*z)+exp(-i*z))/2" "6#,&*&,&-%$expG6#*&%\"iG\"\"\"%\"wGF+F+-F' 6#,$*&F*F+F,F+!\"\"F+F+\"\"#F1F+*&,&-F'6#*&F*F+%\"zGF+F+-F'6#,$*&F*F+F 8F+F1F+F+F2F1F+" }{TEXT -1 1 " " }{TEXT 277 1 "." }{TEXT -1 1 " " } {XPPEDIT 18 0 "(exp(i*w)-exp(-i*w))/(2*i);" "6#*&,&-%$expG6#*&%\"iG\" \"\"%\"wGF*F*-F&6#,$*&F)F*F+F*!\"\"F0F**&\"\"#F*F)F*F0" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = (exp(i*z+i*w)+exp(i*z-i*w)-exp(-i*z+i*w)-exp(-i*z- i*w))/(4*i)+(exp(i*z+i*w)-exp(i*z-i*w)+exp(-i*z+i*w)-exp(-i*z-i*w))/(4 *i);" "6#/%!G,&*&,*-%$expG6#,&*&%\"iG\"\"\"%\"zGF.F.*&F-F.%\"wGF.F.F.- F)6#,&*&F-F.F/F.F.*&F-F.F1F.!\"\"F.-F)6#,&*&F-F.F/F.F7*&F-F.F1F.F.F7-F )6#,&*&F-F.F/F.F7*&F-F.F1F.F7F7F.*&\"\"%F.F-F.F7F.*&,*-F)6#,&*&F-F.F/F .F.*&F-F.F1F.F.F.-F)6#,&*&F-F.F/F.F.*&F-F.F1F.F7F7-F)6#,&*&F-F.F/F.F7* &F-F.F1F.F.F.-F)6#,&*&F-F.F/F.F7*&F-F.F1F.F7F7F.*&FCF.F-F.F7F." } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (2*exp(i*z+i*w)-2*exp(-i*z-i*w))/( 4*i);" "6#/%!G*&,&*&\"\"#\"\"\"-%$expG6#,&*&%\"iGF)%\"zGF)F)*&F/F)%\"w GF)F)F)F)*&F(F)-F+6#,&*&F/F)F0F)!\"\"*&F/F)F2F)F8F)F8F)*&\"\"%F)F/F)F8 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(exp(i*(z+w))-exp(-i*(z+w)))/(2*i) " "6#/%!G*&,&-%$expG6#*&%\"iG\"\"\",&%\"zGF,%\"wGF,F,F,-F(6#,$*&F+F,,& F.F,F/F,F,!\"\"F5F,*&\"\"#F,F+F,F5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ sin(z+w);" "6#/%!G-%$sinG6#,&%\"zG\"\"\"%\"wGF*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "sin(z)*cos(w)+cos(z)*sin(w);\nconvert(%,exp);\nmap(simplify,expand (normal(%)));\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$ sinG6#%\"zG\"\"\"-%$cosG6#%\"wGF)F)*&-F+F'F)-F&F,F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(^##!\"\"\"\"#\"\"\",&-%$expG6#*&%\"zGF)^#F)F)F )*&F)F)F+F'F'F),&*&#F)F(F)-F,6#*&%\"wGF)F0F)F)F)*&F4F)*&F)F)F5F'F)F)F) F)*(F%F),&*&F4F)F+F)F)*&F4F)F1F)F)F),&F5F)F:F'F)F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&*&^##!\"\"\"\"#\"\"\"-%$expG6#*&,&%\"zGF)%\"wGF)F)^ #F)F)F)F)*&^##F)F(F)-F+6#*&^#F'F)F.F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#,&%\"zG\"\"\"%\"wGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Many other standard trigo nometric formulas such as the double \"angle\" formulas:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*2*z = 2*sin*z*cos*z;" "6#/* (%$sinG\"\"\"\"\"#F&%\"zGF&*,F'F&F%F&F(F&%$cosGF&F(F&" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*2*z = cos^2*z -sin^2*z;" "6#/*(%$cosG\"\"\"\"\"#F&%\"zGF&,&*&F%F'F(F&F&*&%$sinGF'F(F &!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "can be deduced \+ from the addition formulas. 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&)o#F*7$F\\aw$!3&eqmpp*RDDF*7$Faaw$!3;Z6&QaqcN#F*7$Ffaw$!3QVaMm:)y;#F* 7$F[bw$!35`9JI9,y>F*7$F`bw$!3oH$GwF=E+P\"F*7$F_cw$!3J3C\"*\\?4Q6F*7$Fdcw$!3%R40Xzb8E*Few7$Ficw$!3 /[N$*)GPP'pFew7$F^dw$!3(**[Y`HOpt%Few7$Fcdw$!3dn`1WHDVAFew7$Fhdw$\"3cD Y&)3Iw([\"Fadu-Fgu6&FiuF]vF+F," 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" "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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 63 "The hyperbolic sine and cosine function s of a complex variable " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 261 1 "z" }{TEXT -1 23 " \+ is complex, we define " }{XPPEDIT 18 0 "sinh*z;" "6#*&%%sinhG\"\"\"%\" zGF%" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "cosh*z" "6#*&%%coshG\"\"\"% \"zGF%" }{TEXT -1 4 " by:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "sinh*z = (exp(z)-exp(-z))/2;" "6#/*&%%sinhG\"\"\"%\"zGF &*&,&-%$expG6#F'F&-F+6#,$F'!\"\"F0F&\"\"#F0" }{TEXT -1 3 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 270 11 "___________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "cosh*z = (exp(z)+exp(-z))/2;" "6#/*&%%coshG\"\" \"%\"zGF&*&,&-%$expG6#F'F&-F+6#,$F'!\"\"F&F&\"\"#F0" }{TEXT -1 3 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 269 12 "____________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "We have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([Diff([sinh*z],z) = cosh*z, ``],[Diff([cosh*z],z) = sinh* z, ``]);" "6#-%*PIECEWISEG6$7$/-%%DiffG6$7#*&%%sinhG\"\"\"%\"zGF.F/*&% %coshGF.F/F.%!G7$/-F)6$7#*&F1F.F/F.F/*&F-F.F/F.F2" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 279 12 "____________" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The Maclaurin series expansions are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sinh*z = z+z^3/3!+z^5/5!+` . . . `+z^(2 *n+1)/(2*n+1)!+` . . . `;" "6#/*&%%sinhG\"\"\"%\"zGF&,.F'F&*&F'\"\"$-% *factorialG6#F*!\"\"F&*&F'\"\"&-F,6#F0F.F&%(~.~.~.~GF&*&)F',&*&\"\"#F& %\"nGF&F&F&F&F&-F,6#,&*&F8F&F9F&F&F&F&F.F&F3F&" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "Sum(z^(2*n+1)/(2*n +1)!,n = 0 .. infinity);" "6#-%$SumG6$*&)%\"zG,&*&\"\"#\"\"\"%\"nGF,F, F,F,F,-%*factorialG6#,&*&F+F,F-F,F,F,F,!\"\"/F-;\"\"!%)infinityG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh*z = 1+z^2/2!+z^4/4+z^6/6!+` . . . \+ `+z^(2*n)/(2*n)!+` . . . `;" "6#/*&%%coshG\"\"\"%\"zGF&,0F&F&*&F'\"\"# -%*factorialG6#F*!\"\"F&*&F'\"\"%F0F.F&*&F'\"\"'-F,6#F2F.F&%(~.~.~.~GF &*&)F'*&F*F&%\"nGF&F&-F,6#*&F*F&F9F&F.F&F5F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "Sum(z^(2*n)/(2*n)!,n = 0 .. infinity);" "6#-%$SumG6$*&)%\"zG*&\"\"#\"\"\"%\"nGF+F+-%*factorial G6#*&F*F+F,F+!\"\"/F,;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "The relations:" }}{PARA 256 "" 0 "" {TEXT -1 6 " \+ " }{XPPEDIT 18 0 "PIECEWISE([sin*i*z = i*sinh*z, ``],[cos*i*z = cosh *z, ``],[sinh*i*z = i*sin*z, ``],[cosh*i*z = cos*z, ``]);" "6#-%*PIECE WISEG6&7$/*(%$sinG\"\"\"%\"iGF*%\"zGF**(F+F*%%sinhGF*F,F*%!G7$/*(%$cos GF*F+F*F,F**&%%coshGF*F,F*F/7$/*(F.F*F+F*F,F**(F+F*F)F*F,F*F/7$/*(F5F* F+F*F,F**&F3F*F,F*F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 271 10 "__________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "can be deduced from the series \+ expansions, or by using the formulas:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*z = (exp(i*z)+exp(-i*z))/2;" "6#/*&%$cosG\"\" \"%\"zGF&*&,&-%$expG6#*&%\"iGF&F'F&F&-F+6#,$*&F.F&F'F&!\"\"F&F&\"\"#F3 " }{TEXT -1 8 " and " }{XPPEDIT 18 0 "sin*z = (exp(i*z)-exp(-i*z))/ (2*i);" "6#/*&%$sinG\"\"\"%\"zGF&*&,&-%$expG6#*&%\"iGF&F'F&F&-F+6#,$*& F.F&F'F&!\"\"F3F&*&\"\"#F&F.F&F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "They are useful in deduc ing properties of hyperbolic functions from the corresponding properti es of trigonometric functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 "If we write " }{XPPEDIT 18 0 "z = x+y*i" "6#/%\"zG,&%\"xG\"\"\"*&%\"yGF'%\"iGF'F'" }{TEXT -1 8 ", where " } {TEXT 280 1 "x" }{TEXT -1 5 " and " }{TEXT 281 1 "y" }{TEXT -1 17 " ar e real, then: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 " " }{XPPEDIT 18 0 "PIECEWISE([sin*z = sin*x*cos h*y+i*cos*x*sinh*y, ``],[cos*z = cos*x*cosh*y+i*sin*x*sinh*y, ``],[sin h*z = sinh*x*cos*y+i*cosh*x*sin*y, ``],[cosh*z = cosh*x*cos*y+i*sinh*x *sin*y, ``]);" "6#-%*PIECEWISEG6&7$/*&%$sinG\"\"\"%\"zGF*,&**F)F*%\"xG F*%%coshGF*%\"yGF*F**,%\"iGF*%$cosGF*F.F*%%sinhGF*F0F*F*%!G7$/*&F3F*F+ F*,&**F3F*F.F*F/F*F0F*F**,F2F*F)F*F.F*F4F*F0F*F*F57$/*&F4F*F+F*,&**F4F *F.F*F3F*F0F*F**,F2F*F/F*F.F*F)F*F0F*F*F57$/*&F/F*F+F*,&**F/F*F.F*F3F* F0F*F**,F2F*F4F*F.F*F)F*F0F*F*F5" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 272 19 "___________________" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "This mea ns that we can compute values of " }{XPPEDIT 18 0 "sin*z" "6#*&%$sinG \"\"\"%\"zGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "cos*z" "6#*&%$cosG\"\" \"%\"zGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sinh*z" "6#*&%%sinhG\"\"\" %\"zGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cosh*z" "6#*&%%coshG\"\" \"%\"zGF%" }{TEXT -1 8 ", where " }{TEXT 282 1 "z" }{TEXT -1 56 " is c omplex, without resorting to the series expansions." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 7 "Example" }{TEXT -1 1 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "zz := 2+I;\nxx := Re(zz);\nyy := Im(zz);\nsin(xx)*cosh(yy)+I* cos(xx)*sinh(yy);\nevalf(evalf(%,15));\nevalf(sin(zz))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#zzG,&\"\"#\"\"\"%\"IGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yyG\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#\"\"#\"\"\"-%%cos hG6#F)F)F)*(%\"IGF)-%$cosGF'F)-%%sinhGF,F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"+^#>JS\"!\"*\"\"\"%\"IG$!+!fi0*[!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&$\"+^#>JS\"!\"*\"\"\"%\"IG$!+!fi0*[!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "zz := 2+I;\nxx := Re(zz);\nyy := Im(zz);\nsinh(xx)*cos(yy)+I*cosh (xx)*sin(yy);\nevalf(evalf(%,15));\nevalf(sinh(zz));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#zzG,&\"\"#\"\"\"%\"IGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yyG\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%%sinhG6#\"\"#\"\"\"-%$co sG6#F)F)F)*(%\"IGF)-%%coshGF'F)-%$sinGF,F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"+T5gf>!\"*\"\"\"%\"IG$\"+8&yd;$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"+T5gf>!\"*\"\"\"%\"IG$\"+8&yd;$F&" }}}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }