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}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 "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 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple Worksheets, 2001" }{TEXT 360 144 "\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 359 197 "\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 356 1 "\n" }{TEXT 256 30 "CHAPTER 6 COMPLEX INTEGRATION" }{TEXT 337 2 "\n\n" }{TEXT 256 39 "Section 6.3 The Cauchy-Goursat Theorem" } {TEXT 338 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 598 " The Cauchy-Goursat theorem states that within certain domains the integral of an analyti c function over a simple closed contour is zero. An extension of this \+ theorem will allow us to replace integrals over certain complicated co ntours with integrals over contours that are easy to evaluate. We will show how to use the technique of partial fractions together with the \+ Cauchy-Goursat theorem to evaluate certain integrals. In Section 6.4 w e will see that the Cauchy-Goursat theorem implies that an analytic fu nction has an antiderivative. To start with, we need to introduce a fe w new concepts. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 " We saw in Section 1.6 that each simple closed contour \+ " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 78 " divides the plane int o two domains. One domain is bounded and is called the " }{TEXT 362 8 "interior" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 55 ", and the other domain is unbounded and is called the " }{TEXT 363 8 "exterior" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "C" "6#%\"CG" } {TEXT -1 53 ". This result is known as the Jordan Curve Theorem. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 " Reall \+ that a domain " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 47 " is a co nnected open set. In particular, if " }{XPPEDIT 18 0 "z[1];" "6#&%\" zG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z[2];" "6#&%\"zG6#\"\" #" }{TEXT -1 29 " are any pair of points in " }{XPPEDIT 18 0 "D;" "6 #%\"DG" }{TEXT -1 61 ", then they can be joined by a curve that lies \+ entirely in " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 13 ". A domain " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 16 " is said to be " } {TEXT 364 16 "simply connected" }{TEXT -1 56 " if it has the property \+ that any simple closed contour " }{XPPEDIT 18 0 "C;" "6#%\"CG" } {TEXT -1 16 " contained in " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT -1 33 " has its interior contained in " }{XPPEDIT 18 0 "D;" "6#%\"DG " }{TEXT -1 126 ". In other words, there are no \"holes'' in a simply connected domain. A domain that is not simply connected is said to b e a " }{TEXT 365 25 "multiply connected domain" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 " Let th e simple closed contour " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 28 " have the parametrization " }{XPPEDIT 18 0 "`C: `;" "6#%$C:~G" } {XPPEDIT 18 0 "z(t) = x(t)+i*y(t);" "6#/-%\"zG6#%\"tG,&-%\"xG6#F'\"\" \"*&%\"iGF,-%\"yG6#F'F,F," }{TEXT -1 7 ", for " }{TEXT 368 1 " " } {TEXT -1 1 "a" }{XPPEDIT 18 0 "`` <= ``;" "6#1%!GF$" }{TEXT -1 1 "t" } {XPPEDIT 18 0 "`` <= ``" "6#1%!GF$" }{TEXT -1 8 "b. If " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 43 " is parametrized so that the interio r of " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 26 " is kept on the le ft as " }{XPPEDIT 18 0 "z(t);" "6#-%\"zG6#%\"tG" }{TEXT -1 16 " move s around " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 21 ", then we say \+ that " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 21 " is oriented in th e " }{TEXT 366 8 "positive" }{TEXT -1 39 " (counterclockwise) sense; o therwise, " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 14 " is oriented \+ " }{TEXT 367 10 "negatively" }{TEXT -1 7 ". If " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 32 " is positively oriented, then " }{XPPEDIT 18 0 "-C;" "6#,$%\"CG!\"\"" }{TEXT -1 25 " is negatively oriented." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 " Green's theorem, an i mportant result from the calculus of real variables, tells us how to e valuate the line integral of real-valued functions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 256 30 "Theorem 6.4 (Green's Theorem)" }{TEXT 369 6 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 407 5 "Let " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 371 65 " be a simple closed contour with positive orientation, and let " } {XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT 376 44 " be the domain that forms the interior of " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 372 3 ". " }} {PARA 0 "" 0 "" {TEXT 408 4 "If " }{XPPEDIT 18 0 "P" "6#%\"PG" } {TEXT 374 7 " and " }{XPPEDIT 18 0 "Q" "6#%\"QG" }{TEXT 375 75 " ar e continuous and have continuous partial derivatives at all points on \+ " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 373 7 " and " }{XPPEDIT 18 0 "R" "6#%\"RG" }{TEXT 377 9 ", then \n" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "int(P(x,y),x = C .. ` `)+int(Q(x,y),y = C .. ` `) = int(int(diff(Q( x,y),x)-diff(P(x,y),y),x = R .. ` `),y);" "6#/,&-%$intG6$-%\"PG6$%\"xG %\"yG/F+;%\"CG%\"~G\"\"\"-F&6$-%\"QG6$F+F,/F,;F/F0F1-F&6$-F&6$,&-%%dif fG6$-F56$F+F,F+F1-F?6$-F)6$F+F,F,!\"\"/F+;%\"RGF0F," }{TEXT 370 3 " . \+ " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 " We are now ready to state the main result of this \+ section. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 37 "Theorem 6.5 (Cauchy-Goursat Theorem)" }{TEXT 378 8 " \+ Let " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 379 44 " be ana lytic in a simply connected domain " }{XPPEDIT 18 0 "D" "6#%\"DG" } {TEXT 382 4 ". " }}{PARA 0 "" 0 "" {TEXT 409 4 "If " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 381 43 " is a simple closed contour that lies \+ in " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 383 10 ", then " } {XPPEDIT 18 0 "int(f(z),z = C .. ` `) = 0;" "6#/-%$intG6$-%\"fG6#%\"zG /F*;%\"CG%\"~G\"\"!" }{TEXT 380 3 " . " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 361 93 "Load Maple's \"residue\" procedure.\n Make sure this is done only ONCE during a Maple session.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "readlib(residue):" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 23 "E xample 6.12, Page 229." }{TEXT 339 22 " Let us recall that " } {XPPEDIT 18 0 "exp(z)" "6#-%$expG6#%\"zG" }{TEXT 263 4 " , " } {XPPEDIT 18 0 "cos(z)" "6#-%$cosG6#%\"zG" }{TEXT 264 9 " , and " } {XPPEDIT 18 0 "z^n" "6#)%\"zG%\"nG" }{TEXT 265 115 " , where n is a po sitive \ninteger, are all entire functions and have continuous derivat ives. For illustration set " }{XPPEDIT 18 0 "n = 5" "6#/%\"nG\"\"&" } {TEXT 266 49 " ,\nand choose the contour to be the unit circle " } {XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 267 2 ": " }{XPPEDIT 18 0 "abs(z) = 1" "6#/-%$absG6#%\"zG\"\"\"" }{TEXT 268 108 " with positive orientat ion.\nThe Cauchy-Goursat theorem implies that for any simple closed co ntour we have:\n" }{TEXT 256 3 "(a)" }{TEXT 341 2 " " }{XPPEDIT 18 0 "int(exp(z), z=C..` `) = 0" "6#/-%$intG6$-%$expG6#%\"zG/F*;%\"CG%$~~ ~G\"\"!" }{TEXT -1 14 " , " }{TEXT 256 3 "(b)" }{TEXT -1 2 " " }{XPPEDIT 18 0 "int(cos(z), z=C..` `) = 0" "6#/-%$intG6$-%$cosG 6#%\"zG/F*;%\"CG%$~~~G\"\"!" }{TEXT 261 14 " , " }{TEXT 256 3 "(c)" }{TEXT 342 2 " " }{XPPEDIT 18 0 "int(z^5, z=C..` `) = 0 " "6#/-%$intG6$*$%\"zG\"\"&/F(;%\"CG%$~~~G\"\"!" }{TEXT 262 2 "\n\n" } {TEXT 256 3 "(a)" }{TEXT 340 19 " First, consider " }{XPPEDIT 18 0 " int(exp(z), z=C..` `) = 0" "6#/-%$intG6$-%$expG6#%\"zG/F*;%\"CG%$~~~ G\"\"!" }{TEXT 269 46 ", which can be verified with the computation: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 538 "dz :='dz':f:='f':F:='F':g:='g' :t:='t':T:='T':z:='z':Z:='Z':z1:='z1':\nf := z -> exp(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-deriva tive is:`;\ng := t -> simplify(subs(T=t,int(f(z(T))*z1(T), T))):\n`g(t ) ` = 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) ` = expand(g1 - g0);\nI nt(f(z),z=C..``) = expand(g1 - g0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 3 "(b)" }{TEXT -1 11 " Se cond, " }{XPPEDIT 18 0 "int(cos(z), z=C..` `) = 0" "6#/-%$intG6$-%$ cosG6#%\"zG/F*;%\"CG%$~~~G\"\"!" }{TEXT -1 5 " , c" }{TEXT 270 37 "an be verified with the computation:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 538 "dz :='dz':f:='f':F:='F':g:='g':t:='t':T:='T':z:='z':Z:='Z':z1:= 'z1':\nf := z -> cos(z):\n`f(z) ` = f(z);\nz := t -> exp(I*t):\n`C: z (t) ` = z(t);\n`f(z(t)) ` = f(z(t));\nz1 := t -> subs(T=t,diff(z(T), T )):\n`dz = z '(t) dt ` = z1(t), `dt`;\nInt(f(z),z=C..``) = Int(f(z(t)) *z1(t),t=0..2*pi);\n`The anti-derivative is:`;\ng := t -> simplify(sub s(T=t,int(f(z(T))*z1(T), T))):\n`g(t) ` = g(t);\ng1 := g(2*Pi):\ng0 : = g(0):\n`g(2*Pi) ` = expand(g1),` and `,`g(0) ` = expand(g0);\n`g(2 *Pi) - g(0) ` = expand(g1 - g0);\nInt(f(z),z=C..``) = expand(g1 - g0) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 3 "(c)" }{TEXT -1 10 " Third, " }{XPPEDIT 18 0 "int(z^5, z =C..` `) = 0" "6#/-%$intG6$*$%\"zG\"\"&/F(;%\"CG%$~~~G\"\"!" }{TEXT -1 5 " , c" }{TEXT 271 37 "an be verified with the computation:\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 535 "dz :='dz':f:='f':F:='F':g:='g':t:= 't':T:='T':z:='z':Z:='Z':z1:='z1':\nf := z -> z^5:\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(t) ` = 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) ` = expand(g1 - g0);\nInt(f(z), z=C..``) = expand(g1 - g0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 " We want to be able to replace integrals over certain complicated co ntours with integrals that are easy to evaluate. If " }{XPPEDIT 18 0 " C[1];" "6#&%\"CG6#\"\"\"" }{TEXT -1 99 " is a simple closed contour th at can be \"continuously deformed\" into another simple closed contour " }{XPPEDIT 18 0 "C[2];" "6#&%\"CG6#\"\"#" }{TEXT -1 40 " without pas sing through a point where " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 62 " is not analytic, then the value of the contour integral of " } {XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 7 " over " }{XPPEDIT 18 0 "C[1] ;" "6#&%\"CG6#\"\"\"" }{TEXT -1 47 " is the same as the value of the i ntegral of " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 7 " over " } {XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT -1 47 ". To be precise, \+ we state the following result." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 37 "Theorem 6.6 (Deformation of Contour)" }{TEXT 384 3 " " }}{PARA 0 "" 0 "" {TEXT 410 5 "Let " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\" " }{TEXT 385 7 " and " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" } {TEXT 386 63 " be two simple closed positively oriented contours such that " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" }{TEXT 387 20 " lie s interior to " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT 388 4 " . " }}{PARA 0 "" 0 "" {TEXT 411 4 "If " }{XPPEDIT 18 0 "f(z)" "6#- %\"fG6#%\"zG" }{TEXT 389 27 " is analytic in a domain " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 393 22 " that contains both " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" }{TEXT 390 7 " and " }{XPPEDIT 18 0 " C[2]" "6#&%\"CG6#\"\"#" }{TEXT 391 37 " and the region between them, \+ then " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 412 7 " " }{XPPEDIT 18 0 "int(f(z),z = C[1] .. ` `) = int(f(z),z = C[2 ] .. ` `);" "6#/-%$intG6$-%\"fG6#%\"zG/F*;&%\"CG6#\"\"\"%\"~G-F%6$-F(6 #F*/F*;&F.6#\"\"#F1" }{TEXT 392 2 " ." }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 293 " We now state an important result th at is implied by the deformation of contour theorem. This result will \+ occur several times in the theory to be developed and is an important \+ tool for computations. You may want to compare the proof of this corol lary with your solution to Exercise 6.2.23. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 13 "Corollary 6.1" }{TEXT -1 8 " Let " }{XPPEDIT 18 0 "z[0];" "6#&%\"zG6#\"\"!" }{TEXT -1 37 " denote a fixed complex val ue. If " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 66 " is a simple c losed contour with positive orientation such that " }{XPPEDIT 18 0 "z [0]" "6#&%\"zG6#\"\"!" }{TEXT -1 20 " lies interior to " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 10 ", then " }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "int(1/(z-z[0]),z = C .. ` `);" "6#- %$intG6$*&\"\"\"F',&%\"zGF'&F)6#\"\"!!\"\"F-/F);%\"CG%\"~G" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "2*pi*i;" "6#*(\"\"#\"\"\"%#piGF%%\"iGF%" } {TEXT -1 5 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 " " {TEXT -1 10 " " }{XPPEDIT 18 0 "int(1/((z-z[0])^n),z = C .. ` `);" "6#-%$intG6$*&\"\"\"F'),&%\"zGF'&F*6#\"\"!!\"\"%\"nGF./F*;%\"C G%\"~G" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT -1 9 ", where " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 23 " is any intege r except " }{XPPEDIT 18 0 "n = 1;" "6#/%\"nG\"\"\"" }{TEXT -1 4 ". \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 23 "Extra Eample, Page 232. " }{TEXT 343 8 " Let " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" } {TEXT 333 37 " denote a fixed complex value. If " }{XPPEDIT 18 0 "C " "6#%\"CG" }{TEXT 334 67 " is a \nsimple closed contour with positiv e orientation such that " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" } {TEXT 335 20 " lies interior to " }{XPPEDIT 18 0 "C" "6#%\"CG" } {TEXT 336 8 " , then\n" }{TEXT 256 3 "(a)" }{TEXT 344 3 " " } {XPPEDIT 18 0 "int(1/(z - z[0]), z=C..` `) = 2*pi*i" "6#/-%$intG6$* &\"\"\"F(,&%\"zGF(&F*6#\"\"!!\"\"F./F*;%\"CG%$~~~G*(\"\"#F(%#piGF(%\"i GF(" }{TEXT 260 9 " and\n" }{TEXT 256 3 "(b)" }{TEXT 345 3 " " } {XPPEDIT 18 0 "int(1/(z - z0)^n, z=C..` `) = 0" "6#/-%$intG6$*&\"\" \"F(),&%\"zGF(%#z0G!\"\"%\"nGF-/F+;%\"CG%$~~~G\"\"!" }{TEXT 272 11 " , where " }{XPPEDIT 18 0 "n <> 1" "6#0%\"nG\"\"\"" }{TEXT 273 18 " i s an integer.\n\n" }{TEXT 257 17 "For illustration," }{TEXT 357 28 " w e use a circle of radius " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 274 15 " centered at " }{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT 275 3 " .\n" }{TEXT 256 3 "(a)" }{TEXT 346 14 " Show that " } {XPPEDIT 18 0 "int(1/(z - z[0]), z=C..` `) = 2*pi*i" "6#/-%$intG6$* &\"\"\"F(,&%\"zGF(&F*6#\"\"!!\"\"F./F*;%\"CG%$~~~G*(\"\"#F(%#piGF(%\"i GF(" }{TEXT 276 3 " . " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 557 "dz :='dz ':f:='f':F:='F':g:='g':t:='t':\nT:='T':z:='z':Z:='Z':z0:='z0':z1:='z1' :\nf := z -> 1/(z - z0):\n`f(z) ` = f(z);\nz := t -> z0 + exp(I*t):\n` C: z(t) ` = z(t);\n`f(z(t)) ` = f(z(t));\nz1 := t -> subs(T=t,diff(z( T), T)):\n`dz = z '(t) dt ` = z1(t), `dt`;\nInt(f(z),z=C..``) = Int(f( z(t))*z1(t),t=0..2*pi);\n`The anti-derivative is:`;\ng := t -> simplif y(subs(T=t,int(f(z(T))*z1(T), T))):\n`g(t) ` = g(t);\ng1 := g(2*Pi): \ng0 := g(0):\n`g(2*Pi) ` = expand(g1),` and `,`g(0) ` = expand(g0); \n`g(2*Pi) - g(0) ` = expand(g1 - g0);\nInt(f(z),z=C..``) = expand(g1 - g0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 3 "(b)" }{TEXT 347 14 " Show that " }{XPPEDIT 18 0 " int(1/(z - z0)^5, z=C..` `) = 0" "6#/-%$intG6$*&\"\"\"F(*$,&%\"zGF( %#z0G!\"\"\"\"&F-/F+;%\"CG%$~~~G\"\"!" }{TEXT 277 2 " \n" }{TEXT 257 17 "For illustration," }{TEXT 358 28 " we use a circle of radius " } {XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 279 15 " centered at " } {XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT 278 3 " .\n" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 619 "dz :='dz':f:='f':F:='F':g:='g':t:='t':\nT:= 'T':z:='z':Z:='Z':z0:='z0':z1:='z1':\nf := z -> 1/(z - z0)^5:\n`f(z) ` = f(z);\nz := t -> z0 + 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);\nInt(f(z) ,z=C..``) = Int(simplify(f(z(t))*z1(t)),t=0..2*pi);\n`The anti-derivat ive is:`;\ng := t -> simplify(subs(T=t,int(f(z(T))*z1(T), T))):\n`g(t) ` = g(t);\ng1 := g(2*Pi):\ng0 := g(0):\n`g(2*Pi) ` = expand(g1),` a nd `,`g(0) ` = expand(g0);\n`g(2*Pi) - g(0) ` = expand(g1 - g0);\nIn t(f(z),z=C..``) = expand(g1 - g0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 " The deformation of contour theorem is an extension of \+ the Cauchy-Goursat theorem to a doubly connected domain in the followi ng sense. Let " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT -1 27 " be a domai n that contains " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" }{TEXT 394 7 " and " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT -1 47 " and the region between them. Then the contour " }{XPPEDIT 18 0 "C = C[2]- C[1];" "6#/%\"CG,&&F$6#\"\"#\"\"\"&F$6#F)!\"\"" }{TEXT -1 52 " is a pa rametrization of the boundary of the region " }{XPPEDIT 18 0 "R;" "6#% \"RG" }{TEXT -1 19 " that lies between " }{XPPEDIT 18 0 "C[1]" "6#&%\" CG6#\"\"\"" }{TEXT 395 7 " and " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\" \"#" }{TEXT -1 23 " so that the points of " }{XPPEDIT 18 0 "R" "6#%\"R G" }{TEXT -1 20 " lie to the left of " }{XPPEDIT 18 0 "C" "6#%\"CG" } {TEXT -1 12 " as a point " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 14 " moves around " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 8 ". Hence " } {XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 46 " is a positive orientation o f the boundary of " }{XPPEDIT 18 0 "R" "6#%\"RG" }{TEXT -1 32 ", and T heorem 6.6 implies that " }{XPPEDIT 18 0 "int(f(z),z = C .. ` `) = 0; " "6#/-%$intG6$-%\"fG6#%\"zG/F*;%\"CG%\"~G\"\"!" }{TEXT -1 4 " . " }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 214 " We can extend Theorem 6.6 to multiply connected domains with more than one \"hole.\" The proof, w hich is left for the reader, involves the introduction of several cuts and is similar to the proof of Theorem 6.6. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 256 46 "Theorem 6.7 (Extended Cauchy-Goursat Theorem)" } {TEXT 396 3 " " }}{PARA 0 "" 0 "" {TEXT 413 6 "Let " }{XPPEDIT 18 0 "C, C[1], C[2], `...`, C[n]" "6'%\"CG&F#6#\"\"\"&F#6#\"\"#%$...G&F#6 #%\"nG" }{TEXT 397 73 " be simple closed positively oriented contour s with the property that " }{XPPEDIT 18 0 "C[k]" "6#&%\"CG6#%\"kG" } {TEXT 398 28 " lies interior to C for " }{XPPEDIT 18 0 "k=1, 2, `. ..`, n" "6&/%\"kG\"\"\"\"\"#%$...G%\"nG" }{TEXT 399 30 " and the set \+ of interior to " }{XPPEDIT 18 0 "C[k]" "6#&%\"CG6#%\"kG" }{TEXT 400 52 " has no points in common with the set interior to " }{XPPEDIT 18 0 "C[j]" "6#&%\"CG6#%\"jG" }{TEXT 401 6 " if " }{XPPEDIT 18 0 "k< >j" "6#0%\"kG%\"jG" }{TEXT 402 9 " . Let " }{XPPEDIT 18 0 "f(z[0]); " "6#-%\"fG6#&%\"zG6#\"\"!" }{TEXT 403 27 " be analytic on a domain \+ " }{XPPEDIT 18 0 "D;" "6#%\"DG" }{TEXT 406 64 " that contains all the contours and the region between C and " }{XPPEDIT 18 0 "C[1]+C[2]+` ...`+C[n]" "6#,*&%\"CG6#\"\"\"F'&F%6#\"\"#F'%$...GF'&F%6#%\"nGF'" } {TEXT 404 11 " , then \n" }}{PARA 0 "" 0 "" {TEXT -1 8 " " } {XPPEDIT 18 0 "int(f(z),z = C .. ` `) = sum(int(f(z),z = C[k] .. ` `), k = 1 .. n);" "6#/-%$intG6$-%\"fG6#%\"zG/F*;%\"CG%\"~G-%$sumG6$-F%6$-F (6#F*/F*;&F-6#%\"kGF./F:;\"\"\"%\"nG" }{TEXT -1 5 " . " }{TEXT 405 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 256 23 "Example 6.14, Page 233." }{TEXT 348 15 " Show that \+ " }{XPPEDIT 18 0 "int(2*z/ (z^2 + 2), z=C..` `) = 4*pi*i" "6#/-%$i ntG6$*(\"\"#\"\"\"%\"zGF),&*$F*F(F)F(F)!\"\"/F*;%\"CG%$~~~G*(\"\"%F)%# piGF)%\"iGF)" }{TEXT 280 10 " ,\nwhere " }{XPPEDIT 18 0 "C" "6#%\"CG " }{TEXT 294 17 " is the circle " }{XPPEDIT 18 0 "C" "6#%\"CG" } {TEXT 295 2 ": " }{XPPEDIT 18 0 "abs(z)= 2" "6#/-%$absG6#%\"zG\"\"#" } {TEXT 296 35 " taken with positive orientation.\n" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 64 "f:='f': F:='F': z:='z':\nf := z -> 2*z/(z^2 + 2):\n `f(z) ` = f(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 143 "First, split the function up into parti al fractions involving linear terms in the denominators. Don't worry \+ \nabout the subroutine. It splits " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6# %\"zG" }{TEXT 297 74 " up into partial fractions. The result will be an equivalent function " }{XPPEDIT 18 0 "F(z)" "6#-%\"FG6#%\"zG" } {TEXT 298 37 " . \nThe list of terms added to form " }{XPPEDIT 18 0 " F(z)" "6#-%\"FG6#%\"zG" }{TEXT 300 62 " have singularities in the sam e order in the list of points " }{XPPEDIT 18 0 "Z" "6#%\"ZG" }{TEXT 299 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 588 "Zn := sort([solve(den om(f(z))=0, z)]):\nRn := array(1..nops(Zn)):\nSn := array(1..nops(Zn)) :\nF1 := 0:\nfor i from 1 to nops(Zn) do \n if i=1 then p:=1 fi; \n if 10 then\n Z[p]:=Zn[i]; R[p] :=Rn[i]; S[p]:=Sn[i]; p:=p+1 fi;\nod:\n`f(z) ` = f(z);\n`f(z) ` = F1; \nprint(`singularities =`, Z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 42 "Determine which singularities lie inside " }{XPPEDIT 18 0 "C" "6# %\"CG" }{TEXT 302 3 ": " }{XPPEDIT 18 0 "abs(z)= 2" "6#/-%$absG6#%\"z G\"\"#" }{TEXT 301 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "prin t(z[1]=Z[1], abs(z[1])<2, evalb(evalf(abs(Z[1]))<2));\nprint(z[2]=Z[2] , abs(z[2])<2, evalb(evalf(abs(Z[2]))<2));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 283 37 "Since both singu larities lie inside " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 303 52 " , \+ add the corresponding term in the \nsum forming " }{XPPEDIT 18 0 "f( z)" "6#-%\"fG6#%\"zG" }{TEXT 304 51 " times constant in the numerator that term times " }{XPPEDIT 18 0 "2*pi*i" "6#*(\"\"#\"\"\"%#piGF%%\" iGF%" }{TEXT 284 4 " . \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "val := 2*Pi*I*numer(S[1]) + 2*Pi*I*numer(S[2]):\nInt(f(z),z=C..``) = val;" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 23 "Example 6.15, Page 234." }{TEXT 349 15 " Show that \+ " }{XPPEDIT 18 0 "int(2*z/(z^2 + 2),z=C..` `) = 2*pi*i" "6#/-%$intG6 $*(\"\"#\"\"\"%\"zGF),&*$F*F(F)F(F)!\"\"/F*;%\"CG%$~~~G*(F(F)%#piGF)% \"iGF)" }{TEXT 285 10 " ,\nwhere " }{XPPEDIT 18 0 "C" "6#%\"CG" } {TEXT 305 17 " is the circle " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 307 2 ": " }{XPPEDIT 18 0 "abs(z - i) = 1" "6#/-%$absG6#,&%\"zG\"\"\"% \"iG!\"\"F)" }{TEXT 306 35 " taken with positive orientation.\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "f:='f': F:='F': z:='z':\nf := z -> \+ 2*z/(z^2 + 2):\n`f(z) ` = f(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 286 143 "First, split the function up into partial fractions involving lin ear terms in the denominators. Don't worry \nabout the subroutine. It splits " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 308 73 " up into partial fractions. The result will be an equivalent function \+ " }{XPPEDIT 18 0 "F(z)" "6#-%\"FG6#%\"zG" }{TEXT 309 37 " . \nThe list of terms added to form " }{XPPEDIT 18 0 "F(z)" "6#-%\"FG6#%\"zG" } {TEXT 311 62 " have singularities in the same order in the list of po ints " }{XPPEDIT 18 0 "Z" "6#%\"ZG" }{TEXT 310 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 588 "Zn := sort([solve(denom(f(z))=0, z)]):\nRn := a rray(1..nops(Zn)):\nSn := array(1..nops(Zn)):\nF1 := 0:\nfor i from 1 \+ to nops(Zn) do \n if i=1 then p:=1 fi;\n if 10 then\n Z[p]:=Zn[i]; R[p]:=Rn[i]; S[p]:=Sn[i]; p:=p+ 1 fi;\nod:\n`f(z) ` = f(z);\n`f(z) ` = F1;\nprint(`singularities =`, \+ Z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 287 42 "Determine which singularities lie inside " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 312 3 ": " }{XPPEDIT 18 0 "abs(z - i) = 1" "6#/-%$absG6#,&%\"zG\"\"\"%\"iG!\"\"F)" }{TEXT 313 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "print(z[1]=Z[1], abs(z[1]-I)<2, \n ev alb(evalf(abs(Z[1]-I))<1));\nprint(z[2]=Z[2], abs(z[2]-I)<2, \n e valb(evalf(abs(Z[2]-I))<1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 46 "Since only the first singulari ty lies inside " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 314 44 " , add \+ the first term in the sum \nforming " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG 6#%\"zG" }{TEXT 315 51 " times constant in the numerator that term ti mes " }{XPPEDIT 18 0 "2*pi*i" "6#*(\"\"#\"\"\"%#piGF%%\"iGF%" }{TEXT 289 4 " . \n" }}{PARA 0 "" 0 "" {TEXT 351 75 "Remark. Sometimes Maple will form the list of values in a different order." }}{PARA 0 "" 0 " " {TEXT 352 79 "It is always necessary to visually inspect the above r esults before proceeding." }}{PARA 0 "" 0 "" {TEXT 353 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "val := 2*Pi*I*numer(S[1]):\nInt(f(z),z=C. .``) = val;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 23 "Example 6.16, Page 234." }{TEXT 350 15 " Sho w that " }{XPPEDIT 18 0 "int((z - 2)/(z^2 - 2), z=C..` `) = - 6*pi *i" "6#/-%$intG6$*&,&%\"zG\"\"\"\"\"#!\"\"F*,&*$F)F+F*F+F,F,/F);%\"CG% $~~~G,$*(\"\"'F*%#piGF*%\"iGF*F," }{TEXT 290 10 " ,\nwhere " } {XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 316 53 " is the figure eight shown in the text on page 135.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "f:=' f': F:='F': z:='z':\nf := z -> (z - 2)/(z^2 - z):\n`f(z) ` = f(z);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 143 "First, split the function up into partial fractions involving linear terms in the denominators. Don't worry \nabout the subroutine . It splits " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 318 73 " up into partial fractions. The result will be an equivalent functio n " }{XPPEDIT 18 0 "F(z)" "6#-%\"FG6#%\"zG" }{TEXT 319 37 " . \nThe l ist of terms added to form " }{XPPEDIT 18 0 "F(z)" "6#-%\"FG6#%\"zG" }{TEXT 321 62 " have singularities in the same order in the list of p oints " }{XPPEDIT 18 0 "Z" "6#%\"ZG" }{TEXT 320 2 " \n" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 588 "Zn := sort([solve(denom(f(z))=0, z)]):\nRn := array(1..nops(Zn)):\nSn := array(1..nops(Zn)):\nF1 := 0:\nfor i from \+ 1 to nops(Zn) do \n if i=1 then p:=1 fi;\n if 10 then\n Z[p]:=Zn[i]; R[p]:=Rn[i]; S[p]:=Sn[i]; p:= p+1 fi;\nod:\n`f(z) ` = f(z);\n`f(z) ` = F1;\nprint(`singularities =` , Z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 58 "Determine which singularities corresponds to the cont our " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" }{TEXT 322 7 " and \+ " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT 323 60 " . The secon d term will correspond \nto the singularity at " }{XPPEDIT 18 0 "z = \+ 1" "6#/%\"zG\"\"\"" }{TEXT 325 21 " used with contour " }{XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT 324 56 " and the first term corr esponds to the singularity at " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"! " }{TEXT 326 21 " \nused with contour " }{XPPEDIT 18 0 "C[1]" "6#&%\" CG6#\"\"\"" }{TEXT 327 11 " . Since " }{XPPEDIT 18 0 "C[2]" "6#&%\"C G6#\"\"#" }{TEXT 328 72 " has positive orientation, we add the second term in the sum forming \n" }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 329 54 " times constant in the numerator of that term times \+ " }{XPPEDIT 18 0 "2*pi*i" "6#*(\"\"#\"\"\"%#piGF%%\"iGF%" }{TEXT 292 19 " . And since the " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" } {TEXT 330 76 " has negative orientation, \nwe subtract the first term in the sum forming " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 331 54 " times constant in the numerator of that term times " } {XPPEDIT 18 0 "2*pi*i" "6#*(\"\"#\"\"\"%#piGF%%\"iGF%" }{TEXT 332 4 " \+ . \n" }}{PARA 0 "" 0 "" {TEXT 354 75 "Remark. Sometimes Maple will fo rm the list of values in a different order." }}{PARA 0 "" 0 "" {TEXT 355 79 "It is always necessary to visually inspect the above results b efore proceeding." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 72 "val := 2*Pi*I*numer(S[2]) - 2*Pi*I*numer(S[1]):\nIn t(f(z),z=C..``) = val;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 293 19 "End of Section 6.3." }}}}{MARK "0 0 0" 13 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }