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Mathews Rus sell W. Howell\nmathews@fullerton.edu howell@westmont.edu\n\nCompl imentary software to accompany the textbook:" }}{PARA 257 "" 0 "" {TEXT 257 82 "COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9" }{TEXT 292 197 "\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 289 1 "\n" }{TEXT 256 29 "CHAPTER 2 COMPLEX FUNCTIONS" }{TEXT 283 2 "\n\n" }{TEXT 256 48 "Section 2.2 Transformations and Linear Mappings " }{TEXT 284 1 "\n" }}{PARA 257 "" 0 "" {TEXT 291 88 " We not take o ur first look at the geometric interpretation of a complex function. I f " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT 294 58 " is the domain of def inition of the real-valued functions " }{XPPEDIT 18 0 "u(x,y);" "6#-% \"uG6$%\"xG%\"yG" }{TEXT 295 5 " and " }{XPPEDIT 18 0 "v(x,y);" "6#-% \"vG6$%\"xG%\"yG" }{TEXT 296 31 ", then the system of equations " } {XPPEDIT 18 0 "u = u(x,y);" "6#/%\"uG-F$6$%\"xG%\"yG" }{TEXT 297 5 " a nd " }{XPPEDIT 18 0 "v = v(x,y);" "6#/%\"vG-F$6$%\"xG%\"yG" }{TEXT 298 44 " describes a transformation or mapping from " }{XPPEDIT 18 0 " D;" "6#%\"DG" }{TEXT 299 36 " in the xy-plane into the uv-plane. " } {TEXT -1 0 "" }{TEXT 300 24 "Therefore, the function " }{XPPEDIT 18 0 "f(z) = u(x,y)+i*v(x,y);" "6#/-%\"fG6#%\"zG,&-%\"uG6$%\"xG%\"yG\"\"\"* &%\"iGF.-%\"vG6$F,F-F.F." }{TEXT 301 63 " can be considered as a mappi ng or transformation from the set " }{XPPEDIT 18 0 "D;" "6#%\"DG" } {TEXT 302 31 " in the z-plane onto the range " }{XPPEDIT 18 0 "R;" "6# %\"RG" }{TEXT 303 19 " in the w-plane. \n" }}{PARA 0 "" 0 "" {TEXT -1 7 " If " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 43 " is a subs et of the domain of definition " }{XPPEDIT 18 0 "D;" "6#%\"DG" } {TEXT -1 16 ", then the set " }{XPPEDIT 18 0 "B = \{`w = f(z): `*z*ep silon*A\};" "6#/%\"BG<#**%+w~=~f(z):~G\"\"\"%\"zGF(%(epsilonGF(%\"AGF( " }{TEXT -1 16 " is called the " }{TEXT 257 5 "image" }{TEXT -1 13 " \+ of the set " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 7 ", and " } {XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 13 " is said to " }{TEXT 257 12 "map A onto B" }{TEXT -1 119 ". The image of a single point is a s ingle point, and the image of the entire domain D is the range R. \+ The mapping " }{XPPEDIT 18 0 "w = f(z);" "6#/%\"wG-%\"fG6#%\"zG" } {TEXT -1 15 " is said to be " }{TEXT 257 13 "from A into S" }{TEXT -1 18 " if the image of " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 19 " i s contained in " }{XPPEDIT 18 0 "S;" "6#%\"SG" }{TEXT -1 62 ". The i nverse image of a point w is the set of all points " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 6 " in " }{XPPEDIT 18 0 "D" "6#%\"DG" } {TEXT -1 13 " such that " }{XPPEDIT 18 0 "w = f(z)" "6#/%\"wG-%\"fG6 #%\"zG" }{TEXT -1 156 ". The inverse image of a point may be one poin ts, several points, or none at all. If the latter case occurs, then t he point w is not in the range of f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " The function " }{XPPEDIT 18 0 "f " "6#%\"fG" }{TEXT -1 54 " is said to be one-to-one if it maps distin ct points " }{XPPEDIT 18 0 "z[1] <> z[2]" "6#0&%\"zG6#\"\"\"&F%6#\"\"# " }{TEXT -1 23 " onto distinct points " }{XPPEDIT 18 0 "f(z[1]) <> f( z[2])" "6#0-%\"fG6#&%\"zG6#\"\"\"-F%6#&F(6#\"\"#" }{TEXT -1 7 ". If \+ " }{XPPEDIT 18 0 "w = f(z)" "6#/%\"wG-%\"fG6#%\"zG" }{TEXT -1 16 " ma ps the set " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 31 " one-to-one \+ and onto the set " }{XPPEDIT 18 0 "B" "6#%\"BG" }{TEXT -1 25 ", then for each w in " }{XPPEDIT 18 0 "B" "6#%\"BG" }{TEXT -1 34 " there exists exactly one point " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 6 " in " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 13 " such that " } {XPPEDIT 18 0 "w = f(z)" "6#/%\"wG-%\"fG6#%\"zG" }{TEXT -1 52 ". Then \+ loosely speaking, we can solve the equation " }{XPPEDIT 18 0 "w = f(z )" "6#/%\"wG-%\"fG6#%\"zG" }{TEXT -1 18 " by solving for " } {XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 20 " as a function of " } {XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 16 ". That is, the " }{TEXT 257 16 "inverse function" }{TEXT -1 2 " " }{XPPEDIT 18 0 "z = g(w);" "6#/%\"zG-%\"gG6#%\"wG" }{TEXT -1 50 " can be found, and the followin g equations hold: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "g(f(z)) = z;" "6#/-%\"gG6#-%\"fG6# %\"zGF*" }{TEXT -1 11 " for all " }{XPPEDIT 18 0 "z*epsilon*A;" "6#* (%\"zG\"\"\"%(epsilonGF%%\"AGF%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "f( g(w)) = w;" "6#/-%\"fG6#-%\"gG6#%\"wGF*" }{TEXT -1 11 " for all " } {XPPEDIT 18 0 "w*epsilon*B;" "6#*(%\"wG\"\"\"%(epsilonGF%%\"BGF%" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 " We now turn our attention to the investigation of some \+ elementary mappings. Let " }{XPPEDIT 18 0 "B = a+i*b;" "6#/%\"BG,&% \"aG\"\"\"*&%\"iGF'%\"bGF'F'" }{TEXT -1 59 " denote a fixed complex n umber. Then the transformation " }{XPPEDIT 18 0 "`w = T(z) = z + B ` = x+a+i*(y+b);" "6#/%2w~=~T(z)~=~z~+~B~G,(%\"xG\"\"\"%\"aGF'*&%\"iGF' ,&%\"yGF'%\"bGF'F'F'" }{TEXT -1 74 " is a one-to-one mapping of the z -plane onto the w-plane and is called a " }{TEXT 257 11 "translation" }{TEXT -1 83 ". This transformation can be visualized as a rigid tran slation whereby the point " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 35 " is displaced through the vector " }{XPPEDIT 18 0 "a+i*b" "6#,&% \"aG\"\"\"*&%\"iGF%%\"bGF%F%" }{TEXT -1 23 " to its new position " } {XPPEDIT 18 0 "w = T(z);" "6#/%\"wG-%\"TG6#%\"zG" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The inver se mapping is given by " }{XPPEDIT 18 0 "`w = `*T^`-1`*`(z) = w - B ` = u-a+i*(v-b);" "6#/*(%%w~=~G\"\"\")%\"TG%#-1GF&%-(z)~=~w~-~B~GF&,(% \"uGF&%\"aG!\"\"*&%\"iGF&,&%\"vGF&%\"bGF.F&F&" }{TEXT -1 1 " " } {XPPEDIT 18 0 "T^`-1`;" "6#)%\"TG%#-1G" }{TEXT -1 17 " and shows that \+ " }{XPPEDIT 18 0 "T" "6#%\"TG" }{TEXT -1 61 " is a one-to-one mappin g from the z-plane onto the w-plane. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 304 119 "Load Maple's \"elimina te\" and \"conformal mapping\" procedures.\nMake sure this is done onl y ONCE during a Maple session." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "readlib(eliminate):\nwith(plots):" }{TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 290 1 "\n" }{TEXT 256 21 "Example 2.6, Page 55." }{TEXT 285 27 " Show that the functio n " }{XPPEDIT 18 0 "f(z) = i z" "6#/-%\"fG6#%\"zG*&%\"iG\"\"\"F'F*" } {TEXT 282 16 " maps the line " }{XPPEDIT 18 0 "y = x + 1" "6#/%\"yG,& %\"xG\"\"\"F'F'" }{TEXT 281 17 " onto the line " }{XPPEDIT 18 0 "v = -u - 1" "6#/%\"vG,&%\"uG!\"\"\"\"\"F'" }{TEXT 280 2 " ." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 329 "f:='f': x: ='x': X:='X': y:='y': Y:='Y': z:='z': Z:='Z':\nassume(X, real); assume (Y, real);\nZ := X + I*Y:\nf := z -> I*z:\n`f(z) ` = f(z);\n`Find the \+ image of the line y = x + 1`;\neqns := \{u = Re(f(Z)), v = Im(f(Z)), \+ y = x + 1\}:\neqns2 := (subs(X=x,Y=y,eqns)): eqns2;\n`Eliminate x and \+ y from these equations.`;\neliminate(eqns2,\{x,y\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 260 34 "Thus we see that the solution is " }{XPPEDIT 18 0 "u + v + 1 = 0" "6#/,(% \"uG\"\"\"%\"vGF&F&F&\"\"!" }{TEXT 278 6 " or " }{XPPEDIT 18 0 "v = \+ -u - 1" "6#/%\"vG,&%\"uG!\"\"\"\"\"F'" }{TEXT 279 2 " ." }}}{EXCHG {PARA 257 "" 0 "" {TEXT 265 1 "\n" }{TEXT 256 21 "Example 2.9, Page 58 ." }{TEXT 286 40 " Show that the linear transformation " }{XPPEDIT 18 0 "w = i z + i" "6#/%\"wG,&*&%\"iG\"\"\"%\"zGF(F(F'F(" }{TEXT 277 28 " maps the right half plane " }{XPPEDIT 18 0 "Re(z) > 1" "6#2\"\" \"-%#ReG6#%\"zG" }{TEXT 276 29 " onto the upper half plane " } {XPPEDIT 18 0 "Im(w) > 2" "6#2\"\"#-%#ImG6#%\"wG" }{TEXT 275 2 " ." }} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 538 " f:='f': u:='u': v:='v': w:='w': x:='x': y:='y': z:='z':\nf := z -> I*z + I:\n`w ` = f(z);\n`u + I v ` = f(x + I*y);\n`u + I v ` = evalc(f(x \+ + I*y)); ` `;\n`Solve for z in terms of w.`;\nsolset := expand(solv e(W = f(z), z)):\ng := w -> subs(W=w,solset):\n`z ` = g(w);\n`x + I y \+ ` = g(u + I*v);\n`x + I y ` = evalc(g(u + I*v));\n`We will use the sub stitutions:`;\neqns := \{x=v-1, y=-u\}: eqns; ` `;\n`Now find the imag e of the right half plane.`;\nineq := Re(z) > 1: ineq;\nineq := x > 1: ineq;\nineq := subs(eqns,ineq): ineq;\nineq := ineq + (1<1): ineq;" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 261 39 "This solution is the upper half plane " }{XPPEDIT 18 0 "2 < Im(w)" "6#2\"\"#-%#ImG6#%\"wG" }{TEXT 274 2 " ." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "f:='f': z:='z':\nf := z -> I*z + I:\n`f(z) ` = f(z);\nconformal(f(z), z=1-6*I..5+4*I,\n ti tle=`w = I*z + I`,\n grid=[9,11],numxy=[9,11],\n scaling=constrained ,\n labels=[`u `,`v `],\n view=[-4.25..6.25,-0.25..6.25]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 22 "Example 2.10, Page 60. " }{TEXT 287 41 " Show that the image of the open disk " }{XPPEDIT 18 0 "abs(z + 1 + i) < 1" "6#2-%$absG6#,(%\"zG\"\"\"F)F)%\"iGF)F)" } {TEXT 273 28 " under the transformation " }{XPPEDIT 18 0 "w = (3 - 4 i)*z + 6 + 2*i" "6#/%\"wG,(*&,&\"\"$\"\"\"*&\"\"%F)%\"iGF)!\"\"F)%\"z GF)F)\"\"'F)*&\"\"#F)F,F)F)" }{TEXT 271 20 " is the open disk " } {XPPEDIT 18 0 "abs(w + 1 - 3*i) < 5" "6#2-%$absG6#,(%\"wG\"\"\"F)F)*& \"\"$F)%\"iGF)!\"\"\"\"&" }{TEXT 272 2 " ." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 651 "f:='f': u:='u': v:='v': \+ w:='w': x:='x': y:='y': z:='z':\nf := z -> (3 - 4*I)*z + 6 + 2*I:\n`w \+ ` = f(z);\n`u + I v ` = f(x + I*y);\n`u + I v ` = evalc(f(x + I*y)); ` `;\n`Solve for z in terms of w.`;\nsolset := expand(solve(W = f(z) , z)):\ng := w -> subs(W=w,solset):\n`z ` = g(w);\n`x + I y ` = g(u + \+ I*v);\n`x + I y ` = evalc(g(u + I*v));\n`We will use the substitutions :`;\neqns := \{x=3*u/25-2/5-4*v/25, y=3*v/25-6/5+4*u/25\}:\neqns; ` `; \n`Now find the image of the disk.`;\nineq := abs(z+1+I)^2 < 1: ineq; \nineq := (x+1)^2 + (y+1)^2 < 1: ineq;\nineq := subs(eqns,ineq): ineq; \nineq := map(expand,ineq): ineq;\nineq := ineq - (2/5<2/5): ineq;\nin eq := 25*ineq: ineq;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 257 "" 0 "" {TEXT 262 19 "Which is the disk " }{XPPEDIT 18 0 "abs(w + 1 - 3*i) < 5" "6#2-%$absG6#,(%\"wG\"\"\"F)F)*&\"\"$F)%\" iGF)!\"\"\"\"&" }{TEXT 270 17 " in the w-plane." }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "f:='f': F:='F': z:='z':\n f := z -> (3 - 4*I)*z + 6 + 2*I:\n`f(z) ` = f(z);\nF := z -> subs(Z=z- 1-I,f(Z)):\nconformal(F(Re(z)*exp(I*Im(z))), z=0.01..1+I*2*Pi,\n titl e=`w = (3-4i)*z + 6 + 2i`,\n grid=[15,15], numxy=[50,50],\n scaling= constrained,\n labels=[`u `,`v `],\n view=[-6..4,-2..8]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 22 "Example 2.11, Page 60. " }{TEXT 288 48 " Show that the image of the right half plane " } {XPPEDIT 18 0 "Re(z) > 1" "6#2\"\"\"-%#ReG6#%\"zG" }{TEXT 267 35 " un der the linear transformation " }{XPPEDIT 18 0 "w = (-1 + i) *z - 2 + 3*i" "6#/%\"wG,(*&,&\"\"\"!\"\"%\"iGF(F(%\"zGF(F(\"\"#F)*&\"\"$F(F*F( F(" }{TEXT 268 21 " is the half plane " }{XPPEDIT 18 0 "v > u + 7" " 6#2,&%\"uG\"\"\"\"\"(F&%\"vG" }{TEXT 269 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 624 "f:='f': u:='u': v:='v': w:='w': x:='x': y:='y': z:=' z':\nf := z -> (-1 + I)*z - 2 + 3*I:\n`w ` = f(z);\n`u + I v ` = f(x + I*y);\n`u + I v ` = evalc(f(x + I*y)); ` `;\n`Solve for z in terms \+ of w.`;\nsolset := expand(solve(W = f(z), z)):\ng := w -> subs(W=w,so lset):\n`z ` = g(w);\n`x + I y ` = g(u + I*v);\n`x + I y ` = evalc(g(u + I*v));\n`We will use the substitutions:`;\neqns := \{x=-u/2-5/2+v/2 , y=-v/2+1/2-u/2\}:\neqns; ` `;\n`Now find the image of the right half plane.`;\nineq := Re(z) > 1: ineq;\nineq := x > 1: ineq;\nineq := sub s(eqns,ineq): ineq;\nineq := ineq + (5/2<5/2): ineq;\nineq := 2*ineq: \+ ineq;\nineq := ineq + (u 7 + u" "6#2,&\"\"(\"\"\"%\"uGF&%\"vG" } {TEXT 266 17 " in the w-plane." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "f:='f': z:='z':\nf := z -> (-1 + I)*z - 2 \+ + 3*I:\n`f(z) ` = f(z);\nconformal(f(z), z=1-6*I..5+7*I,\n title=`w = (-1+I)*z - 2 + 3*I`,\n grid=[9,14], numxy=[9,14],\n scaling=constra ined,\n labels=[`u `,` v`],\n view=[-14.25..3.25,-3.25..14.25]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 264 19 "End of Se ction 2.2." }}}}{MARK "0 0 0" 9 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }