{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 "" 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 "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 "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 "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 2" -1 257 1 {CSTYLE "" -1 -1 "Symbol" 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" -1 258 1 {CSTYLE "" -1 -1 "Gene va" 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 273 144 "\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 272 197 "\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 270 1 "\n" }{TEXT 256 30 "CHAPTER 6 COMPLEX INTEGRATION" }{TEXT 266 2 "\n\n" }{TEXT 256 52 "Section 6.4 The Fundamental Theorems of Integ ration" }{TEXT 267 3 "\n\n\n" }{TEXT -1 7 " Let " }{XPPEDIT 18 0 "f; " "6#%\"fG" }{TEXT -1 44 " be analytic in the simply connected domain \+ " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 59 ". The theorems in this s ection show that an antiderivative " }{XPPEDIT 18 0 "F;" "6#%\"FG" } {TEXT -1 147 " can be constructed by contour integration. A consequenc e will be the fact that in a simply connected domain, the integral of \+ an analytic function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 27 " al ong any contour joining " }{XPPEDIT 18 0 "z[0];" "6#&%\"zG6#\"\"!" } {TEXT -1 4 " to " }{XPPEDIT 18 0 "z[1];" "6#&%\"zG6#\"\"\"" }{TEXT -1 40 " is the same, and its value is given by " }{XPPEDIT 18 0 "F(z[1])- F(z[0])" "6#,&-%\"FG6#&%\"zG6#\"\"\"F*-F%6#&F(6#\"\"!!\"\"" }{TEXT -1 127 ". Because of this, we will be able to use the antiderivative form ulas from Calculus to compute the value of definite integrals." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 55 "Theorem 6.8 (Indefinite Integrals or Anti-derivatives)" } {TEXT 275 8 " \n " }}{PARA 0 "" 0 "" {TEXT 280 5 "Let " } {XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 278 46 " be analytic in the simply connected domain " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 279 9 " . If " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT 284 34 " is a fixed value in D and if " }{XPPEDIT 18 0 "C" "6#%\"CG" } {TEXT 285 44 " is any contour in D with initial point " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT 283 54 " and terminal point z, \+ then the function given by \n" }}{PARA 0 "" 0 "" {TEXT 294 6 " \+ " }{XPPEDIT 18 0 "F(z) = int(f(z), z=z[0]..z)" "6#/-%\"FG6#%\"zG-%$int G6$-%\"fG6#F'/F';&F'6#\"\"!F'" }{TEXT 282 30 " , is analytic in D \+ and " }{XPPEDIT 18 0 "`F '(z)` = f(z)" "6#/%'F~'(z)G-%\"fG6#%\"zG" } {TEXT 281 2 " ." }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "If we set " }{XPPEDIT 18 0 "z = z[ 1];" "6#/%\"zG&F$6#\"\"\"" }{TEXT -1 121 " in Theorem 6.8, then we obt ain the following familiar result for evaluating a definite integral o f an analytic function." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 256 33 "Theorem 6.9 (Definite Integrals)" } {TEXT 274 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 5 "Let " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 293 44 " be analytic in a simply connected domain " }{XPPEDIT 18 0 " D" "6#%\"DG" }{TEXT 291 7 ". If " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6# \"\"!" }{TEXT 289 7 " and " }{XPPEDIT 18 0 "z[1]" "6#&%\"zG6#\"\"\" " }{TEXT 290 21 " are two points in " }{XPPEDIT 18 0 "D" "6#%\"DG" } {TEXT 292 8 ", then\n" }}{PARA 0 "" 0 "" {TEXT 295 6 " " } {XPPEDIT 18 0 "int(f(z), z=z[0]..z[1) = F(z[1]) - F(z[0])" "6#/-%$intG 6$-%\"fG6#%\"zG/F*;&F*6#\"\"!&F*6#\"\"\",&-%\"FG6#&F*6#F2F2-F56#&F*6#F /!\"\"" }{TEXT 286 13 " , where " }{XPPEDIT 18 0 "F(z)" "6#-%\"FG6 #%\"zG" }{TEXT 287 28 " is any antiderivative of " }{XPPEDIT 18 0 "f (z)" "6#-%\"fG6#%\"zG" }{TEXT 288 2 " ." }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 23 " Example 6.17, Page 240." }{TEXT 277 36 " Evaluate the definite integ ral " }{XPPEDIT 18 0 "int(1/(2*sqrt(z)), z=4..8+6*i) = 1 + i" "6#/-% $intG6$*&\"\"\"F(*&\"\"#F(-%%sqrtG6#%\"zGF(!\"\"/F.;\"\"%,&\"\")F(*&\" \"'F(%\"iGF(F(,&F(F(F7F(" }{TEXT 276 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 389 "f:='f': F:='F': g:='g': z:='z':\nf := z -> 1/(2*z^(1 /2)):\na := 4:\nb := 8+6*I:\ng := z -> subs(Z=z,int(f(Z),Z)):\n`f(z) ` = f(z);\n`a ` = a, ` b ` = b;\n`g(z) = `, Int(f(z), z) = g(z);\n`g( b) ` = expand(g(b)),` and `,`g(a) ` = expand(g(a));\n`g(b) - g(a) ` \+ = g(b) - g(a);\n`g(b) - g(a) ` = evalc(g(b) - g(a));\n`g(b) - g(a) ` = simplify(g(b) - g(a),power);\nInt(f(z), z=a..b) = int(f(z), z=a..b); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 1 "\n" }{TEXT 256 23 "Example 6.18, Page 241." }{TEXT 268 36 " Evaluate the definite integral " }{XPPEDIT 18 0 "int(cos(z), \+ z=1..i) = -sin(1) + i*sinh(1)" "6#/-%$intG6$-%$cosG6#%\"zG/F*;\"\"\"% \"iG,&-%$sinG6#F-!\"\"*&F.F--%%sinhG6#F-F-F-" }{TEXT 261 2 " ." } {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 331 "f:='f': F:='F': \+ g:='g': z:='z':\nf := z -> cos(z):\na := 1:\nb := I:\ng := z -> subs(Z =z,int(f(Z),Z)):\n`f(z) ` = f(z);\n`a ` = a, ` b ` = b;\n`g(z) = `, I nt(f(z), z) = g(z);\n`g(b) ` = expand(g(b)),` and `,`g(a) ` = expand (g(a));\n`g(b) - g(a) ` = g(b) - g(a);\n`g(b) - g(a) ` = evalc(g(b) - \+ g(a));\nInt(f(z), z=a..b) = int(f(z), z=a..b);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 32 "Example 6.1 9 and 6.20, Page 241." }{TEXT 269 36 " Evaluate the definite integra l " }{XPPEDIT 18 0 "int(1/z, z=C..` `) = 2*pi*i" "6#/-%$intG6$*&\" \"\"F(%\"zG!\"\"/F);%\"CG%$~~~G*(\"\"#F(%#piGF(%\"iGF(" }{TEXT 262 10 " ,\nwhere " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 263 22 " is the uni t circle " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 264 2 ": " }{XPPEDIT 18 0 "abs(z) = 1" "6#/-%$absG6#%\"zG\"\"\"" }{TEXT 265 36 " , taken w ith positive orientation\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 533 "dz : ='dz': f:='f': F:='F': t:='t': T:='T': z:='z': z1:='z1':\nf := z -> 1/ z:\n`f(z) ` = f(z);\nz := t -> exp(I*t):\n`C: z(t) ` = z(t);\n`f(z(t) ) ` = f(z(t));\nz1 := t -> subs(T=t,diff(z(T), T)):\n`dz = z '(t) dt ` = z1(t), `dt`;\nInt(f(z),z=C..` `) = Int(f(z(t))*z1(t),t=0..2*pi);\n` The anti-derivative is:`;\ng := t -> simplify(subs(T=t,int(f(z(T))*z1( T), T))):\n`g(z) = ` = g(t);\ng1 := g(2*Pi):\ng0 := g(0):\n`g(2*Pi) ` \+ = expand(g1),` and `,`g(0) ` = expand(g0);\n`g(2*Pi) - g(0) ` = exp and(g1 - g0);\nInt(f(z), z=C..` `) = expand(g1 - g0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 19 "End of Section 6.4." }}}}{MARK "0 0 0" 25 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }