{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 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 275 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 280 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 285 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 290 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 1 10 0 0 0 0 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 "" 1 10 0 0 0 0 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 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 300 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 301 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "" 1 10 0 0 0 0 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 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 306 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 308 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 310 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 311 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 315 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 316 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 317 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 318 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 319 "" 1 10 0 0 0 0 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 0" -1 257 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 }{PSTYLE "R3 Font 2" -1 258 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 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple Worksheets, 2001" }{TEXT 307 144 "\n(c) John H. Mathews Rus sell W. Howell\nmathews@fullerton.edu howell@westmont.edu\n\nCompl imentary software to accompany the textbook:" }}{PARA 257 "" 0 "" {TEXT 257 82 "COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9" }{TEXT 306 197 "\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 257 "" 0 "" {TEXT 299 1 "\n" }{TEXT 256 42 "CHAPTER 3 ANALYTIC and HARMONIC FUNCTIONS" }{TEXT 291 2 "\n\n" }{TEXT 256 31 "Section 3.3 Harmonic Functions" } {TEXT 292 1 "\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 " Let \+ " }{XPPEDIT 18 0 "phi(x,y);" "6#-%$phiG6$%\"xG%\"yG" }{TEXT -1 54 " b e a real-valued function of the two real variables " }{XPPEDIT 18 0 "x ;" "6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 37 ". The partial differential equation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "phi[xx]( x,y)+phi[yy](x,y) = 0;" "6#/,&-&%$phiG6#%#xxG6$%\"xG%\"yG\"\"\"-&F'6#% #yyG6$F+F,F-\"\"!" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is known as L" }{TEXT 313 17 "aplace' s equation" }{TEXT -1 60 " and is sometimes referred to as the potenti al equation. If " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "phi[x];" "6#&%$phiG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[y];" "6#&%$phiG6#%\"yG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "ph i[xx];" "6#&%$phiG6#%#xxG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[xy];" "6#&%$phiG6#%#xyG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "phi[yy];" "6#& %$phiG6#%#yyG" }{TEXT -1 28 " are all continuous and if " }{XPPEDIT 18 0 "phi(x,y)" "6#-%$phiG6$%\"xG%\"yG" }{TEXT -1 36 " satisfies Lapla ce's equation, then " }{XPPEDIT 18 0 "phi(x,y)" "6#-%$phiG6$%\"xG%\"yG " }{TEXT -1 13 " is called a " }{TEXT 314 17 "harmonic function" } {TEXT -1 435 ". Harmonic functions are important in the areas of appli ed mathematics, engineering, and mathematical physics. They are used t o solve problems involving steady state temperatures, two-dimensional \+ electrostatics, and ideal fluid flow. In Chapter 10 we will see how co mplex analysis techniques can be used to solve some problems involving harmonic functions. We begin with an important theorem relating analy tic and harmonic functions. " }{TEXT 301 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 312 97 "Load Maple's \+ \"contourplot\" procedure.\nMake sure this is done only ONCE during a Maple session.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots): " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "Theorem \+ 3.8" }{TEXT 311 3 " " }{TEXT -1 5 "Let " }{XPPEDIT 18 0 "f(x+i*y) = u(x,y)+i*v(x,y);" "6#/-%\"fG6#,&%\"xG\"\"\"*&%\"iGF)%\"yGF)F),&-%\"uG 6$F(F,F)*&F+F)-%\"vG6$F(F,F)F)" }{TEXT -1 41 " be an analytic functio n in the domain " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 2 ". " } {TEXT 308 61 " Assume that all of the second-order partial derivatives of " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"xG%\"yG" }{TEXT 309 7 " \+ and " }{XPPEDIT 18 0 "v(x,y)" "6#-%\"vG6$%\"xG%\"yG" }{TEXT 310 17 " \+ are continuous. " }{TEXT -1 11 " Then both " }{XPPEDIT 18 0 "u;" "6#% \"uG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v;" "6#%\"vG" }{TEXT -1 28 " are harmonic functions in " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 86 ". In other words, the real and imaginary parts of an analytic func tion are harmonic. " }}}{EXCHG {PARA 257 "" 0 "" {TEXT 300 1 "\n" } {TEXT 256 22 "Example for Page 115." }{TEXT 293 36 " Show that the \+ complex function \n" }{XPPEDIT 18 0 "f(z) = x^2 + y^2 + i*2*x*y" "6# /-%\"fG6#%\"zG,(*$%\"xG\"\"#\"\"\"*$%\"yGF+F,**%\"iGF,F+F,F*F,F.F,F," }{TEXT 269 5 " is " }{TEXT 257 7 "NOWHERE" }{TEXT 302 11 " analytic. \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 517 "U:='U': V:='V': x:='x': y:=' y':\nU := proc(x,y) x^2 + y^2 end:\nV := proc(x,y) 2*x*y end:\n`F( x,y)` = U(x,y) + I*V(x,y);\n`U(x,y)` = U(x,y);\n`V(x,y)` = V(x,y); ` ` ;\n`Look at the Cauchy-Riemann equations.`;\n`Ux(x,y)` = diff(U(x,y),x );\n`Vy(x,y)` = diff(V(x,y),y);\nprint(`Ux = Vy `,diff(U(x,y),x) = d iff(V(x,y),y),\n evalb(diff(U(x,y),x) = diff(V(x,y),y)));\n` `;\n`Uy( x,y)` = diff(U(x,y),y);\n`Vx(x,y)` = diff(V(x,y),x);\nprint(`Uy = -Vx \+ `,diff(U(x,y),y) = - diff(V(x,y),x),\n evalb(diff(U(x,y),y) = - dif f(V(x,y),x)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 257 "" 0 "" {TEXT 260 12 "Therefore, " }{XPPEDIT 18 0 "f (z)" "6#-%\"fG6#%\"zG" }{TEXT 270 31 " is differentiable only when \+ " }{XPPEDIT 18 0 "y = 0" "6#/%\"yG\"\"!" }{TEXT 271 11 " . Since " } {XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 272 5 " is " }{TEXT 257 3 "NOT" }{TEXT 303 43 " differentiable in any open neighborhood, \+ " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 273 5 " is " }{TEXT 257 7 "NOWHERE" }{TEXT 304 10 " analytic." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 31 " If we are given a function " }{XPPEDIT 18 0 "u(x,y) " "6#-%\"uG6$%\"xG%\"yG" }{TEXT -1 34 " that is harmonic in the domai n " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT -1 48 " and if we can find a nother harmonic function " }{XPPEDIT 18 0 "v(x,y)" "6#-%\"vG6$%\"xG% \"yG" }{TEXT -1 42 ", such that the partial derivatives for " } {XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "v; " "6#%\"vG" }{TEXT -1 51 " satisfy the Cauchy-Riemann equations throu ghout " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT -1 21 ", then we say tha t " }{XPPEDIT 18 0 "v(x,y)" "6#-%\"vG6$%\"xG%\"yG" }{TEXT -1 6 " is a " }{TEXT 315 18 "harmonic conjugate" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"xG%\"yG" }{TEXT -1 38 ". It then follows tha t the function " }{XPPEDIT 18 0 "f(x+i*y) = u(x,y)+i*v(x,y);" "6#/-% \"fG6#,&%\"xG\"\"\"*&%\"iGF)%\"yGF)F),&-%\"uG6$F(F,F)*&F+F)-%\"vG6$F(F ,F)F)" }{TEXT -1 18 " is analytic in " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 268 1 "\n" }{TEXT 256 23 "Example 3.11, Page 1 15." }{TEXT 294 19 " Show that both " }{XPPEDIT 18 0 "U(x,y) = x^2 - y^2" "6#/-%\"UG6$%\"xG%\"yG,&*$F'\"\"#\"\"\"*$F(F+!\"\"" }{TEXT 277 7 " and " }{XPPEDIT 18 0 "V(x,y) = 2*x*y" "6#/-%\"VG6$%\"xG%\"y G*(\"\"#\"\"\"F'F+F(F+" }{TEXT 274 30 " are harmonic functions, and \+ " }{XPPEDIT 18 0 "V;" "6#%\"VG" }{TEXT 316 30 " is the harmonic conjug ate of " }{XPPEDIT 18 0 "U;" "6#%\"UG" }{TEXT 317 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "f:='f': U: ='U': V:='V': w:='w': x:='x': y:='y': z:='z':\nf := z -> z^2:\n`f(z) ` = f(z);\n`f(z) is an analytic function.`;\n`f(x + I y) ` = f(x+I*y); \n`f(x + I y) ` = evalc(f(x+I*y)); ` `;\nU := proc(x,y) x^2 - y^2 en d:\nV := proc(x,y) 2*x*y end:\n`The real and imaginary parts are har monic functions.`;\n`Re(f(z) = U(x,y)` = U(x,y);\n`Im(f(z) = V(x,y)` = V(x,y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 261 27 "And we can show that both " }{XPPEDIT 18 0 "U(x ,y)" "6#-%\"UG6$%\"xG%\"yG" }{TEXT 275 7 " and " }{XPPEDIT 18 0 "V(x ,y)" "6#-%\"VG6$%\"xG%\"yG" }{TEXT 276 29 " satisfy Laplace's equatio n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 439 "`U(x,y) ` = U(x,y); `V(x,y) ` = V(x,y);\n`Verify Laplace's equati on.`;\n`Uxx(x,y) ` = diff(U(x,y),x$2); \n`Uyy(x,y) ` = diff(U(x,y),y$2 );\nprint(`0 = Uxx + Uyy `,\n 0 = diff(U(x,y),x$2) + diff(U(x,y),y$ 2),\n evalb(0 = diff(U(x,y),x$2) + diff(U(x,y),y$2)));\n`Vxx(x,y) ` = diff(V(x,y),x$2); \n`Vyy(x,y) ` = diff(V(x,y),y$2);\nprint(`0 = Vxx + Vyy `,\n 0 = diff(V(x,y),x$2) + diff(V(x,y),y$2),\n evalb(0 = dif f(V(x,y),x$2) + diff(V(x,y),y$2)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 262 14 "Hence, both " } {XPPEDIT 18 0 "U(x,y)" "6#-%\"UG6$%\"xG%\"yG" }{TEXT 278 8 " and " }{XPPEDIT 18 0 "V(x,y)" "6#-%\"VG6$%\"xG%\"yG" }{TEXT 279 25 " are ha rmonic functions." }}}{EXCHG {PARA 257 "" 0 "" {TEXT 267 1 "\n" } {TEXT 256 23 "Example 3.12, Page 115." }{TEXT 295 19 " Show that bot h " }{XPPEDIT 18 0 "U(x,y) = x^3 - 3*x*y^2" "6#/-%\"UG6$%\"xG%\"yG, &*$F'\"\"$\"\"\"*(F+F,F'F,F(\"\"#!\"\"" }{TEXT 280 8 " and " } {XPPEDIT 18 0 "V(x,y) = 3*x^2*y - y^3" "6#/-%\"VG6$%\"xG%\"yG,&*(\"\"$ \"\"\"*$F'\"\"#F,F(F,F,*$F(F+!\"\"" }{TEXT 281 30 " are harmonic func tions, and " }{XPPEDIT 18 0 "V;" "6#%\"VG" }{TEXT 318 30 " is the harm onic conjugate of " }{XPPEDIT 18 0 "U;" "6#%\"UG" }{TEXT 319 5 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 371 "f:='f': U:='U': V:='V': w:='w': x:='x': y:='y': z:='z':\nf := z - > z^3:\n`f(z) ` = f(z);\n`f(z) is an analytic function.`;\n`f(x + I y) ` = f(x+I*y);\n`f(x + I y) ` = evalc(f(x+I*y)); ` `;\nU := proc(x,y) \+ x^3 - 3*x*y^2 end:\nV := proc(x,y) 3*x^2*y - y^3 end:\n`The real a nd imaginary parts are harmonic functions.`;\n`Re(f(z)) = U(x,y)` = U( x,y);\n`Im(f(z)) = V(x,y)` = V(x,y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 263 27 "And we can show that both " }{XPPEDIT 18 0 "U(x,y)" "6#-%\"UG6$%\"xG%\"yG" }{TEXT 282 7 " and " }{XPPEDIT 18 0 "V(x,y)" "6#-%\"VG6$%\"xG%\"yG" }{TEXT 283 28 " satisfy Laplace's equation" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 439 "`U(x,y) ` = U(x,y); `V(x,y) ` = V( x,y);\n`Verify Laplace's equation.`;\n`Uxx(x,y) ` = diff(U(x,y),x$2); \+ \n`Uyy(x,y) ` = diff(U(x,y),y$2);\nprint(`0 = Uxx + Uyy `,\n 0 = di ff(U(x,y),x$2) + diff(U(x,y),y$2),\n evalb(0 = diff(U(x,y),x$2) + dif f(U(x,y),y$2)));\n`Vxx(x,y) ` = diff(V(x,y),x$2); \n`Vyy(x,y) ` = diff (V(x,y),y$2);\nprint(`0 = Vxx + Vyy `,\n 0 = diff(V(x,y),x$2) + dif f(V(x,y),y$2),\n evalb(0 = diff(V(x,y),x$2) + diff(V(x,y),y$2)));" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 264 13 "Hence, both " }{XPPEDIT 18 0 "U(x,y)" "6#-%\"UG6$%\"xG% \"yG" }{TEXT 285 7 " and " }{XPPEDIT 18 0 "V(x,y)" "6#-%\"VG6$%\"xG% \"yG" }{TEXT 284 25 " are harmonic functions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 256 41 "Theorem 3.9 (Construction of a conjugate)" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Let " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"xG%\"yG" } {TEXT -1 20 " be harmonic in an " }{XPPEDIT 18 0 "epsilon;" "6#%(epsi lonG" }{TEXT -1 29 "-neighborhood of the point (" }{XPPEDIT 18 0 "x[0 ],y[0];" "6$&%\"xG6#\"\"!&%\"yG6#F&" }{TEXT -1 53 "). Then there exis ts a conjugate harmonic function " }{XPPEDIT 18 0 "v(x,y)" "6#-%\"vG6 $%\"xG%\"yG" }{TEXT -1 42 " defined in this neighborhood such that \+ " }{XPPEDIT 18 0 "f(x+i*y) = u(x,y)+i*v(x,y)" "6#/-%\"fG6#,&%\"xG\"\" \"*&%\"iGF)%\"yGF)F),&-%\"uG6$F(F,F)*&F+F)-%\"vG6$F(F,F)F)" }{TEXT -1 28 " is an analytic function. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 266 1 "\n" }{TEXT 256 7 "METHOD ." }{TEXT 296 2 " " }{TEXT 257 38 "Construction of the harmonic conju gate" }{TEXT 305 5 " of " }{XPPEDIT 18 0 "U(x,y)" "6#-%\"UG6$%\"xG%\" yG" }{TEXT 286 73 ".\nActivate the following procedure before doing Ex ample 3.13, Page 117.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "conj:= 'conj': U:='U': V:='V': x:='x': y:='y':\nconj := proc(U)\n local lap ,v1,v2,v3,v4;\n lap := diff(U,x$2)+diff(U,y$2);\n v1 := int(diff(U,x ), y);\n v2 := - diff(U,y) - diff(v1,x);\n v3 := int(v2,x); \n v4 : = v1 + v3;\n if lap=0 then \n RETURN(v4) \n else\n RETURN(`U(x ,y) was Not harmonic.`)\n fi\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 23 "Example 3.13, Page 117." }{TEXT 297 13 " Show that " } {XPPEDIT 18 0 "U(x,y) = x*y^3 - x^3*y" "6#/-%\"UG6$%\"xG%\"yG,&*&F' \"\"\"*$F(\"\"$F+F+*&F'F-F(F+!\"\"" }{TEXT 287 59 " is a harmonic fun ction and find the harmonic conjugate " }{XPPEDIT 18 0 "V(x,y)" "6#- %\"VG6$%\"xG%\"yG" }{TEXT 288 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 337 "U := proc(x,y) x*y^3 - x^3*y end:\n`U(x,y) ` = U(x,y);\n`Verify Laplace's equation.`;\n`Uxx(x,y) ` = diff(U(x,y),x$2); \n`Uyy(x,y) ` \+ = diff(U(x,y),y$2);\nprint(`0 = Uxx + Uyy `,\n 0 = diff(U(x,y),x$2) + diff(U(x,y),y$2),\n evalb(0 = diff(U(x,y),x$2) + diff(U(x,y),y$2)) );\n` `; `The harmonic conjugate is:`;\nV := conj(U(x,y)):\n`V(x,y) ` \+ = V;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 23 "Example 3.14, Pag e 119." }{TEXT 298 14 " Show that " }{XPPEDIT 18 0 "U(x,y) = x^2 - \+ y^2" "6#/-%\"UG6$%\"xG%\"yG,&*$F'\"\"#\"\"\"*$F(F+!\"\"" }{TEXT 289 56 " is the scalar potential function for the fluid flow: " } {XPPEDIT 18 0 "V(x,y) = 2*x - i*2*y" "6#/-%\"VG6$%\"xG%\"yG,&*&\"\"#\" \"\"F'F,F,*(%\"iGF,F+F,F(F,!\"\"" }{TEXT 290 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 433 "f:='f': F:='F': U:='U': V:='V': x:='x': \nX:='X ': y:='y': Y:='Y': z:='z': Z:='Z':\nassume(X,real); assume(Y,real);\nF := z -> z^2:\n`F(z) ` = F(z);\nf := z -> subs(Z=z, diff(F(Z), Z)):\n` f(z) = F '(z)`;\n`f(z) ` = f(z);\n`f(x + I y) ` = f(x+I*y);\nv := conj ugate(f(X+I*Y)):\nV := proc(x,y) subs(\{X=x,Y=y\},v) end:\n`V(x,y) ` = V(x,y);\n`F(x + I y) ` = F(x+I*y);\n`F(x + I y) ` = evalc(F(x+I*y)) ;\nU := proc(x,y) x^2-y^2 end:\n`U(x,y) ` = U(x,y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "v:='v': x:='x': y:='y':\nv := proc(x,y) 2*x*y end:\n`v(x,y) ` = v(x,y);\ncontourplot(v(x,y), x=0..5, y=0..5,\n title=`The streamli nes 2xy = C`,\n scaling=constrained,\n axes=boxed, grid=[30,30]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 265 19 "End of Secti on 3.3." }}}}{MARK "0 0 0" 25 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }