{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 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 1 10 0 0 0 0 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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Genev a" 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 Font 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 294 144 "\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 293 197 "\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 292 1 "\n" }{TEXT 256 31 "CHAPTER 5 ELEMENTARY FUNCTIONS" }{TEXT 280 2 "\n\n" }{TEXT 256 59 "Section 5.5 Inverse Trigonometric and Hyperbo lic Functions" }{TEXT 281 2 "\n\n" }{TEXT -1 187 " We expressed the \+ trigonometric and hyperbolic functions in Section 5.4 in terms of the \+ exponential function. In this section we look at their inverses. When \+ we solve equations such as " }{XPPEDIT 18 0 "w = sin(z);" "6#/%\"wG-%$ sinG6#%\"zG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 254 ", we will obtain formulas that involve the logarithm. Since tr igonometric and hyperbolic functions are all periodic, they are many-t o-one, hence their inverses are necessarily multivalued. The formulas \+ for the inverse trigonometric functions are given by" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 " arcsin(z)" "6#-%'arcsinG6#%\"zG" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "- i*log(i*z+sqrt(1-z^2));" "6#,$*&%\"iG\"\"\"-%$logG6#,&*&F%F&%\"zGF&F&- %%sqrtG6#,&F&F&*$F,\"\"#!\"\"F&F&F3" }{TEXT -1 6 ", " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "arccos(z)" "6#-%'arccosG6#%\"zG" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "-i*log(z+i*sqrt(1-z^2));" "6#,$*&%\"iG\"\"\"-%$logG6#,&%\"zGF&*& F%F&-%%sqrtG6#,&F&F&*$F+\"\"#!\"\"F&F&F&F3" }{TEXT -1 11 ", and \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }{XPPEDIT 18 0 "arctan(z)" "6#-%'arctanG6#%\"zG" }{TEXT -1 5 " = \+ " }{XPPEDIT 18 0 "i/2;" "6#*&%\"iG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "log((i+z)/(i-z));" "6#-%$logG6#*&,&%\"iG\"\"\"%\"zGF)F) ,&F(F)F*!\"\"F," }{TEXT -1 11 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The derivatives of any branch o f the functions above can be found by use of the chain rule:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " } {XPPEDIT 18 0 "arcsin(z)" "6#-%'arcsinG6#%\"zG" }{TEXT -1 5 " = " } {XPPEDIT 18 0 "1/sqrt(1-z^2)" "6#*&\"\"\"F$-%%sqrtG6#,&F$F$*$%\"zG\"\" #!\"\"F," }{TEXT -1 6 " , " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "arccos(z)" "6#-%'a rccosG6#%\"zG" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "1/sqrt(1-z^2)" "6#* &\"\"\"F$-%%sqrtG6#,&F$F$*$%\"zG\"\"#!\"\"F," }{TEXT -1 10 " , and \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }{XPPEDIT 18 0 "arctan(z)" "6#-%'arctanG6#%\"zG" }{TEXT -1 5 " = \+ " }{XPPEDIT 18 0 "1/(1+z^2);" "6#*&\"\"\"F$,&F$F$*$%\"zG\"\"#F$!\"\"" }{TEXT -1 5 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 295 103 "Load Maple's \"conformal mapping\" procedure. \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 260 1 "\n" }{TEXT 256 25 " Example 5.11, Page 196." }{TEXT 282 22 " Find the value of " }{XPPEDIT 18 0 "arc sin(sqrt(2))" "6#-%'arcsinG6#-%%sqrtG6#\"\"#" }{TEXT 270 3 " .\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "w:='w': z:='z': Z:='Z':\nZ := sqrt (2):\n`z ` = z;\nw := -I*log(I*Z + (1-Z^2)^(1/2)):\n-I*log(I*z + (1-z^ 2)^(1/2)) = w;\nw := evalc(w):\n-I*log(I*z + (1-z^2)^(1/2)) = w; ` `; \nw := arcsin(Z):\n`arcsin(z) ` = w;\nw := evalc(w):\n`arcsin(z) ` = w ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 23 "Example 5.13, Page 198. " }{TEXT 283 21 " Find the value of " }{XPPEDIT 18 0 "arctanh(1 + 2 \+ i)" "6#-%(arctanhG6#,&\"\"\"F'*&\"\"#F'%\"iGF'F'" }{TEXT 271 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "w:='w': z:='z': Z:='Z':\nZ := 1 \+ + I*2:\n`z ` = Z;\nw := 1/2*log((1+Z)/(1-Z)):\nlog((1+z)/(1-z))/2 = w; \nw := evalc(w):\nlog((1+z)/(1-z))/2 = w; ` `;\nw := arctanh(Z):\n`arc tan(z) ` = w;\nw := evalc(w):\n`arctanh(z) ` = w;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 16 "The Inverse Sine" }{TEXT 284 13 "\n\nFormula: \+ " }{XPPEDIT 18 0 "arcsin(z) = - i*Log(i*z + sqrt(1 - z^2))" "6#/-%'a rcsinG6#%\"zG,$*&%\"iG\"\"\"-%$LogG6#,&*&F*F+F'F+F+-%%sqrtG6#,&F+F+*$F '\"\"#!\"\"F+F+F7" }{TEXT 272 74 " ,\nMaple can \"almost\" derive this formula. Can you tell the difference ?\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 367 "w:='w': W:='W': z:='z':\neqn := z = sin(w): eqn;\neq n := z = (exp(I*w)-exp(-I*w))/(2*I): eqn;\neqn := eqn*exp(I*w): eqn;\n eqn := expand(eqn): eqn;\n`Substitute:`;\nW = exp(I*w);\neqn := subs(e xp(I*w)= W, eqn): eqn;\nsolset := solve(eqn, W):\neqn := W = expand(so lset[2]): eqn;\neqn := I*w = log(expand(solset[2])): eqn;\neqn := w = \+ -I*log(expand(solset[2])): eqn;\nw = arcsin(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 11 "Formula \+ " }{XPPEDIT 18 0 "arcsin(z) = - i*Log(i*z + sqrt(1 - z^2))" "6#/-%'arc sinG6#%\"zG,$*&%\"iG\"\"\"-%$LogG6#,&*&F*F+F'F+F+-%%sqrtG6#,&F+F+*$F' \"\"#!\"\"F+F+F7" }{TEXT 273 72 " , is correct, we can verify this gra phically.\n(At least for values of " }{XPPEDIT 18 0 "z" "6#%\"zG" } {TEXT 274 27 " in the upper half plane " }{XPPEDIT 18 0 "Im(z) > 0" "6#2\"\"!-%#ImG6#%\"zG" }{TEXT 275 4 " .)\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "f:='f': z:='z':\nf := z -> - I*log(I*z + sqrt(1 - z^ 2)):\n`f(z) ` = f(z);\nconformal(f(z), z = -5+I*0.0001..5+I*10,\n tit le=`w = - I*log(I*z + sqrt(1 - z^2))`,\n grid=[11,11], numxy=[50,50], \n scaling=constrained,\n labels=[`u`,`v `],\n view=[-1.7..1.7,0.. 3.4]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "`f(z) = arcsin(z )`;\nconformal(arcsin(z), z = -5+I*0.0001..5+I*10,\n title=`w = arcsi n(z)`,\n grid=[11,11], numxy=[50,50],\n scaling=constrained,\n labe ls=[`u`,`v `],\n view=[-1.7..1.7,0..3.4]);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 154 "REMARK. If y ou mess around with the square root it will be wrong. The portion tha t is supposed be \nin the first quadrant appears symmetrically through " }{XPPEDIT 18 0 "z = pi/2" "6#/%\"zG*&%#piG\"\"\"\"\"#!\"\"" } {TEXT 262 26 " in the fourth quadrant.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "f:'f':\nf := z -> - I*log(I*z + I*sqrt(z^2 - 1)):\n` f(z) ` = f(z);\nconformal(f(z), z = -5+I*0.0001..5+I*10,\n title=`w = - I*log(I*z + I*sqrt(z^2 - 1))`,\n grid=[11,11], numxy=[50,50],\n s caling=constrained,\n labels=[`u`,`v`]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 18 "The Inverse Cosine" } {TEXT 286 12 "\n\nFormula " }{XPPEDIT 18 0 "arccos(z) = - i*Log( z + i*sqrt(1 - z^2))" "6#/-%'arccosG6#%\"zG,$*&%\"iG\"\"\"-%$LogG6#,&F'F+ *&F*F+-%%sqrtG6#,&F+F+*$F'\"\"#!\"\"F+F+F+F7" }{TEXT 263 109 " is cor rect, we can verify this graphically.\n(At least for values of z in th e upper half plane Im(z) > 0.)\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "f:='f': z:='z':\nf := z -> - I*log(z + I*sqrt(1 - z^2)):\n`f(z) ` = f(z);\nconformal(f(z), z = -Pi/2+I*0.0001..Pi/2+I*3,\n title=`w = \+ - I*log(z + I*sqrt(1 - z^2))`,\n grid=[11,11], numxy=[50,50],\n scal ing=constrained,\n labels=[` u`,`v `],\n view=[0..3.15,-2..0]);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "`f(z) = arccos(z)`;\nconf ormal(arccos(z), z = -Pi/2+I*0.0001..Pi/2+I*3,\n title=`w = arccos(z) `,\n grid=[11,11], numxy=[50,50],\n scaling=constrained,\n labels=[ ` u`,`v `],\n view=[0..3.15,-2..0]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 66 "REMARK. If you m ess around with the square root it will be wrong.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "f:='f': z:='z':\nf := z -> - I*log(z + sqrt(z^2 - 1)):\n`f(z) ` = f(z);\nconformal(f(z), z = -Pi/2+I*0.0001..Pi/2+I*3, \n title=`w = - I*log(z + sqrt(z^2 - 1))`,\n grid=[20,20], numxy=[10 0,100],\n scaling=constrained,\n labels=[`u`,`v`]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 19 "The Inverse Tangent" }{TEXT 287 11 "\n Formula " }{XPPEDIT 18 0 "arctan(z) = i/2*Log((i + z)/(i - z))" "6#/ -%'arctanG6#%\"zG*(%\"iG\"\"\"\"\"#!\"\"-%$LogG6#*&,&F)F*F'F*F*,&F)F*F 'F,F,F*" }{TEXT 276 71 " is correct, we can verify this graphically. \n(At least for values of " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 278 27 " in the upper half plane " }{XPPEDIT 18 0 "Im(z) > 0" "6#2\"\"!- %#ImG6#%\"zG" }{TEXT 277 4 " .)\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 255 "f:='f': z:='z':\nf := z -> I/2 *log((I+z)/(I-z)):\n`f(z) ` = f(z) ;\nconformal(f(z), z = 0.0001+I*0.0001..1.5706+I*3,\n title=`w = I/2 \+ *log((I+z)/(I-z))`,\n grid=[11,11], numxy=[100,100],\n scaling=const rained,\n labels=[` u`,`v `],\n view=[0..2,0..1.6]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "`f(z) = arctan(z)`;\nconformal(arc tan(z), z = 0.0001+I*0.0001..1.5706+I*3,\n title=`w = arctan(z)`,\n \+ grid=[20,20], numxy=[100,100],\n scaling=constrained,\n labels=[` \+ u`,`v `],\n view=[0..2,0..1.6]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 27 "The Inverse Hyperbolic Sine" }{TEXT 288 12 "\n\nFormula \+ " }{XPPEDIT 18 0 "arcsinh(z) = Log(z + sqrt(z^2 + 1))" "6#/-%(arcsin hG6#%\"zG-%$LogG6#,&F'\"\"\"-%%sqrtG6#,&*$F'\"\"#F,F,F,F," }{TEXT 265 46 " is correct, we can verify this graphically.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 242 "f:='f': z:='z':\nf := z -> log(z + sqrt(z^2 + 1)) :\n`f(z) ` = f(z);\nconformal(f(z), z=I*0.0001..10+I*10,\n title=`w = log(z + sqrt(z^2 + 1))`,\n grid=[11,11],numxy=[50,50],\n scaling=co nstrained,\n labels=[` u`,`v `],\n view=[0..4,0..1.6]);" }}} {EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 194 "`f(z) = arcsinh(z)`;\ncon formal(arcsinh(z), z = I*0.0001..10+I*10,\n title=`w = arcsinh(z)`,\n grid=[11,11], numxy=[50,50],\n scaling=constrained,\n labels=[` \+ u`,`v `],\n view=[0..4,0..1.6]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 29 "The Inverse Hyperbolic Cosine" }{TEXT 289 1 "\n" }} {PARA 0 "" 0 "" {TEXT 290 10 "Formula " }{XPPEDIT 18 0 "arccosh(z) = Log(z + sqrt(z^2 - 1))" "6#/-%(arccoshG6#%\"zG-%$LogG6#,&F'\"\"\"-%%s qrtG6#,&*$F'\"\"#F,F,!\"\"F," }{TEXT 266 95 " is correct, we can veri fy this graphically.\nBut it is correct only in quadrants I and IV. \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "f:='f': z:='z':\nf := z -> l og(z + sqrt(z^2 - 1)):\n`f(z) ` = f(z);\nconformal(f(z), z = I*0.0001. .10+I*10,\n title=`w = log(z + sqrt(z^2 - 1))`,\n grid=[11,11], numx y=[50,50],\n scaling=constrained,\n labels=[` u`,`v `],\n view=[ 0..4,0..1.6]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "`f(z) = \+ arccosh(z)`;\nconformal(arccosh(z), z = I*0.0001..10+I*10,\n title=`w = arccosh(z)`,\n grid=[11,11], numxy=[50,50],\n scaling=constrained ,\n labels=[` u`,`v `],\n view=[0..4,0..1.6]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 141 "REMAR K. However, for other places it might not agree!\nFor Example in Quadr ant II and III.\nHere we must use the other branch of square root " }{XPPEDIT 18 0 "arccosh(z) = Log(z - sqrt(z^2 - 1))" "6#/-%(arccoshG6# %\"zG-%$LogG6#,&F'\"\"\"-%%sqrtG6#,&*$F'\"\"#F,F,!\"\"F3" }{TEXT 268 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "f:='f': z:='z':\nf := z -> log(z - sqrt(z^2 - 1)):\n`f(z) ` = f(z);\nconformal(f(z), z = -10+ I*0.0001..-0.0001+I*10,\n title=`w = log(z - sqrt(z^2 - 1))`,\n grid =[11,11], numxy=[50,50],\n scaling=constrained,\n labels=[` u`,`v \+ `],\n view=[0..4,0..3.2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "`f(z) = arccosh(z)`;\nconformal(arccosh(z), z = -10+I*0.0001..- 0.0001+I*10,\n title=`w = arccosh(z)`,\n grid=[11,11], numxy=[50,50] ,\n scaling=constrained,\n labels=[` u`,`v `],\n view=[0..4,0..3 .2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 30 "The Inverse Hyperboli c Tangent" }{TEXT 291 11 "\nFormula " }{XPPEDIT 18 0 "arctanh(z) = L og((1 + z)/(1 - z))/2" "6#/-%(arctanhG6#%\"zG*&-%$LogG6#*&,&\"\"\"F.F' F.F.,&F.F.F'!\"\"F0F.\"\"#F0" }{TEXT 279 46 " is correct, we can veri fy this graphically.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "f:='f': \+ z:='z':\nf := z -> 1/2 *log((1+z)/(1-z)):\n`f(z) ` = f(z);\nconformal( f(z), z = I*0.0001..10+I*10,\n title=`w = 1/2 *log((1+z)/(1-z))`,\n \+ grid=[11,11], numxy=[50,50],\n scaling=constrained,\n labels=[`u`,`v `],\n view=[0..1,0..1.6]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "`f(z) = arctanh(z)`;\nconformal(arctanh(z), z = I*0.0001..10+I* 10,\n title=`w = arctanh(z)`,\n grid=[11,11], numxy=[50,50],\n scal ing=constrained,\n labels=[`u`,`v `],\n view=[0..1,0..1.6]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 19 "End of Section 5 .5." }}}}{MARK "0 0 0" 29 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }