{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 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 301 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 304 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "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 "Mo naco" 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 "Monaco" 1 9 0 0 255 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 297 144 "\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 296 197 "\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 293 1 "\n" }{TEXT 256 42 "CHAPTER 3 ANALYTIC and HARMONIC FUNCTIONS" }{TEXT 288 2 "\n\n" }{TEXT 256 37 "Section 3.1 Differentiable Functio ns" }{TEXT 289 2 "\n\n" }{TEXT -1 281 " Does the notion of a derivati ve of a complex function make sense? If so, how should it be defined, \+ and what does it represent? These and other questions will be the focu s of the next few sections.\n\011\nUsing our imagination, we take our \+ lead from elementary Calculus and define the " }{TEXT 298 11 "derivate of" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 1 " " } {TEXT 299 2 "at" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z[0];" "6#&%\"zG6#\" \"!" }{TEXT -1 14 ", written f '(" }{XPPEDIT 18 0 "z[0];" "6#&%\"zG6# \"\"!" }{TEXT -1 7 "), by " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " f '(" }{XPPEDIT 18 0 "z[0];" "6#&%\"zG6 #\"\"!" }{TEXT -1 6 ") = " }{XPPEDIT 18 0 "Limit((f(z)-f(z[0]))/(z-z [0]),z = z[0]);" "6#-%&LimitG6$*&,&-%\"fG6#%\"zG\"\"\"-F)6#&F+6#\"\"!! \"\"F,,&F+F,&F+6#F1F2F2/F+&F+6#F1" }{TEXT -1 10 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "provided that t he limit exists. When this happens, we say that the function " } {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 5 " is " }{TEXT 300 14 "differ entiable" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "z[0];" "6#&%\"zG6#\"\"!" }{TEXT -1 15 ". If we write " }{XPPEDIT 18 0 "Delta*z = z-z[0];" "6#/ *&%&DeltaG\"\"\"%\"zGF&,&F'F&&F'6#\"\"!!\"\"" }{TEXT -1 53 ", then thi s definition can be expressed in the form " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " f '(" }{XPPEDIT 18 0 "z[0 ];" "6#&%\"zG6#\"\"!" }{TEXT -1 6 ") = " }{XPPEDIT 18 0 "Limit((f(z[ 0]+Delta*z)-f(z[0]))/(Delta*z),Delta*z = 0)" "6#-%&LimitG6$*&,&-%\"fG6 #,&&%\"zG6#\"\"!\"\"\"*&%&DeltaGF0F-F0F0F0-F)6#&F-6#F/!\"\"F0*&F2F0F-F 0F7/*&F2F0F-F0F/" }{TEXT -1 7 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 1 "\n" }{TEXT 256 21 "Example 3.1, Page 94." }{TEXT 290 54 " Use the limit definition t o find the derivative of " }{XPPEDIT 18 0 "f(z) = z^3" "6#/-%\"fG6#% \"zG*$F'\"\"$" }{TEXT 274 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "df:='df': dz:='dz': f:='f': z:='z' : z0:='z0':\nf := z ->z^3:\n`f(z) ` = f(z); ` `;\n`Form the difference quotient.`;\ndfdz := (f(z) - f(z0))/(z - z0):\ndf/dz = dfdz; ` `;" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Substitution of " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "z" "6 #%\"zG" }{TEXT -1 20 " is indeterminate.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(z=z0, dfdz);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 56 "The derivative is the limit of the difference quotient.\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Limit(df/dz, z=z0) = limit(dfdz, z= z0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 65 "The difference quotient \+ can be simplified before taking a limit.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "df/dz = dfdz;\nQ := simplify(dfdz):\ndf/dz = Q;\n`f ' (z0) = `, Limit(df/dz, z=z0) = subs(z=z0, Q);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Which is known to be the formula for the derivative of " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 2 " ." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 1 "\n" }{TEXT 256 21 "Example 3.2, Page 94." }{TEXT 291 41 " Use \+ the limit definition to show that " }{XPPEDIT 18 0 "f(z) = conjugate( z)" "6#/-%\"fG6#%\"zG-%*conjugateG6#F'" }{TEXT 275 29 " is NOWHERE di fferentiable.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f:='f': z:='z': \nf := z -> conjugate(z):\n`f(z) ` = f(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 48 "Form the difference quotient using a change in " } {XPPEDIT 18 0 "dx" "6#%#dxG" }{TEXT 277 6 " in " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 276 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 423 "dx :='dx': dX:='dX': dy:='dy': dY:='dY': \nX:='X': X0:='X0': Y:='Y': Y0:= 'Y0':\nvar := \{dX='dx',dY='dy',X='x',X0='x0',Y='y',Y0='y0'\}:\nassume (dX,real); assume(dY,real);\nassume(X,real); assume(Y,real);\nassume( X0,real); assume(Y0,real);\ndFdX := (f(X0 + dX + I*Y0) - f(X0 + I*Y0)) /\n (X0 + dX + I*Y0 - X0 - I*Y0):\ndfdx := subs(var, dFdX):\n`f(z0 +dx) ` = f(z0+dx),` and `,\n`f(z0) ` = f(z0); ` `;\n`f(z0+dx) - f(z 0)`/dx = dfdx;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 35 "And this limit is easy to compute:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "lim1 := l imit(dfdx , dx=0):\nLimit(df/dx, x=0) = lim1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 48 "Form the difference quotient using a change in " } {XPPEDIT 18 0 "dy" "6#%#dyG" }{TEXT 279 6 " in " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT 278 4 " .\n\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 181 " dFdY := (f(X0 + I*(Y0+dY)) - f(X0 + I*Y0))/\n (X0 + I*(Y0+dY) - X0 - I*Y0):\ndfdy := subs(var, dFdY):\n`f(z0+Idy) - f(z0)`/`I dy` = dfdy; \n`f(z0+Idy) - f(z0)`/`I dy` = simplify(dfdy);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 35 "And this limit is easy to compute:\n" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 54 "lim2 := limit(dfdy , dy=0):\nLimit(df/dy, dy=0 ) = lim2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 44 "The two limits are \+ different for any point " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" } {TEXT 280 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Limit(df/dx, d x=0) <> Limit(df/dy, dy=0);\nlim1 <> lim2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 50 "Since t he two limits are different for any point " }{XPPEDIT 18 0 "z=z[0]" " 6#/%\"zG&F$6#\"\"!" }{TEXT 287 4 " , " }{XPPEDIT 18 0 "diff(f(z), z) " "6#-%%diffG6$-%\"fG6#%\"zGF)" }{TEXT 282 26 " does NOT exist for an y " }{XPPEDIT 18 0 "z= z[0]" "6#/%\"zG&F$6#\"\"!" }{TEXT 281 3 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 25 "Definition 3.1: Analyti c" }{TEXT 302 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 34 "We say that the complex function " }{XPPEDIT 18 0 "f; " "6#%\"fG" }{TEXT -1 5 " is " }{TEXT 301 8 "analytic" }{TEXT -1 15 " at the point " }{XPPEDIT 18 0 "z[0];" "6#&%\"zG6#\"\"!" }{TEXT -1 26 " provided there is some " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilon G" }{TEXT -1 15 ">0 such that " }{XPPEDIT 18 0 "`f '`(z);" "6#-%$f~' G6#%\"zG" }{TEXT -1 18 " exists for all " }{XPPEDIT 18 0 "z*epsilon* D[epsilon](z[0]);" "6#*(%\"zG\"\"\"%(epsilonGF%-&%\"DG6#F&6#&F$6#\"\"! F%" }{TEXT -1 20 ". In other words, " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 38 " must be differentiable not only at " }{XPPEDIT 18 0 " z[0];" "6#&%\"zG6#\"\"!" }{TEXT -1 34 ", but also at all points in som e " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 18 "-neighborh ood of " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 2 ". " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 23 "D efinition 3.1: Entire" }{TEXT 303 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "f;" "6#%\"fG " }{TEXT -1 47 " is analytic on the whole complex plane then " } {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 13 " said to be " }{TEXT 304 6 "entire" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 0 "" }} {PARA 0 "" 0 "" {TEXT 256 41 "The rules for differentiation on page 96 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "c:='c': f:='f': F:='F': z:='z':\nF := z -> c*f(z): \nprint(`Derivative of a scalar multiple.`);\n`F(z)` = F(z);\n`F '(z)` = diff(F(z), z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "a:='a ': b:='b': f:='f': g:='g': H:='H': z:='z':\nH := z -> a*f(z) + b*g(z) :\nprint(`Derivative of a linear combination multiple.`);\n`H(z)` = H( z);\n`H '(z)` = diff(H(z), z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "f:='f': g:='g': P:='P': z:='z':\nP := z -> f(z)*g(z):\nprint (`Derivative of the product of two functions.`);\n`P(z)` = P(z);\n`P ' (z)` = diff(P(z), z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "f :='f': F:='F': g:='g': Q:='Q': z:='z':\nQ := z -> f(z)/g(z):\nprint(` Derivative of the quotient of two functions.`);\n`Q(z)` = Q(z);\n`Q '( z)` = normal(diff(Q(z), z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "f:='f': g:='g': H:='H': z:='z':\nH := z -> f(g(z)):\nprint(`De rivative of the composition of two functions.`);\n`H(z)` = H(z);\n`H ' (z)` = diff(H(z), z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 21 "E xample 3.3, Page 96." }{TEXT 292 43 " Use the rules to find the deriv ative of " }{XPPEDIT 18 0 "f(z)= z^2 + i*2*z +3" "6#/-%\"fG6#%\"zG,(* $F'\"\"#\"\"\"*(%\"iGF+F*F+F'F+F+\"\"$F+" }{TEXT 283 57 " , \nand the n use this result to find the derivative of " }{XPPEDIT 18 0 "(f(z))^ 4 = (z^2 + i*2*z + 3)^4" "6#/*$-%\"fG6#%\"zG\"\"%*$,(*$F(\"\"#\"\"\"*( %\"iGF.F-F.F(F.F.\"\"$F.F)" }{TEXT 284 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "f:='f': g:= 'g': h:='h': w:='w': z:='z':\nf := z -> \+ z^2 + I*2*z + 3:\ng := z -> z^4:\nh := z -> g(f(z)):\n`f(z)` = f(z);\n `f '(z)` = diff(f(z), z); ` `;\n`g(z)` = g(z);\n`g '(z)` = diff(g(z), \+ z); ` `;\n`h(z) = g(f(z))` = g(f(z));\n`h '(z)` = diff(h(z), z);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 48 "Theorem 3.1 (Differentiable i mplies continuous)" }{TEXT 305 2 " " }{MPLTEXT 1 0 0 "" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 24 " is differentiable at " }{XPPEDIT 18 0 "z[0];" "6#&%\"zG6#\"\"!" }{TEXT -1 8 ", then " } {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 20 " is continuous at " } {XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 30 "Theorem \+ 3.2 (L'Hopital's Rule)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Assume " }{XPPEDIT 18 0 "f;" "6#%\"f G" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "g;" "6#%\"gG" }{TEXT -1 19 " \+ are analytic at " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 15 ". If we have " }{XPPEDIT 18 0 "f(z[0]) = 0;" "6#/-%\"fG6#&%\"zG6 #\"\"!F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "g(z[0]) = 0;" "6#/-%\"gG6#& %\"zG6#\"\"!F*" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "`g'(`*z[0]*`)` <> 0;" "6#0*(%$g'(G\"\"\"&%\"zG6#\"\"!F&%\")GF&F*" }{TEXT -1 8 ", then \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }{XPPEDIT 18 0 "Limit(f(z)/g(z),z = z[0]);" "6#-%&LimitG6$*&-%\"fG6# %\"zG\"\"\"-%\"gG6#F*!\"\"/F*&F*6#\"\"!" }{TEXT -1 6 " = " } {XPPEDIT 18 0 "Limit(`f '(z)`/`g '(z)`,z = z[0]);" "6#-%&LimitG6$*&%'f ~'(z)G\"\"\"%'g~'(z)G!\"\"/%\"zG&F,6#\"\"!" }{TEXT -1 7 " . " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 40 "E xample about L'Hopital's Rule, Page 98." }{TEXT 295 1 "\n" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "f:='f': g:='g': z:='z': \nf := z -> z^2 + z - 1 - 3*I:\ng := z -> z^2 - 2*z + 2:\n`f(z) ` = f( z);\n`g(z) ` = g(z); ` `;\n`f(z)/g(z) ` = f(z)/g(z);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 24 "Direct substitution of " }{XPPEDIT 18 0 "z = 1 + i" "6#/%\"zG,&\"\"\"F&%\"iGF&" }{TEXT 285 19 " is indeterm inate." }{MPLTEXT 1 0 0 "" }{TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f(1+I)/g(1+I);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 7 "Find " }{XPPEDIT 18 0 "limit(f(z)/g(z), z =1+i)" "6#-%&limitG6$*& -%\"fG6#%\"zG\"\"\"-%\"gG6#F*!\"\"/F*,&F+F+%\"iGF+" }{TEXT 286 3 " .\n " }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Limit(f(z)/g (z), z=1+I) = limit(f(z)/g(z), z=1+I);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 51 "Which agrees with the L'Hopital rule computation. \n" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "f1:='f1': g1:= 'g1': Z:='Z':\nf1 := z -> subs(Z=z, diff(f(Z), Z)):\ng1 := z -> subs(Z =z, diff(g(Z), Z)):\n`f '(z)/g '(z)` = f1(z)/g1(z); ` `;\n`f '(1+I)` = f1(1+I);\n`g '(1+I)` = g1(1+I); ` `;\n`f '(1+I)/g '(1+I)` = f1(1+I)/g 1(1+I);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 40 "Theorem 3.3 (The first identity theorem)" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let \+ " }{XPPEDIT 18 0 "f = u+i*v;" "6#/%\"fG,&%\"uG\"\"\"*&%\"iGF'%\"vGF'F' " }{TEXT -1 41 " be an analytic function in the domain " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 20 ". Suppose for all " }{XPPEDIT 18 0 "z*epsilon*D;" "6#*(%\"zG\"\"\"%(epsilonGF%%\"DGF%" }{TEXT -1 8 " t hat " }{XPPEDIT 18 0 "abs(f(z)) = K;" "6#/-%$absG6#-%\"fG6#%\"zG%\"KG " }{TEXT -1 10 ", where " }{XPPEDIT 18 0 "K;" "6#%\"KG" }{TEXT -1 17 " is a constant. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " } {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 15 " constant in " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 19 "End of Section 3.1." } }}}{MARK "0 0 0" 24 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }