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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 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 0 "" 0 "" {TEXT 301 1 "\n" }{TEXT 256 29 "CHAPTER 2 COMPLEX FUNCTIONS" }{TEXT 294 2 "\n\n" }{TEXT 256 34 "Section 2.4 Limits and Continuity" }{TEXT 295 13 "\n\n Let " }{XPPEDIT 18 0 "u = u(x,y)" "6#/%\"uG-F$6$%\"xG% \"yG" }{TEXT 272 55 " be a real-valued function of the two real varia bles " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 273 7 " and " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT 274 4 " . " }{TEXT -1 12 "Recall that " } {XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 9 " has the " }{XPPEDIT 18 0 "u [0];" "6#&%\"uG6#\"\"!" }{TEXT -1 4 " as " }{TEXT 309 1 "(" }{XPPEDIT 18 0 "x,y;" "6$%\"xG%\"yG" }{TEXT 308 1 ")" }{TEXT -1 12 " approaches \+ " }{TEXT 311 1 "(" }{XPPEDIT 18 0 "x[0],y[0];" "6$&%\"xG6#\"\"!&%\"yG6 #F&" }{TEXT 310 1 ")" }{TEXT -1 28 " provided that the value of " } {XPPEDIT 18 0 "u(x,y);" "6#-%\"uG6$%\"xG%\"yG" }{TEXT -1 55 " can be m ade to get as close as we please to the value " }{XPPEDIT 18 0 "u(x[0] ,y[0]);" "6#-%\"uG6$&%\"xG6#\"\"!&%\"yG6#F)" }{TEXT -1 11 " by taking \+ " }{TEXT 313 1 "(" }{XPPEDIT 18 0 "x,y;" "6$%\"xG%\"yG" }{TEXT 312 1 " )" }{TEXT -1 29 " to be sufficiently close to " }{TEXT 315 1 "(" } {XPPEDIT 18 0 "x[0],y[0];" "6$&%\"xG6#\"\"!&%\"yG6#F&" }{TEXT 314 1 ") " }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 302 1 "\n" }{TEXT 256 22 "Example 2.14, Page 69." }{TEXT 296 15 " The function " } {XPPEDIT 18 0 "u(x,y) = 2*x^3/(x^2+y^2);" "6#/-%\"uG6$%\"xG%\"yG*(\"\" #\"\"\"*$F'\"\"$F+,&*$F'F*F+*$F(F*F+!\"\"" }{TEXT 260 17 " has the li mit " }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT 275 6 " as (" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 276 2 ", " }{XPPEDIT 18 0 "y" "6#%\"yG" } {TEXT 277 14 ") approaches (" }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT 278 2 ", " }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT 279 4 ") .\n" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 234 "t:='t': u:='u': x:='x': y:='y':\nu := proc(x, y) 2*x^3/(x^2+y^2) end:\n`u(x,y) ` = u(x,y); ` `;\nlim1 := limit(u(x ,y), x=0):\nlim2 := limit(lim1, y=0):\n`limit u(x,y) as x->0 ` = li m1; `and`;\n`limit u(x,y) as x->0 and y->0 ` = lim2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "`u(x,y) ` = u(x,y); ` `;\nlim1 := \+ limit(u(x,y), y=0):\nlim2 := limit(lim1, x=0):\n`limit u(x,y) as y- >0 ` = lim1; `and`;\n`limit u(x,y) as y->0 and x->0 ` = lim2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "U := subs(\{x=r*cos(t),y=r* sin(t)\},u(x,y)):\n`u(r cos t,r sin t) ` = U; ` `;\nlim1 := limit(U, r =0):\n`limit u(r cos t,r sin t) as r->0 ` = lim1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 54 "So, al ong all lines through the origin, the limit is " }{XPPEDIT 18 0 "0" " 6#\"\"!" }{TEXT 280 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 1 "\n" }{TEXT 256 22 "Example 2.15, Page 70." }{TEXT 297 17 " The function \+ " }{XPPEDIT 18 0 "u(x,y) = x*y / (x^2 + y^2)" "6#/-%\"uG6$%\"xG%\"yG* (F'\"\"\"F(F*,&*$F'\"\"#F**$F(F-F*!\"\"" }{TEXT 262 8 " \ndoes " } {TEXT 257 3 "NOT" }{TEXT 304 19 " have a limit as (" }{XPPEDIT 18 0 " x" "6#%\"xG" }{TEXT 281 2 ", " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT 282 16 ") approaches (" }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT 283 2 ", " }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT 284 4 ") .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "t:='t': u:='u': x:='x': y:='y':\nu := proc(x,y) \+ x*y/(x^2+y^2) end:\n`u(x,y) ` = u(x,y); ` `;\nlim1 := limit(u(x,y), x =0):\nlim2 := limit(lim1, y=0):\n`limit u(x,y) as x->0 ` = lim1; `a nd`;\n`limit u(x,y) as x->0 and y->0 ` = lim2;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 164 "`u(x,y) ` = u(x,y); ` `;\nlim1 := limit(u(x ,y), y=0):\nlim2 := limit(lim1, x=0):\n`limit u(x,y) as y->0 ` = li m1; `and`;\n`limit u(x,y) as y->0 and x->0 ` = lim2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "U := subs(\{x=r*cos(t),y=r*sin(t) \},u(x,y)):\n`u(r cos t,r sin t) ` = U; ` `;\nlim1 := limit(U, r=0):\n `limit u(r cos t,r sin t) as r->0 ` = simplify(lim1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 59 " Since this value is dependent on the angle of approach to " } {XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT 285 3 ", \n" }{XPPEDIT 18 0 "u(x,y) = x*y /(x^2 + y^2)" "6#/-%\"uG6$%\"xG%\"yG*(F'\"\"\"F(F*,&*$F'\"\"#F* *$F(F-F*!\"\"" }{TEXT 264 7 " does " }{TEXT 257 3 "NOT" }{TEXT 303 20 " have a limit as (" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 289 2 ", " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT 288 16 ") approaches (" } {XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT 287 2 ", " }{XPPEDIT 18 0 "0" "6#\" \"!" }{TEXT 286 3 ") ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "T heorem 2.1" }{TEXT 316 3 " " }{TEXT -1 5 "Let " }{XPPEDIT 18 0 "f(z ) = u(x,y)+i*v(x,y);" "6#/-%\"fG6#%\"zG,&-%\"uG6$%\"xG%\"yG\"\"\"*&%\" iGF.-%\"vG6$F,F-F.F." }{TEXT -1 65 " be a complex function that is de fined in some neighborhood of " }{XPPEDIT 18 0 "z[0];" "6#&%\"zG6#\" \"!" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 19 "except perhaps a t " }{XPPEDIT 18 0 "z[0] = x[0]+i*y[0];" "6#/&%\"zG6#\"\"!,&&%\"xG6#F '\"\"\"*&%\"iGF,&%\"yG6#F'F,F," }{TEXT -1 9 ". Then " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "Limit(f(z),z = z[0]);" "6#-%&LimitG6$-%\"fG6#%\"zG/F)&F)6#\"\"!" } {TEXT -1 5 " = " }{XPPEDIT 18 0 "w[0] = u[0]+i*v[0];" "6#/&%\"wG6#\" \"!,&&%\"uG6#F'\"\"\"*&%\"iGF,&%\"vG6#F'F,F," }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "if and on ly if" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }{XPPEDIT 18 0 "Limit(u(x,y),`(x,y)` = (x[0], y[0])) = u[0];" " 6#/-%&LimitG6$-%\"uG6$%\"xG%\"yG/%&(x,y)G6$&F*6#\"\"!&F+6#F1&F(6#F1" } {TEXT -1 9 " and " }{XPPEDIT 18 0 "Limit(v(x,y),`(x,y)` = (x[0], y [0])) = v[0];" "6#/-%&LimitG6$-%\"vG6$%\"xG%\"yG/%&(x,y)G6$&F*6#\"\"!& F+6#F1&F(6#F1" }{TEXT -1 4 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 274 " Limits of complex functions ar e formally the same as in the case of real functions, and the sum, dif ference, product, and quotient of functions have limits given by the s um, difference, product, and quotient of the respective limits. These \+ proofs are left as exercises. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 1 "\n" }{TEXT 256 22 "Example 2.17, Page 73." }{TEXT 298 10 " Find \+ " }{XPPEDIT 18 0 "limit(f(z), z=1+i)" "6#-%&limitG6$-%\"fG6#%\"zG/F), &\"\"\"F,%\"iGF," }{TEXT 290 9 " for " }{XPPEDIT 18 0 "f(z) = z^2 - 2*z + 1" "6#/-%\"fG6#%\"zG,(*$F'\"\"#\"\"\"*&F*F+F'F+!\"\"F+F+" } {TEXT 265 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "f:='f': z:='z ':\nf := z -> z^2 - 2*z + 1:\n`f(z) ` = f(z); ` `;\n`limit f(z) as \+ z->1+i ` = limit(f(z), z=1+I);\n `Also, the value of f(1+i) is:`;\n`f( 1+i) ` = f(1+I);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 1 "\n" }{TEXT 256 22 "Example 2.18, Page 75. " }{TEXT 299 51 " Show that the polynomial function given by\n \+ " }{XPPEDIT 18 0 "P(z) = a[0] + a[1]*z + a[2]*z^2" "6#/-%\"PG6#%\"zG,( &%\"aG6#\"\"!\"\"\"*&&F*6#F-F-F'F-F-*&&F*6#\"\"#F-*$F'F4F-F-" }{TEXT -1 9 " + ... + " }{XPPEDIT 18 0 "a[n]*z^n" "6#*&&%\"aG6#%\"nG\"\"\")% \"zGF'F(" }{TEXT 267 30 "\nis continuous at each point " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT 291 24 " in the complex plane.\n " }{TEXT 257 17 "For illustration," }{TEXT 305 9 " we use " } {XPPEDIT 18 0 "n = 5" "6#/%\"nG\"\"&" }{TEXT 293 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "P:='P': z:='z': z0:='z0':\nP := z -> sum('a[ k]'*z^k, 'k'=0..5):\n`P(z) ` = P(z);\nlim := limit(P(z), z=z0):\n`limi t P(z) as z->z0 ` = lim;\n`Also, the value of P(z0) is:`;\n`P(z0) ` \+ = P(z0); ` `;\n`P(z0) = limit P(z) as z->z0 `;\nevalb(limit(P(z), z= z0) = P(z0));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 22 "Example 2.19, Page 76." }{TEXT 300 9 " \+ Find " }{XPPEDIT 18 0 "limit(f(z), z=1+i)" "6#-%&limitG6$-%\"fG6#%\" zG/F),&\"\"\"F,%\"iGF," }{TEXT 292 9 " for " }{XPPEDIT 18 0 "f(z) \+ = (z^2 - 2*i) / (z^2 - 2*z + 2)" "6#/-%\"fG6#%\"zG*&,&*$F'\"\"#\"\" \"*&F+F,%\"iGF,!\"\"F,,(*$F'F+F,*&F+F,F'F,F/F+F,F/" }{TEXT 268 4 " . \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "f:='f': z:='z':\nf := z -> ( z^2 - 2*I)/(z^2 - 2*z + 2):\n`f(z) ` = f(z);\nfun := f(1+I):\n`f(1+I) \+ ` = undefined;\n`However,`;\nlim := limit(f(z) ,z=1+I):\n`limit f(z) \+ as z->1+i ` = lim;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "f: ='f': F:='F': z:='z': Z :='Z':\nf := z -> (z^2 - 2*I)/(z^2 - 2*z + 2): \nfact := factor(f(Z)):\nF := z -> subs(Z=z,fact):\n`f(z) ` = f(z);\n` Simplify the function.`;\n`F(z) ` = F(z); ` `;\n`Evaluate F(z) at z = 1+i`;\n`F(1+i) ` = F(1+I);" }}}{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 2.4." }}}}{MARK "0 0 0" 29 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }