{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "Define" -1 256 "Times" 1 12 163 163 163 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "Emphasis" -1 257 "Times" 1 12 128 0 128 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "normal C" -1 258 "Times" 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Menu" -1 259 "" 0 0 163 163 163 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 260 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 261 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 268 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 269 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 277 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 284 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 285 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 292 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 293 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 296 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 300 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 301 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 305 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 306 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 307 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 308 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Geneva" 1 10 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 "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Monaco" 1 9 255 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 "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Symbol" 1 10 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 "R3 Fo nt 0" -1 258 1 {CSTYLE "" -1 -1 "Geneva" 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 {EXCHG {PARA 258 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple Worksheets, 2001" }{TEXT 304 144 "\n(c) John H. Mathews Rus sell W. Howell\nmathews@fullerton.edu howell@westmont.edu\n\nCompl imentary software to accompany the textbook:" }}{PARA 258 "" 0 "" {TEXT 257 82 "COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9" }{TEXT 303 197 "\nJones and Bartlett Publi shers, Inc., 40 Tall Pine Drive, Sudbury, MA 01776\nTel e. (800) 832-0034; FAX: (508) 443-8000, E-mail: mkt@jbpu b.com, http://www.jbpub.com/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 1 "\n" }{TEXT 256 31 "CHAPTER 5 ELEMENTARY FUNCTIONS" }{TEXT 293 2 "\n\n" }{TEXT 256 51 "Section 5.4 Trigonometric and Hyperbolic Func tions" }{TEXT 294 2 "\n\n" }}{PARA 0 "" 0 "" {TEXT -1 261 " Given th e success we had in using power series to define the complex exponenti al, we have reason to believe this approach will be fruitful for other elementary functions as well. The power series expansions for the rea l-valued sine and cosine functions are " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "sin(x) = Sum ((-1)^n/(2*n+1)!*x^(2*n+1),n = 0 .. infinity);" "6#/-%$sinG6#%\"xG-%$S umG6$*(),$\"\"\"!\"\"%\"nGF.-%*factorialG6#,&*&\"\"#F.F0F.F.F.F.F/)F', &*&F6F.F0F.F.F.F.F./F0;\"\"!%)infinityG" }{TEXT -1 23 " and \+ " }{XPPEDIT 18 0 "cos(x) = Sum((-1)^n/(2*n)!*x^(2*n),n = 0 .. \+ infinity);" "6#/-%$cosG6#%\"xG-%$SumG6$*(),$\"\"\"!\"\"%\"nGF.-%*facto rialG6#*&\"\"#F.F0F.F/)F'*&F5F.F0F.F./F0;\"\"!%)infinityG" }{TEXT -1 4 " . " }{TEXT 306 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 305 103 "Load Maple's \"conformal mapping\" pro cedure.\nMake sure this is done only ONCE during a Maple session.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Th us it is natural to make the following definitions for the complex sin e and cosine. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 25 "Definition 5.5, Page 182." }{TEXT 296 2 " " } {TEXT -1 0 "" }{TEXT 307 15 "The series for " }{XPPEDIT 18 0 "cos(z)" "6#-%$cosG6#%\"zG" }{TEXT 273 7 " is: " }{XPPEDIT 18 0 "cos(z) = sum ((-1)^n*z^(2*n)/(2*n)!, n=0..infinity)" "6#/-%$cosG6#%\"zG-%$sumG6$*() ,$\"\"\"!\"\"%\"nGF.)F'*&\"\"#F.F0F.F.-%*factorialG6#*&F3F.F0F.F//F0; \"\"!%)infinityG" }{TEXT 272 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "f:='f': F:='F': p:='p': P:='P': z:='z': Z:='Z':\nf := z -> cos( z):\nF := series(f(Z), Z=0, 12):\nP := convert(F, polynom):\np := z -> subs(Z=z, P):\n`f(z) ` = f(z);\n`f(Z) ` = F;\n`p(z) ` = p(z);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 37 "The general term for the series for " }{XPPEDIT 18 0 "cos(z) " "6#-%$cosG6#%\"zG" }{TEXT 274 6 " is:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "a:='a': n:='n': S:='S': z:='z':\na := n -> (-1)^n*z^ (2*n)/(2*n)!:\n`f(z) = cos(z)`;\n`An ` = a(n);\n`The sum of five terms :`;\nS5 := sum(a(n), n=0..4):\n`S5(z) ` = S5;\n`The sum of infinitely \+ many terms:`;\nS := sum(a(n), n=0..infinity):\n`S(z) = `, Sum(a(n), n=0..infinity) = S;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 256 25 "Definition 5.5, Page 182." } {TEXT 297 3 " " }{TEXT -1 0 "" }{TEXT 308 15 "The series for " } {XPPEDIT 18 0 "sin(z)" "6#-%$sinG6#%\"zG" }{TEXT 276 7 " is: " } {XPPEDIT 18 0 "sin(z) = sum((-1)^n*z^(2*n+1)/(2*n+1)!, n=0..infinity) " "6#/-%$sinG6#%\"zG-%$sumG6$*(),$\"\"\"!\"\"%\"nGF.)F',&*&\"\"#F.F0F. F.F.F.F.-%*factorialG6#,&*&F4F.F0F.F.F.F.F//F0;\"\"!%)infinityG" } {TEXT 275 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "g:='g': G:='G ': q:='q': Q:='Q': z:='z': Z:='Z':\ng := z -> sin(z):\nG := series(g(Z ), Z=0, 11):\nQ := convert(G, polynom):\nq := z -> subs(Z=z, Q):\n`g(z ) ` = g(z);\n`g(Z) ` = G;\n`q(z) ` = q(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 37 "The general term for the series for " }{XPPEDIT 18 0 "sin(z)" "6#-%$sinG6#%\"zG" } {TEXT 292 6 " is:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "b:='b': n: ='n': S:='S': z:='z':\nb := n -> (-1)^n*z^(2*n+1)/(2*n+1)!:\n`f(z) = s in(z)`;\n`Bn ` = b(n);\n`The sum of five terms:`;\nS5 := sum(b(n), n=0 ..4):\n`S5(z) ` = S5;\n`The sum of infinitely many terms:`;\nS := sum (b(n), n=0..infinity):\n`S(z) = `, Sum(b(n), n=0..infinity) = S;" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "Theorem 5.4" }{TEXT -1 17 " T he functions " }{XPPEDIT 18 0 "sin(z);" "6#-%$sinG6#%\"zG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "cos(z);" "6#-%$cosG6#%\"zG" }{TEXT -1 45 " are entire functions, with the properties " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "d iff(sin(z),z) = cos(z);" "6#/-%%diffG6$-%$sinG6#%\"zGF*-%$cosG6#F*" } {TEXT -1 19 " and " }{XPPEDIT 18 0 "diff(cos(z),z) = -si n(z);" "6#/-%%diffG6$-%$cosG6#%\"zGF*,$-%$sinG6#F*!\"\"" }{TEXT -1 5 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 42 "Properties of the trigonometric functions." }{TEXT 302 21 " The derivative of " }{XPPEDIT 18 0 "cos(z)" "6#-%$cosG6#%\" zG" }{TEXT 278 6 " is " }{XPPEDIT 18 0 "- sin(z)" "6#,$-%$sinG6#%\"z G!\"\"" }{TEXT 277 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "`f(z ) ` = f(z);\n`f '(z) ` = diff(f(z), z);\n`p '(z) ` = diff(p(z), z);\n` -q'(z) ` = -q(z);\n`-g(z) ` = -g(z);\n`f '(z) = -g(z)`;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 56 " Remark. In order to demonstrate that the derivative of " }{XPPEDIT 18 0 "sin(z)" "6#-%$sinG6#%\"zG" }{TEXT 279 6 " is " }{XPPEDIT 18 0 "cos(z)" "6#-%$cosG6#%\"zG" }{TEXT 280 48 " ,\nit is necessary to adju st the length of the " }{XPPEDIT 18 0 "sin(z)" "6#-%$sinG6#%\"zG" } {TEXT 281 10 " series.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 390 "g:='g ': G:='G': q:='q': Q:='Q': z:='z':\ng := z -> sin(z):\nG := series(g(Z ), Z=0, 11):\nQ := convert(G, polynom):\nq := z -> subs(Z=z, Q):\nf:=' f': F:='F': p:='p': P:='P': z:='z':\nf := z -> cos(z):\nF := series(f( Z), Z=0, 10):\nP := convert(F, polynom):\np := z -> subs(Z=z, P):\n`g( z) ` = g(z);\n`g '(z) ` = diff(g(z), z);\n`q '(z) ` = diff(q(z), z);\n `p(z) ` = p(z);\n`f(z) ` = f(z);\n`g '(z) = f(z)`;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 " To establish additional properties, it will \+ be useful to express " }{XPPEDIT 18 0 "sin(z);" "6#-%$sinG6#%\"zG" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "cos(z);" "6#-%$cosG6#%\"zG" } {TEXT -1 25 " in the cartesian form " }{XPPEDIT 18 0 "u+i*v;" "6#,&% \"uG\"\"\"*&%\"iGF%%\"vGF%F%" }{TEXT -1 111 ". (Additionally, the app lications in Chapters 9 and 10 will use these formulas.) We begin by \+ observing that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "exp(-i*z) = cos(z)+i*sin(z);" "6#/ -%$expG6#,$*&%\"iG\"\"\"%\"zGF*!\"\",&-%$cosG6#F+F**&F)F*-%$sinG6#F+F* F*" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "exp(-i*z) = cos(z)-i*sin(z); " "6#/-%$expG6#,$*&%\"iG\"\"\"%\"zGF*!\"\",&-%$cosG6#F+F**&F)F*-%$sinG 6#F+F*F," }{TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 31 "which are then used to obtain " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " } {XPPEDIT 18 0 "sin(z);" "6#-%$sinG6#%\"zG" }{TEXT -1 5 " = " } {XPPEDIT 18 0 "1/(2*i);" "6#*&\"\"\"F$*&\"\"#F$%\"iGF$!\"\"" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "exp(i*z)-exp(-i*z);" "6#,&-%$expG6#*&%\"iG\" \"\"%\"zGF)F)-F%6#,$*&F(F)F*F)!\"\"F/" }{TEXT -1 8 ") " }}{PARA 0 "" 0 "" {TEXT -1 21 " = " }{XPPEDIT 18 0 "1/(2*i); " "6#*&\"\"\"F$*&\"\"#F$%\"iGF$!\"\"" }{TEXT -1 1 "(" }{XPPEDIT 18 0 " exp(i*(x+i*y))-exp(-i*(x+i*y));" "6#,&-%$expG6#*&%\"iG\"\"\",&%\"xGF)* &F(F)%\"yGF)F)F)F)-F%6#,$*&F(F),&F+F)*&F(F)F-F)F)F)!\"\"F4" }{TEXT -1 8 ") " }}{PARA 0 "" 0 "" {TEXT -1 21 " = " } {XPPEDIT 18 0 "1/(2*i);" "6#*&\"\"\"F$*&\"\"#F$%\"iGF$!\"\"" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "exp(-y)*(cos(x)+i*sin(x))-exp(-y)*(cos(x)-i*s in(x));" "6#,&*&-%$expG6#,$%\"yG!\"\"\"\"\",&-%$cosG6#%\"xGF+*&%\"iGF+ -%$sinG6#F0F+F+F+F+*&-F&6#,$F)F*F+,&-F.6#F0F+*&F2F+-F46#F0F+F*F+F*" } {TEXT -1 8 ") " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 29 "from which we conclude that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "sin(z); " "6#-%$sinG6#%\"zG" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "sin(x)*cosh(y )+i*cos(x)*sinh(y);" "6#,&*&-%$sinG6#%\"xG\"\"\"-%%coshG6#%\"yGF)F)*(% \"iGF)-%$cosG6#F(F)-%%sinhG6#F-F)F)" }{TEXT -1 6 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "In an similar fash ion it can be shown that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "cos(z);" "6#-%$cosG6#%\"zG " }{TEXT -1 5 " = " }{XPPEDIT 18 0 "cos(x)*cosh(y)-i*sin(x)*sinh(y); " "6#,&*&-%$cosG6#%\"xG\"\"\"-%%coshG6#%\"yGF)F)*(%\"iGF)-%$sinG6#F(F) -%%sinhG6#F-F)!\"\"" }{TEXT -1 6 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 256 21 "Figure 5.7, Page 187." }{TEXT 298 29 " Graph the tra nsformation " }{XPPEDIT 18 0 "w = sin(z)" "6#/%\"wG-%$sinG6#%\"zG" } {TEXT 282 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "f:='f': z:='z ':\nf := z -> sin(z):\n`f(z) ` = f(z);\nconformal(f(z), z=-Pi/2-I*2..P i/2+I*2,\n title=`w = sin(z)`,\n grid=[9,9],numxy=[100,100],\n scal ing=constrained,\n labels=[` u`,`v `],\n view=[-4..4,-4..4]);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 1 "\n" }{TEXT 256 21 "Example for Page 186." }{TEXT 299 14 " Verify that " }{XPPEDIT 18 0 "cos(z) = 0" "6#/-%$cosG6#%\"zG\"\"!" } {TEXT 283 20 " , if and only if " }{XPPEDIT 18 0 "z = (n + 1/2)*pi" "6#/%\"zG*&,&%\"nG\"\"\"*&F(F(\"\"#!\"\"F(F(%#piGF(" }{TEXT 264 8 " \+ for " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 284 14 " an integer.\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "x:='x': y:='y':\ncos(x + I*y) = `0 `;\nevalc(cos(x + I*y)) = `0 `;\n`Solve the equations:`;\neqns := \{c os(x)*cosh(y) = 0, -sin(x)*sinh(y) = 0\}: eqns;\nsolset := solve(eqns \+ ,\{x,y\}): solset;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 34 "Remark. It is assumed that both " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 285 7 " and " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT 286 82 " are real numbers. Hence, of the four solut ions, the only valid \nsolutions are " }{XPPEDIT 18 0 "x = pi/2" "6#/ %\"xG*&%#piG\"\"\"\"\"#!\"\"" }{TEXT 266 3 " , " }{XPPEDIT 18 0 "y = 0 " "6#/%\"yG\"\"!" }{TEXT 287 7 " and " }{XPPEDIT 18 0 "x = -pi/2" "6 #/%\"xG,$*&%#piG\"\"\"\"\"#!\"\"F*" }{TEXT 267 3 " , " }{XPPEDIT 18 0 "y = 0" "6#/%\"yG\"\"!" }{TEXT 288 48 " . The other solutions are obt ained by adding " }{XPPEDIT 18 0 "n*pi" "6#*&%\"nG\"\"\"%#piGF%" } {TEXT 289 19 " to this result.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "cos(-pi/2) = cos(-Pi/2);\ncos(pi/2) = cos(Pi/2);\ncos(pi/2+pi) = \+ cos(Pi/2+Pi);\ncos(pi/2+2*pi) = cos(Pi/2+2*Pi);\ncos(pi/2+3*pi) = cos( Pi/2+3*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 54 "Or showing that the system of equations is sat isfied.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "eqns;\neval(subs(\{x= Pi/2, y=0\},eqns));\neval(subs(\{x=Pi/2+Pi, y=0\},eqns));\neval(s ubs(\{x=Pi/2+2*Pi,y=0\},eqns));\neval(subs(\{x=Pi/2+3*Pi,y=0\},eqns)); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 21 "Example for Page 185." } {TEXT 300 63 " Establish the trigonometric identity for complex numbe rs:\n " }{XPPEDIT 18 0 "cos(z[1] + z[2]) = cos(z[1])*cos(z[2]) - sin (z[1])*sin(z[2])" "6#/-%$cosG6#,&&%\"zG6#\"\"\"F+&F)6#\"\"#F+,&*&-F%6# &F)6#F+F+-F%6#&F)6#F.F+F+*&-%$sinG6#&F)6#F+F+-F;6#&F)6#F.F+!\"\"" } {TEXT 269 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 531 "x:='x': y:='y ': z:='z': \nZ1 := x[1] + I*y[1]:\nZ2 := x[2] + I*y[2]:\neq1 := cos(Z1 +Z2):\ncos(z[1] + z[2]) = eq1;\neq1 := evalc(eq1):\ncos(z[1] + z[2]) = eq1;\neq1 := expand(evalc(eq1), trig):\ncos(z[1] + z[2]) = eq1; ` `; \neq2 := cos(Z1)*cos(Z2) - sin(Z1)*sin(Z2):\ncos(z[1])*cos(z[2]) - sin (z[1])*sin(z[2]) = eq2;\neq2 := evalc(eq2):\ncos(z[1])*cos(z[2]) - sin (z[1])*sin(z[2]) = eq2;\neq2 := expand(evalc(eq2)):\ncos(z[1])*cos(z[2 ]) - sin(z[1])*sin(z[2]) = eq2; ` `;\n`Does cos(z1 + z1) = cos(z1)cos (z2) - sin(z1)sin(z2) ?`;\nevalb(eq1 = eq2);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 12 "Therefore, " }{XPPEDIT 18 0 "cos(z[1] + z[2]) = cos(z[1])*cos(z[2]) - sin(z[1])*sin (z[2])" "6#/-%$cosG6#,&&%\"zG6#\"\"\"F+&F)6#\"\"#F+,&*&-F%6#&F)6#F+F+- F%6#&F)6#F.F+F+*&-%$sinG6#&F)6#F+F+-F;6#&F)6#F.F+!\"\"" }{TEXT 290 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 1 "\n" }{TEXT 256 15 "Properti es of " }{XPPEDIT 18 0 "sin(z)" "6#-%$sinG6#%\"zG" }{TEXT 256 7 " an d " }{XPPEDIT 18 0 "cos(z)" "6#-%$cosG6#%\"zG" }{TEXT 256 15 " on P age 187." }{TEXT 301 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "`cos (x + I y) ` = evalc(cos(x + I*y)); ` ` ;\n`sin(x + I y) ` = evalc(sin( x + I*y)); ` ` ;\n`tan(x + I y) ` = evalc(tan(x + I*y)); ` ` ;\n`|sin( x + I y)| ` = evalc(abs(sin(x + I*y)));\n`|sin(x + I y)| ` = simplify( evalc(abs(sin(x + I*y))));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 271 19 "End of Section 5.4." }}}}{MARK "0 0 0" 27 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }