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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 302 197 "\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 300 1 "\n" }{TEXT 256 42 "CHAPTER 3 ANALYTIC and HARMONIC FUNCTIONS" }{TEXT 294 2 "\n\n" }{TEXT 256 41 "Section 3.2 The Cauchy-Riemann Equ ations" }{TEXT 295 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 113 " We saw in the last section that computing the derivative of complex functions w ritten in a nice form such as " }{XPPEDIT 18 0 "f(z) = z^2;" "6#/-%\" fG6#%\"zG*$F'\"\"#" }{TEXT -1 110 " is a rather simple task. But life is not so easy, for many times we encounter complex functions written as " }{XPPEDIT 18 0 "f(x+i*y) = u(x,y)+i*v(x,y);" "6#/-%\"fG6#,&%\"x G\"\"\"*&%\"iGF)%\"yGF)F),&-%\"uG6$F(F,F)*&F+F)-%\"vG6$F(F,F)F)" } {TEXT -1 30 ". For example, suppose we had" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "f(x+i*y) = x^3-3*x*y^2+i*(3*x^2*y-y^3)" "6#/-%\"fG6#,&%\"xG\"\"\"*&%\"iGF)%\"y GF)F),(*$F(\"\"$F)*(F/F)F(F)F,\"\"#!\"\"*&F+F),&*(F/F)*$F(F1F)F,F)F)*$ F,F/F2F)F)" }{TEXT -1 6 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 72 "Is there some criterion---perhaps involvi ng the partial derivatives for " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "v;" "6#%\"vG" }{TEXT -1 43 " - - that we can use to determine whether " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 54 " is differentiable, and if so, to find the value of " } {XPPEDIT 18 0 "`f '(z)`;" "6#%'f~'(z)G" }{TEXT -1 2 "?\n" }}{PARA 0 " " 0 "" {TEXT -1 31 "The answer to this question is " }{TEXT 304 3 "yes " }{TEXT -1 109 ", thanks in part to the independent discovery of two \+ important equations relating the partial derivatives of " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "v;" "6#%\"vG " }{TEXT -1 89 " by the French mathematician A. L. Cauchy and the Germ an mathematician G. F. B. Riemann. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 38 "Theorem 3.4 (Cauchy-Riemann equations)" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Suppo se 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 34 " is differentiable at the point " }{XPPEDIT 18 0 "z[0] \+ = x[0]+i*y[0];" "6#/&%\"zG6#\"\"!,&&%\"xG6#F'\"\"\"*&%\"iGF,&%\"yG6#F' F,F," }{TEXT -1 36 " . Then the partial derivatives of " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v;" "6#%\"vG" }{TEXT -1 21 " exist at the point (" }{XPPEDIT 18 0 "x[0],y[0];" "6$&% \"xG6#\"\"!&%\"yG6#F&" }{TEXT -1 10 "), and " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "(3-15) f '(" } {XPPEDIT 18 0 "z[0];" "6#&%\"zG6#\"\"!" }{TEXT -1 6 ") = " } {XPPEDIT 18 0 "u[x]*(x[0], y[0])+i*v[x]*(x[0], y[0]);" "6#,&*&&%\"uG6# %\"xG\"\"\"6$&F(6#\"\"!&%\"yG6#F-F)F)*(%\"iGF)&%\"vG6#F(F)6$&F(6#F-&F/ 6#F-F)F)" }{TEXT -1 6 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "(3-16) f '(" }{XPPEDIT 18 0 "z[0];" "6#&%\"zG6#\"\"!" }{TEXT -1 6 ") = " }{XPPEDIT 18 0 "v[y]*(x[0], y[0 ])-i*u[y]*(x[0], y[0]);" "6#,&*&&%\"vG6#%\"yG\"\"\"6$&%\"xG6#\"\"!&F(6 #F.F)F)*(%\"iGF)&%\"uG6#F(F)6$&F,6#F.&F(6#F.F)!\"\"" }{TEXT -1 6 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Equ ating the real and imaginary parts of gives us the Cauchy-Riemann equa tions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "(3-17) " }{XPPEDIT 18 0 "u[x]*(x[0], y[0]) = v[y]*(x[0], y[0] );" "6#/*&&%\"uG6#%\"xG\"\"\"6$&F(6#\"\"!&%\"yG6#F-F)*&&%\"vG6#F/F)6$& F(6#F-&F/6#F-F)" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "u[y]*(x[0], y [0]) = -v[x]*(x[0], y[0]);" "6#/*&&%\"uG6#%\"yG\"\"\"6$&%\"xG6#\"\"!&F (6#F.F),$*&&%\"vG6#F,F)6$&F,6#F.&F(6#F.F)!\"\"" }{TEXT -1 4 ". " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "No te carefully some of the implications of this theorem:\n\011\nIf " } {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 20 " differentiable at " } {XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 73 ", then we know th e Cauchy-Riemann equations (3-17) will be satisfied at " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 66 ", and we can use either eq uation (3-15) or (3-16) to evaluate f '(" }{XPPEDIT 18 0 "z[0];" "6#&% \"zG6#\"\"!" }{TEXT -1 3 ").\n" }}{PARA 0 "" 0 "" {TEXT -1 69 "Taking \+ the contrapositive, if equations (3-17) are not satisfied at " } {XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 35 ", then we know au tomatically that " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 5 " is " }{TEXT 305 3 "not" }{TEXT -1 20 " differentiable at " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "On the other hand, just because eq uations (3-17) are satisfied at " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\" \"!" }{TEXT -1 39 ", we cannot necessarily conclude that " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 24 " is differentiable at " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 27 "Derivation o f Theorem 3.4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 801 "dx:='dx': dy:='dy': f:='f': U:='U': V:='V': x:='x': \+ y:='y':\nassume(x,real);\nassume(y,real);\nf := proc(x,y) U(x,y) + I* V(x,y) end:\n`f(x + I y) = U(x,y) + I V(x,y)`; ` `;\ndfdx := evalc( (f(x+dx,y)-f(x,y))/dx):\ndf/dx = subs(\{x='x',y='y'\}, dfdx); ` `;\nL1 := limit((f(x+dx,y)-f(x,y))/dx, dx=0):\nL1:=expand(L1):\ndfdy := eval c((f(x,y+dy)-f(x,y))/(I*dy)):\ndf/dy = subs(\{x='x',y='y'\}, dfdy);\nL 2 := limit((f(x,y+dy)-f(x,y))/(I*dy), dy=0):\nL2:=expand(L2):\nLimit(d f/dx, dx=0) = subs(\{x='x',y='y'\}, L1);\nLimit(df/dy, dy=0) = subs(\{ x='x',y='y'\}, L2); ` `;\nR1 := subs(I=0,expand(L1)):\nR2 := subs(I=0, expand(L2)):\n`The two limits must be the same.`;\nsubs(\{x='x',y='y' \}, L1 = L2); ` `;\nprint(`The Cauchy-Riemann equations are:`);\nsubs( \{x='x',y='y'\}, R1 = R2);\nsubs(\{x='x',y='y'\}, subs(I=1,L1-R1) = su bs(I=1,L2-R2));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 257 "" 0 "" {TEXT 260 40 "These are the Cauchy-Riema nn equations \n" }{XPPEDIT 18 0 "U[x](x[0],y[0]) = V[y](x[0],y[0])" "6 #/-&%\"UG6#%\"xG6$&F(6#\"\"!&%\"yG6#F,-&%\"VG6#F.6$&F(6#F,&F.6#F," } {TEXT 288 7 " and " }{XPPEDIT 18 0 "U[y](x[0],y[0]) = - V[x](x[0],y[ 0])" "6#/-&%\"UG6#%\"yG6$&%\"xG6#\"\"!&F(6#F-,$-&%\"VG6#F+6$&F+6#F-&F( 6#F-!\"\"" }{TEXT 289 2 " ." }}}{EXCHG {PARA 257 "" 0 "" {TEXT 266 1 " \n" }{TEXT 256 21 "Example for Page 104." }{TEXT 296 16 " The functio n " }{XPPEDIT 18 0 "f(z)= z^3" "6#/-%\"fG6#%\"zG*$F'\"\"$" }{TEXT 267 9 " i.e. \n" }{XPPEDIT 18 0 "f(z) = x^3-3*x*y^2+i*(3*x^2*y-y^3) " "6#/-%\"fG6#%\"zG,(*$%\"xG\"\"$\"\"\"*(F+F,F*F,%\"yG\"\"#!\"\"*&%\"i GF,,&*(F+F,*$F*F/F,F.F,F,*$F.F+F0F,F," }{TEXT 268 116 " is known to \+ be differentiable.\nVerify that its derivative satisfies the results o f the Cauchy-Riemann equations.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 520 "f:='f': U:='U': V:='V': x:='x': y:='y': z:='z':\nf := z -> z^3:\n `f(z) ` = f(z);\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`U(x,y) ` = U(x,y);\n`V(x,y) ` = V(x,y); ` `;\nf1 := z -> \+ subs(Z=z,diff(f(Z),Z)):\n`f '(z) ` = f1(z);\n`f '(x + I y) ` = f1(x+I* y);\n`f '(x + I y) ` = evalc(f1(x+I*y)); ` `;\n`f '(z) = Ux(x,y) + i V x(x,y)` =\n diff(U(x,y),x) + I* diff(V(x,y),x);\n`f '(z) = Vy(x,y) - \+ i Uy(x,y)` =\n diff(V(x,y),y) - I* diff(U(x,y),y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 23 "Exam ple 3.6, Page 105." }{TEXT 297 37 " Verify that the complex functio n " }{XPPEDIT 18 0 "f(z) = conjugate(z)^` 2`/z;" "6#/-%\"fG6#%\"zG*&) -%*conjugateG6#F'%#~2G\"\"\"F'!\"\"" }{TEXT 269 5 " is " }{TEXT 257 3 "NOT" }{TEXT 301 11 " analytic.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 871 "f:='f': F:='F': U:='U': V:='V': x:='x': y:='y': z:='z':\nf := z - > conjugate(z)^2/z:\n`f(z) ` = f(z);\n`f(x + I y) ` = f(x+I*y);\nassum e(x,real); assume(y,real);\nw1 := simplify(evalc(f(x+I*y))):\n`f(x + I y)` = subs(\{x='x',y='y'\}, w1); ` `;\nx:='x': y:='y':\nU := proc(x,y ) (x^3-3*x*y^2)/(x^2+y^2) end:\nV := proc(x,y) (-3*x^2*y+y^3)/(x^2+ y^2) end:\nF := proc(x,y) U(x,y) + I*V(x,y) end:\n`F(x,y)` = subs(\{ x='x',y='y'\}, F(x,y));\n`U(x,y)` = subs(\{x='x',y='y'\}, U(x,y));\n`V (x,y)` = subs(\{x='x',y='y'\}, V(x,y)); ` `;\nUx := simplify(diff(U(x, y), x)): `Ux(x,y)` = Ux;\nVy := simplify(diff(V(x,y), y)): `Vy(x,y)` = Vy;\nUy := simplify(diff(U(x,y), y)): `Uy(x,y)` = Uy;\nVx := simpl ify(diff(V(x,y), x)): `Vx(x,y)` = Vx; ` `;\nprint(`0 = Ux - Vy `, 0 \+ = simplify(Ux - Vy),\n 0 = numer(simplify(Ux - Vy)));\nprint(`0 = Uy \+ + Vx `, 0 = simplify(Uy + Vx),\n 0 = numer(simplify(Uy + Vx)));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 261 52 "Hence the function f is differentiable only when " } {XPPEDIT 18 0 "x^2 - y^2 = 0" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'!\"\"\" \"!" }{TEXT 271 7 " and " }{XPPEDIT 18 0 "x*y = 0" "6#/*&%\"xG\"\"\" %\"yGF&\"\"!" }{TEXT 272 82 " or at the origin.\n\nLet's check to see if the Cauchy-Riemann equations hold at (" }{XPPEDIT 18 0 "0" "6#\" \"!" }{TEXT 291 2 ", " }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT 290 28 ") . Using the facts that \n" }{XPPEDIT 18 0 "U(0,0) = 0" "6#/-%\"UG6$\" \"!F'F'" }{TEXT 273 7 " and " }{XPPEDIT 18 0 "V(0,0) = 0" "6#/-%\"VG 6$\"\"!F'F'" }{TEXT 274 55 " , we compute the limits of the differenc e quotients.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "`Ux(0,0)` = li mit((U(x,0)-0)/(x-0), x=0);\n`Vy(0,0)` = limit((V(0,y)-0)/(y-0), y=0 );\n`Uy(0,0)` = limit((U(0,y)-0)/(y-0), y=0);\n`-Vx(0,0)` = -limit(( V(x,0)-0)/(x-0), x=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 257 "" 0 "" {TEXT 262 5 "So, " }{XPPEDIT 18 0 "U[x](0,0) = V[y](0,0)" "6#/-&%\"UG6#%\"xG6$\"\"!F*-&%\"VG6#%\"yG6$F*F*" }{TEXT 275 7 " and " }{XPPEDIT 18 0 "U[y](0,0) = -V[x](0,0)" "6#/-&%\"UG6#% \"yG6$\"\"!F*,$-&%\"VG6#%\"xG6$F*F*!\"\"" }{TEXT 276 56 ".and we see t hat the Cauchy-Riemann \nequations hold at " }{XPPEDIT 18 0 "z=0" "6# /%\"zG\"\"!" }{TEXT 277 8 " . But " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6# %\"zG" }{TEXT 278 21 " is not-analytic at (" }{XPPEDIT 18 0 "0" "6#\" \"!" }{TEXT 293 2 ", " }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT 292 49 ") b ecause the following two limits are distinct.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "`Along the x-axis: `, Limit(df/dz, dz=0) =\nlimit(( F(x,0)-0)/(x-0),x=0);\n`Along the line x=t, y=t: `, Limit(df/dz, dz=0 ) =\nlimit((F(t,t)-0)/(t+I*t-0),t=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 263 1 "\n" }{TEXT 256 22 "Example 3.8, Page 108." }{TEXT 298 51 " Use the Cauchy-Riemann e quations to show that \n" }{XPPEDIT 18 0 "f(z) = exp(-y)*(cos x) + \+ i*exp(-y)*(sin x)" "6#/-%\"fG6#%\"zG,&*&-%$expG6#,$%\"yG!\"\"\"\"\"*&% $cosGF0%\"xGF0F0F0*(%\"iGF0-F+6#,$F.F/F0*&%$sinGF0F3F0F0F0" }{TEXT 279 30 " is differentiable for all " }{XPPEDIT 18 0 "z = x + i*y" " 6#/%\"zG,&%\"xG\"\"\"*&%\"iGF'%\"yGF'F'" }{TEXT 280 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 498 "U:='U': V:='V': x:='x': y:='y':\nU := pr oc(x,y) exp(-y)*cos(x) end:\nV := proc(x,y) exp(-y)*sin(x) end:\n`U( x,y)` = U(x,y);\n`V(x,y)` = V(x,y); ` `;\n`Ux(x,y)` = diff(U(x,y),x); \n`Vy(x,y)` = diff(V(x,y),y);\ndiff(U(x,y),x) = diff(V(x,y),y);\nevalb (diff(U(x,y),x) = diff(V(x,y),y)); ` `;\n`Uy(x,y)` = diff(U(x,y),y);\n `Vx(x,y)` = diff(V(x,y),x);\ndiff(U(x,y),y) = - diff(V(x,y),x);\nevalb (diff(U(x,y),y) = - diff(V(x,y),x)); ` `;\n`f(x+iy)` = U(x,y) + I*V(x, y);\n`f '(x+iy)` = diff(U(x,y),x) + I*diff(V(x,y),x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 264 54 "The Cauchy-Riemann equations hold everywhere, so that\n" } {XPPEDIT 18 0 "f(z) = exp(-y)*(cos x) + i*exp(-y)*(sin x)" "6#/-%\"f G6#%\"zG,&*&-%$expG6#,$%\"yG!\"\"\"\"\"*&%$cosGF0%\"xGF0F0F0*(%\"iGF0- F+6#,$F.F/F0*&%$sinGF0F3F0F0F0" }{TEXT 281 23 " is analytic for all \+ " }{XPPEDIT 18 0 "z = x + i*y" "6#/%\"zG,&%\"xG\"\"\"*&%\"iGF'%\"yGF'F '" }{TEXT 282 2 " ." }}}{EXCHG {PARA 257 "" 0 "" {TEXT 270 1 "\n" } {TEXT 256 22 "Example 3.9, Page 108." }{TEXT 299 33 " Show that the co mplex function " }{XPPEDIT 18 0 "f(z) = x^3 + 3*x*y^2 + i*(y^3 \+ + 3*x^2*y)" "6#/-%\"fG6#%\"zG,(*$%\"xG\"\"$\"\"\"*(F+F,F*F,%\"yG\"\"#F ,*&%\"iGF,,&*$F.F+F,*(F+F,*$F*F/F,F.F,F,F,F," }{TEXT 283 67 "\nis diff erentiable only at points that lie on the coordinate axes.\n" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 477 "U:='U': V:='V' : x:='x': y:='y':\nU := proc(x,y) x^3 + 3*x*y^2 end:\nV := proc(x,y) \+ y^3 + 3*x^2*y end:\n`U(x,y)` = U(x,y);\n`V(x,y)` = V(x,y); ` `;\n`Ux( x,y)` = diff(U(x,y),x);\n`Vy(x,y)` = diff(V(x,y),y);\nprint(`0 = Ux - \+ Vy `,\n 0 = diff(U(x,y),x) - diff(V(x,y),y),\nevalb(0 = diff(U(x,y), x) - diff(V(x,y),y))); ` `;\n`Uy(x,y)` = diff(U(x,y),y);\n`Vx(x,y)` = \+ diff(V(x,y),x); \nprint(`0 = Uy + Vx `,\n 0 = diff(U(x,y),y) + diff (V(x,y),x),\nevalb(0 = diff(U(x,y),y) + diff(V(x,y),x)));" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 265 43 "The Cauchy-Riemann equations hold only if " }{XPPEDIT 18 0 "xy = 0" "6#/%#xyG\"\"!" }{TEXT 284 13 " . So t hat " }{XPPEDIT 18 0 "f(z) = x^3 + 3*x*y^2 + i*(y^3 + 3*x^2*y) " "6#/-%\"fG6#%\"zG,(*$%\"xG\"\"$\"\"\"*(F+F,F*F,%\"yG\"\"#F,*&%\"iGF, ,&*$F.F+F,*(F+F,*$F*F/F,F.F,F,F,F," }{TEXT 287 25 " \nis analytic only when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT 286 6 " or " } {XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT 285 59 " , which occurs at points that lie on the coordinate axes." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 306 19 "End of Section 3 .2." }{TEXT -1 0 "" }}}}{MARK "0 0 0" 25 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }