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{TEXT 257 41 "COMPLEX ANALYSIS: Maple Worksheets, 2001" }{TEXT 349 144 "\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 348 197 "\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 346 1 "\n" }{TEXT 256 30 "CHAPTER 6 COMPLEX INTEGRATION" }{TEXT 329 2 "\n\n" }{TEXT 256 43 "Section 6.2 Contours and Contour Integrals" } {TEXT 330 1 "\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 " In S ection 6.1 we learned how to evaluate integrals of the form " } {XPPEDIT 18 0 "int(f(t),t = a .. b)" "6#-%$intG6$-%\"fG6#%\"tG/F);%\"a G%\"bG" }{TEXT -1 10 ", where " }{XPPEDIT 18 0 "f;" "6#%\"fG" } {TEXT -1 25 " was complex-valued and " }{XPPEDIT 18 0 "[a, b];" "6#7$ %\"aG%\"bG" }{TEXT -1 43 " was an interval on the real axis (so that \+ " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 16 " was real, with " } {XPPEDIT 18 0 "t*epsilon;" "6#*&%\"tG\"\"\"%(epsilonGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 72 "). In this section we shall define and evaluate integrals of the form " } {XPPEDIT 18 0 "int(f(z),z = C .. ` `)" "6#-%$intG6$-%\"fG6#%\"zG/F);% \"CG%\"~G" }{TEXT -1 10 ", where " }{XPPEDIT 18 0 "f;" "6#%\"fG" } {TEXT -1 25 " is complex-valued and " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 37 " is a contour in the plane (so that " }{XPPEDIT 18 0 "z ;" "6#%\"zG" }{TEXT -1 19 " is complex, with " }{XPPEDIT 18 0 "z*epsi lon*C;" "6#*(%\"zG\"\"\"%(epsilonGF%%\"CGF%" }{TEXT -1 144 "). Our ma in result is Theorem 6.1, which will show how to transform the latter \+ type of integral into the kind we investigated in Section 6.1. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 " We wil l use concepts first introduced in Section 1.6. Recall that to represe nt a curve " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 47 " in the pla ne we use the parametric notation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "`C: `;" "6#%$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 350 1 " " } {TEXT -1 1 "a" }{XPPEDIT 18 0 "`` <= ``;" "6#1%!GF$" }{TEXT -1 1 "t" } {XPPEDIT 18 0 "`` <= ``" "6#1%!GF$" }{TEXT -1 1 "b" }{TEXT 351 3 ", \+ " }{TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "x(t);" "6#-%\"xG6#%\"tG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y(t);" "6#-%\"yG6#%\"tG" }{TEXT -1 252 " are continuous functions. We will place a few more restriction s on the type of curve that we will be studying. The following discus sion will lead to the concept of a contour, which is a type of curve t hat is adequate for the study of integration. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 16 " Recall that " }{XPPEDIT 18 0 "C;" "6#%\"CG" } {TEXT -1 5 " is " }{TEXT 352 6 "simple" }{TEXT -1 43 " if it does not cross itself, which means " }{XPPEDIT 18 0 "z(t[1]) <> z(t[2]);" "6# 0-%\"zG6#&%\"tG6#\"\"\"-F%6#&F(6#\"\"#" }{TEXT -1 12 " whenever " } {XPPEDIT 18 0 "t[1] <> t[2];" "6#0&%\"tG6#\"\"\"&F%6#\"\"#" }{TEXT -1 12 ". A curve " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 26 " with t he property that " }{XPPEDIT 18 0 "z(a) = z(b);" "6#/-%\"zG6#%\"aG-F% 6#%\"bG" }{TEXT -1 7 " is a " }{TEXT 353 12 "closed curve" }{TEXT -1 7 ". If " }{XPPEDIT 18 0 "z(a) = z(b)" "6#/-%\"zG6#%\"aG-F%6#%\"bG" }{TEXT -1 55 " is the only point of intersection, then we say that \+ " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 7 " is a " }{TEXT 354 19 "s imple closed curve" }{TEXT -1 20 ". As the parameter " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 26 " increases from the value " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 14 " to the value " }{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 13 ", the point " }{XPPEDIT 18 0 "z(t);" "6#-%\"z G6#%\"tG" }{TEXT -1 16 " starts at the " }{TEXT 356 13 "initial point " }{TEXT -1 2 " " }{XPPEDIT 18 0 "z(a)" "6#-%\"zG6#%\"aG" }{TEXT -1 26 ", moves along the curve " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 22 ", and ends up at the " }{TEXT 355 15 "terminal point " }{TEXT -1 1 " " }{XPPEDIT 18 0 "z(b)" "6#-%\"zG6#%\"bG" }{TEXT -1 7 ". If \+ " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 19 " is simple, then " } {XPPEDIT 18 0 "z(t)" "6#-%\"zG6#%\"tG" }{TEXT -1 27 " moves continuou sly from " }{XPPEDIT 18 0 "z(a)" "6#-%\"zG6#%\"aG" }{TEXT -1 6 " to \+ " }{XPPEDIT 18 0 "z(b)" "6#-%\"zG6#%\"bG" }{TEXT -1 5 " as " } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 38 " increases, and the curve is given an " }{TEXT 357 11 "orientation" }{TEXT -1 55 ", which we indic ate by drawing arrows along the curve. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 32 " The complex-valued function " }{XPPEDIT 18 0 "z(t) \+ = x(t)+i*y(t)" "6#/-%\"zG6#%\"tG,&-%\"xG6#F'\"\"\"*&%\"iGF,-%\"yG6#F'F ,F," }{TEXT -1 16 " is said to be " }{TEXT 358 14 "differentiable" } {TEXT -1 4 " on " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 10 " if both " }{XPPEDIT 18 0 "x(t)" "6#-%\"xG6#%\"tG" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "y(t)" "6#-%\"yG6#%\"tG" }{TEXT -1 25 " are di fferentiable for " }{TEXT 359 1 " " }{TEXT -1 1 "a" }{XPPEDIT 18 0 "`` <= ``;" "6#1%!GF$" }{TEXT -1 1 "t" }{XPPEDIT 18 0 "`` <= ``" "6#1%!GF $" }{TEXT -1 4 "b. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 45 "Here we require the one-sided derivatives of " } {XPPEDIT 18 0 "x(t)" "6#-%\"xG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y(t)" "6#-%\"yG6#%\"tG" }{TEXT -1 86 " to exist at the endpoints of the interval. As we saw in Section 6.1, the derivative " } {XPPEDIT 18 0 "`z'`;" "6#%#z'G" }{TEXT -1 6 " is " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 " `C:`;" "6#%#C:G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`z'(t)` = `x'(t)`+i*` y'(t)`;" "6#/%&z'(t)G,&%&x'(t)G\"\"\"*&%\"iGF'%&y'(t)GF'F'" }{TEXT -1 7 " for " }{TEXT 360 1 " " }{TEXT -1 1 "a" }{XPPEDIT 18 0 "`` <= ``; " "6#1%!GF$" }{TEXT -1 1 "t" }{XPPEDIT 18 0 "`` <= ``" "6#1%!GF$" } {TEXT -1 6 "b. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The curve " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 16 " \+ is said to be " }{TEXT 361 6 "smooth" }{TEXT -1 5 " if " }{XPPEDIT 18 0 "`z'`;" "6#%#z'G" }{TEXT -1 51 " is continuous and nonzero on t he interval. If " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 27 " is a \+ smooth curve, then " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 46 " has a nonzero tangent vector at each point " }{XPPEDIT 18 0 "`z'(t)`" "6 #%&z'(t)G" }{TEXT -1 33 ", which is given by the vector " }{XPPEDIT 18 0 "`z'(t)`" "6#%&z'(t)G" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " The " }{TEXT 362 14 "opposite curve" }{TEXT -1 2 " " }{XPPEDIT 18 0 "-C;" "6#,$%\"CG! \"\"" }{TEXT -1 105 " traces out the same set of points in the plane \+ but in the reverse order, and it has the parametrization" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "`-C:`;" "6#%$-C:G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z[2](t) = x(- t)+i*y(-t);" "6#/-&%\"zG6#\"\"#6#%\"tG,&-%\"xG6#,$F*!\"\"\"\"\"*&%\"iG F1-%\"yG6#,$F*F0F1F1" }{TEXT -1 9 ", for " }{TEXT 363 1 " " } {XPPEDIT 18 0 "-b;" "6#,$%\"bG!\"\"" }{XPPEDIT 18 0 "`` <= ``;" "6#1%! GF$" }{TEXT -1 1 "t" }{XPPEDIT 18 0 "`` <= ``" "6#1%!GF$" }{XPPEDIT 18 0 "-a;" "6#,$%\"aG!\"\"" }{TEXT 364 3 ". " }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " A curv e " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 84 " that is constructed \+ by joining finitely many smooth curves end to end is called a " } {TEXT 365 7 "contour" }{TEXT -1 8 ". Let " }{XPPEDIT 18 0 "C[1],C[2] ,`...`,C[n];" "6&&%\"CG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG" }{TEXT 366 1 " " }{TEXT -1 9 " denote " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 58 " smooth curves such that the terminal point of the curve " } {XPPEDIT 18 0 "C[k];" "6#&%\"CG6#%\"kG" }{TEXT -1 39 " coincides with the initial point of " }{XPPEDIT 18 0 "C[k+1];" "6#&%\"CG6#,&%\"kG\" \"\"F(F(" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "`k = `;" "6#%%k~=~G" } {XPPEDIT 18 0 "1,2,`...`,n-1;" "6&\"\"\"\"\"#%$...G,&%\"nGF#F#!\"\"" } {TEXT -1 26 ". We express the contour " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 19 " by the equation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "C = C[1]+C[2]+`... `+C[n];" "6#/%\"CG,*&F$6#\"\"\"F(&F$6#\"\"#F(%$...GF(&F$6#%\"nGF(" } {TEXT -1 7 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 25 "A synonym for contour is " }{TEXT 367 4 "path" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 347 1 "\n" }{TEXT 256 23 "Example 6.5, Page 208." } {TEXT 331 56 " Let us find a parameterization of the polygonal path \+ " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 264 8 " from " }{XPPEDIT 18 0 "-1+i" "6#,&\"\"\"!\"\"%\"iGF$" }{TEXT 265 6 " to " }{XPPEDIT 18 0 " 3-i" "6#,&\"\"$\"\"\"%\"iG!\"\"" }{TEXT 266 62 " \nconsisting of the \+ three line segments:\n Segment 1 from " }{XPPEDIT 18 0 "-1+i" "6#, &\"\"\"!\"\"%\"iGF$" }{TEXT 267 6 " to " }{XPPEDIT 18 0 "-1" "6#,$\" \"\"!\"\"" }{TEXT 268 23 " ,\n Segment 2 from " }{XPPEDIT 18 0 "-1 " "6#,$\"\"\"!\"\"" }{TEXT 269 6 " to " }{XPPEDIT 18 0 "1+i" "6#,&\" \"\"F$%\"iGF$" }{TEXT 270 23 " ,\n Segment 3 from " }{XPPEDIT 18 0 "1+i" "6#,&\"\"\"F$%\"iGF$" }{TEXT 271 6 " to " }{XPPEDIT 18 0 "3- i" "6#,&\"\"$\"\"\"%\"iG!\"\"" }{TEXT 272 3 " .\n" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 290 "t:='t':x0:='x0':x1:='x1':y0:='y0':y1:='y1':z:='z': Z:='Z':Z1:='Z1':\nz0 := - 1 + I:\nz1 := - 1:\nx0 := Re(z0): y0 := Im(z 0): x1 := Re(z1): y1 := Im(z1):\nZ1 := t -> x0 + (x1-x0)*t + I*(y0 + ( y1-y0)*t):\n`Find the segment from -1+i to -1.`;\nz[0] = Z1(0),` a nd `,z[1] = Z1(1);\nZ[1](t) = Z1(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%EFind~the~segment~from~~-1+i~~to~~-1.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"zG6#\"\"!^$!\"\"\"\"\"%(~~and~~G/&F%6#F*F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"ZG6#\"\"\"6#%\"tG,&!\"\"F(*&^#F( F(,&F(F(F*F,F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 287 "t:='t' :x0:='x0':x1:='x1':y0:='y0':y1:='y1':z:='z':Z:='Z':Z2:='Z2':\nz0 := - \+ 1:\nz1 := 1 + I:\nx0 := Re(z0): y0 := Im(z0): x1 := Re(z1): y1 := Im(z 1):\nZ2 := t -> x0 + (x1-x0)*t + I*(y0 + (y1-y0)*t):\n`Find the segmen t from -1 to 1+i.`;\nz[0] = Z2(0),` and `,z[0] = Z2(1);\nZ[2](t) \+ = Z2(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DFind~the~segment~from~~- 1~~to~~1+i.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"zG6#\"\"!!\"\"%(~ ~and~~G/F$^$\"\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"ZG6#\"\" #6#%\"tG,(!\"\"\"\"\"*&F(F-F*F-F-*&^#F-F-F*F-F-" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 290 "t:='t':x0:='x0':x1:='x1':y0:='y0':y1:='y1':z: ='z':Z:='Z':Z3:='Z3':\nz0 := 1 + I:\nz1 := 3 - I:\nx0 := Re(z0): y0 := Im(z0): x1 := Re(z1): y1 := Im(z1):\nZ3 := t -> x0 + (x1-x0)*t + I*(y 0 + (y1-y0)*t):\n`Find the segment from 1+i to 3-i.`;\nz[0] = Z3(0) ,` and `,z[1] = Z3(1);\nz[3](t) = Z3(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%EFind~the~segment~from~~1+i~~to~~3-i.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/&%\"zG6#\"\"!^$\"\"\"F)%(~~and~~G/&F%6#F)^$\"\"$! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"zG6#\"\"$6#%\"tG,(\"\"\" F,*&\"\"#F,F*F,F,*&^#F,F,,&F,F,*&F.F,F*F,!\"\"F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 " We are now ready to define the integral of a complex function along a contour " }{XPPEDIT 18 0 "C" "6#%\"CG" } {TEXT -1 35 " in the plane with initial point " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 22 " and terminal point " }{XPPEDIT 18 0 "B;" "6 #%\"BG" }{TEXT -1 78 ". Our approach is to mimic what is done in calc ulus. We create a partition " }{XPPEDIT 18 0 "P = \{A = z[0], z[1], \+ z[2], `...`, z[n] = B\};" "6#/%\"PG<'/%\"AG&%\"zG6#\"\"!&F)6#\"\"\"&F) 6#\"\"#%$...G/&F)6#%\"nG%\"BG" }{TEXT -1 32 " of points that proceed \+ along " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 8 " from " } {XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "B" " 6#%\"BG" }{TEXT -1 28 " and form the differences " }{XPPEDIT 18 0 "D elta*z[k] = z[k]-z[k-1];" "6#/*&%&DeltaG\"\"\"&%\"zG6#%\"kGF&,&&F(6#F* F&&F(6#,&F*F&F&!\"\"F1" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "`k = `; " "6#%%k~=~G" }{XPPEDIT 18 0 "1,2,`...`,n-1;" "6&\"\"\"\"\"#%$...G,&% \"nGF#F#!\"\"" }{TEXT -1 42 ". Between each pair of partition points \+ " }{XPPEDIT 18 0 "z[k-1]" "6#&%\"zG6#,&%\"kG\"\"\"F(!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "z[k]" "6#&%\"zG6#%\"kG" }{TEXT -1 21 " we select a point " }{XPPEDIT 18 0 "c[k]" "6#&%\"cG6#%\"kG" }{TEXT -1 6 " on " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 23 ", where the func tion " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 13 " evaluated. " }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 42 "Definition 6.2: Complex integ ral of f(z)" }{TEXT 368 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "int(f(z),z = C .. \+ ` `)" "6#-%$intG6$-%\"fG6#%\"zG/F);%\"CG%\"~G" }{TEXT -1 5 " = " } {XPPEDIT 18 0 "Limit(` `,n = infinity);" "6#-%&LimitG6$%\"~G/%\"nG%)in finityG" }{XPPEDIT 18 0 "Sum(f(c[k])*Delta*z[k],k = 1 .. n)" "6#-%$Sum G6$*(-%\"fG6#&%\"cG6#%\"kG\"\"\"%&DeltaGF.&%\"zG6#F-F./F-;F.%\"nG" } {TEXT -1 8 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 62 "provided the limit exists in the sense previously discu ssed. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 1 "\n" }{TEXT 256 22 "Example 6.6, Page 209." }{TEXT 332 77 " Use a Riemann sum to construct an approximation for the cont our \nintegral " }{XPPEDIT 18 0 "int(exp(z), z=C..`.`)" "6#-%$intG6$- %$expG6#%\"zG/F);%\"CG%\".G" }{TEXT 273 10 " where " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 274 43 " is a the line segment joining the poi nt " }{XPPEDIT 18 0 "a = 0" "6#/%\"aG\"\"!" }{TEXT 275 6 " to " } {XPPEDIT 18 0 "b = 2 + i*pi/4" "6#/%\"bG,&\"\"#\"\"\"*(%\"iGF'%#piGF' \"\"%!\"\"F'" }{TEXT 261 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 276 "a:='a':b:='b':C:='C':f:='f':F:='F':n:='n':z:='z':Z:='Z':\nF := z \+ -> exp(z):\n`f(z) ` = F(z);\na := 0:\nb := 2 + I*Pi/4:\nn := 8:\n`a ` \+ = a, ` b ` = b, ` n ` = n;\ndz := (b-a)/8:\nZ := k -> a + k*dz: \nc := k -> simplify((Z(k-1)+Z(k))/2):\nSum(f(c[k])*Delta*z[k]) = Sum( F(c(k))*dz);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&f(z)~G-%$expG6#%\"z G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%#a~G\"\"!/%'~~~~b~G,&\"\"#\"\" \"*&^##F*\"\"%F*%#PiGF*F*/%'~~~~n~G\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6#*(-%\"fG6#&%\"cG6#\"\"'\"\"\"%&DeltaGF/&%\"zGF-F/-F%6# *&-%$expG6#,&#\"#6\"\")F/*&^##F;\"#kF/%#PiGF/F/F/,&#F/\"\"%F/*&^##F/\" #KF/FAF/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "RS := sum( F(c(k))*dz, k=1..8):\n`The Riemann sum is:`;\nSum(f(c[k])*Delta*z[k],k =1..8) = RS;\n`The Riemann sum is:`;\nSum(f(c[k])*Delta*z[k],k=1..8) = evalf(RS);" }}{PARA 8 "" 1 "" {TEXT -1 96 "Error, (in sum) summation \+ variable previously assigned, second argument evaluates to 6 = 1 .. 8 \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4The~Riemann~sum~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*(-%\"fG6#&%\"cG6#\"\"'\"\"\"%&Del taGF/&%\"zGF-F//F.;F/\"\")%#RSG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4T he~Riemann~sum~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*(-% \"fG6#&%\"cG6#\"\"'\"\"\"%&DeltaGF/&%\"zGF-F//F.;F/\"\")%#RSG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "Theorem 6.1" }{TEXT 373 12 " \+ Suppose " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 369 83 " i s a continuous complex-valued function defined on a set containing the contour " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 375 3 ". " }}{PARA 0 "" 0 "" {TEXT 382 5 "Let " }{XPPEDIT 18 0 "z(t)" "6#-%\"zG6#%\"tG" } {TEXT 370 30 " be any parameterization of " }{XPPEDIT 18 0 "C" "6#% \"CG" }{TEXT 376 7 " for " }{TEXT -1 1 "a" }{XPPEDIT 18 0 "`` <= ``; " "6#1%!GF$" }{TEXT -1 1 "t" }{XPPEDIT 18 0 "`` <= ``" "6#1%!GF$" } {TEXT -1 1 "b" }{TEXT 371 10 " . Then " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 374 6 " " }{XPPEDIT 18 0 "int(f(z),z = C .. ` `) = int(f(z(t))*`z ' (t)`,t = a .. b);" "6#/-%$intG6$-%\"fG 6#%\"zG/F*;%\"CG%\"~G-F%6$*&-F(6#-F*6#%\"tG\"\"\"%(z~'~(t)GF7/F6;%\"aG %\"bG" }{TEXT 372 3 " . " }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 22 "Example 6.7, Pag e 212." }{TEXT 333 35 " Evaluate the contour integral " }{XPPEDIT 18 0 "int(exp(z), z=C..`.`)" "6#-%$intG6$-%$expG6#%\"zG/F);%\"CG%\".G " }{TEXT 276 9 " \nwhere " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT 277 43 " is a the line segment joining the point " }{XPPEDIT 18 0 "a = 0 " "6#/%\"aG\"\"!" }{TEXT 278 6 " to " }{XPPEDIT 18 0 "b = 2 + i*pi/4 " "6#/%\"bG,&\"\"#\"\"\"*(%\"iGF'%#piGF'\"\"%!\"\"F'" }{TEXT 262 3 " . \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 605 "f:='f': F:='F': g:='g': t:=' t': T:='T': z:='z': z1:='z1':\nf := z -> exp(z):\nF := t -> exp((2 + I *Pi/4)*t):\n`f(z) ` = f(z);\na := 0:\nb := 2 + I*Pi/4:\n`a ` = a, ` \+ b ` = b;\nz := t -> a + (b - a)*t:\n`C: z(t) ` = z(t);\n`f(z(t)) = \+ `,F(t) = evalc(F(t));\nz1 := t -> subs(T=t,diff(z(T), T)):\nInt(f(z),z =C..``) = Int(F(t)*diff(z(t),t),t=0..1);\n`The anti-derivative is:`;\n g := t -> subs(T=t,int(F(T)*z1(T), T)):\n`g(t) ` = g(t);\ng1 := g(1): \ng0 := g(0):\n`g(1) ` = g1, ` g(0) ` = g0;\n`g(1) - g(0) ` = g1 - g0;\nInt(f(z),z=C..``) = g1 - g0;\nInt(f(z),z=C..``) = evalc(g1 - g0) ;\nInt(f(z),z=C..``) = evalf(g1 - g0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&f(z)~G-%$expG6#%\"zG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%#a~ G\"\"!/%'~~~~b~G,&\"\"#\"\"\"*&^##F*\"\"%F*%#PiGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*C:~~z(t)~G*&,&\"\"#\"\"\"*&^##F(\"\"%F(%#PiGF(F(F (%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,f(z(t))~=~~G/-%$expG6#*& ,&\"\"#\"\"\"*&^##F+\"\"%F+%#PiGF+F+F+%\"tGF+,&*&-F&6#,$F1F*F+-%$cosG6 #,$*&F1F+F0F+F.F+F+*(^#F+F+F4F+-%$sinGF9F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$expG6#%\"zG/F*;%\"CG%!G-F%6$*&-F(6#*&,&\" \"#\"\"\"*&^##F7\"\"%F7%#PiGF7F7F7%\"tGF7F7F5F7/F=;\"\"!F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8The~anti-derivative~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'g(t)~~G-%$expG6#*&,&\"\"#\"\"\"*&^##F+\"\"%F+%#Pi GF+F+F+%\"tGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%'g(1)~~G-%$expG6#, &\"\"#\"\"\"*&^##F*\"\"%F*%#PiGF*F*/%)~~g(0)~~GF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%.g(1)~-~g(0)~~G,&-%$expG6#,&\"\"#\"\"\"*&^##F+\"\"%F+ %#PiGF+F+F+F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$exp G6#%\"zG/F*;%\"CG%!G,&-F(6#,&\"\"#\"\"\"*&^##F4\"\"%F4%#PiGF4F4F4F4!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$expG6#%\"zG/F*;%\"C G%!G,(*&-F(6#\"\"#\"\"\"-%%sqrtG6#F3F4#F4F3F4!\"\"*(^#F8F4F1F4F5F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$expG6#%\"zG/F*;%\"CG%!G ^$$\"+u;&[A%!\"*$\"+v;&[A&F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 79 "An alternate method is to use \+ the integral of the real and imaginary parts of " }{XPPEDIT 18 0 "f(z )" "6#-%\"fG6#%\"zG" }{TEXT 279 22 " .\nWARNING. Because " } {XPPEDIT 18 0 "`z'(t)`" "6#%&z'(t)G" }{TEXT 280 117 " is a constant f or this example, we took it outside the integral!\nAlso, this computat ion is very memory intensive !\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 722 "f:='f': F:='F': g:='g': t:='t': T:='T': u:='u': v:='v': z:='z':\n f := z -> exp(z):\nu := t -> exp(2*t)*cos(1/4*t*Pi):\nv := t -> exp(2* t)*sin(1/4*t*Pi):\n`f(z) ` = f(z);\na := 0:\nb := 2 + I*Pi/4:\n`a ` = \+ a, ` b ` = b;\nz := t -> a + (b - a)*t:\n`C: z(t) ` = z(t);\n`f(z( t)) = `,f(z(t)) = u(t) + I*v(t);\nz1 := diff(z(t), t):\nInt(f(z),z=C. .``) = Int((u(t)+I*v(t))*diff(z(t),t),t=0..1);\n`The anti-derivative i s:`;\ng := t -> simplify(subs(T=t,int(u(T), T)))*z1 +\n I*sim plify(subs(T=t,int(v(T), T)))*z1:\n`g(t) ` = g(t); ` `;\ng1 := simpli fy(g(1)):\ng0 := simplify(g(0)):\n`g(1) ` = g1,` and `,\n`g(0) ` = g0;\n`g(1) - g(0) ` = expand(g1 - g0); ` `;\nInt(f(z),z=C..``) = exp and(g1 - g0);\nInt(f(z),z=C..``) = evalf(g1 - g0);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%&f(z)~G-%$expG6#%\"zG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%#a~G\"\"!/%'~~~~b~G,&\"\"#\"\"\"*&^##F*\"\"%F*%#PiGF*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%*C:~~z(t)~G*&,&\"\"#\"\"\"*&^##F(\" \"%F(%#PiGF(F(F(%\"tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,f(z(t))~= ~~G/-%$expG6#*&,&\"\"#\"\"\"*&^##F+\"\"%F+%#PiGF+F+F+%\"tGF+,&*&-F&6#, $F1F*F+-%$cosG6#,$*&F1F+F0F+F.F+F+*(^#F+F+F4F+-%$sinGF9F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$expG6#%\"zG/F*;%\"CG%!G-F%6$*&, &*&-F(6#,$%\"tG\"\"#\"\"\"-%$cosG6#,$*&F7F9%#PiGF9#F9\"\"%F9F9*(^#F9F9 F4F9-%$sinGF " 0 "" {MPLTEXT 1 0 496 "dz :='dz':f: ='f':F:='F':g:='g':t:='t':T:='T':z:='z':Z:='Z':z1:='z1':\nf := z -> 1/ (z - 2):\n`f(z) ` = f(z);\nz := t -> 2 + 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))*diff(z(t),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) - g(0) ` = expand(g1 - g0);\nInt(f(z),z=C..``) = expand( g1 - g0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&f(z)~G*&\"\"\"F&,&%\"z GF&\"\"#!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*C:~~z(t)~G,&\"\" #\"\"\"-%$expG6#*&^#F'F'%\"tGF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %)f(z(t))~G*&\"\"\"F&-%$expG6#*&^#F&F&%\"tGF&!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$/%0dz~=~z~'(t)~dt~G*&^#\"\"\"F'-%$expG6#*&F&F'%\"tGF' F'%#dtG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&%\"zG F(\"\"#!\"\"F,/F*;%\"CG%!G-F%6$^#F(/%\"tG;\"\"!,$%#piGF+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%8The~anti-derivative~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'g(t)~~G*&^#\"\"\"F'%\"tGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1g(2*Pi)~-~g(0)~~G*&^#\"\"#\"\"\"%#PiGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&%\"zGF(\"\"#!\"\"F,/F*;% \"CG%!G*&^#F+F(%#PiGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 22 "E xample 6.9, Page 214." }{TEXT 335 37 " Evaluate the contour integrals of " }{XPPEDIT 18 0 "f(z) = z" "6#/-%\"fG6#%\"zGF'" }{TEXT 288 50 " \+ over a line segment and portion of a parabola.\n" }{TEXT 256 3 "(a) " }{TEXT 344 47 " Use the straight line connecting the points " } {XPPEDIT 18 0 "-1 - i" "6#,&\"\"\"!\"\"%\"iGF%" }{TEXT 290 7 " and \+ " }{XPPEDIT 18 0 "3 + i" "6#,&\"\"$\"\"\"%\"iGF%" }{TEXT 289 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 538 "dz :='dz':f:='f':F:='F':g:='g':t:='t':T:='T':z:='z':Z:='Z':z1:='z1':\nf \+ := z -> z:\n`f(z) ` = f(z);\nz := t -> 2*t + 1 + I*t:\n`C: z(t) ` = z (t);\n`f(z(t)) ` = f(z(t));\nz1 := t -> subs(T=t,diff(z(T), T)):\n`dz \+ = z '(t) dt ` = z1(t), `dt`;\nInt(f(z),z=C..``) = Int(f(z(t))*z1(t),t= -1..1);\nInt(f(z),z=C..``) = Int(evalc(f(z(t))*z1(t)),t=-1..1);\n`The \+ anti-derivative is:`;\ng := t -> simplify(subs(T=t,int(f(z(T))*z1(T), \+ T))):\n`g(t) ` = evalc(g(t));\ng1 := g(1):\ng0 := g(-1):\n`g(1) - g(- 1) ` = expand(g1 - g0);\nInt(f(z),z=C..``) = expand(g1 - g0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%&f(z)~G%\"zG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*C:~~z(t)~G,(%\"tG\"\"#\"\"\"F(*&^#F(F(F&F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%)f(z(t))~G,(%\"tG\"\"#\"\"\"F(*&^#F( F(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0dz~=~z~'(t)~dt~G^$\"\"# \"\"\"%#dtG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$%\"zG/F';%\"C G%!G-F%6$*&^$\"\"#\"\"\"F1,(%\"tGF0F1F1*&^#F1F1F3F1F1F1/F3;!\"\"F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$%\"zG/F';%\"CG%!G-F%6$,(%\"t G\"\"$\"\"#\"\"\"*&^#F2F2,&F/\"\"%F2F2F2F2/F/;!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8The~anti-derivative~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'g(t)~~G,(*$)%\"tG\"\"#\"\"\"#\"\"$F)*&F)F*F(F*F**&^# F*F*,&F&F)F(F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%/g(1)~-~g(-1)~ ~G^$\"\"%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$%\"zG/F';% \"CG%!G^$\"\"%\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 256 3 "(b)" }{TEXT 345 55 " Use the port ion of a parabola connecting the points " }{XPPEDIT 18 0 "-1 - i" "6# ,&\"\"\"!\"\"%\"iGF%" }{TEXT 292 7 " and " }{XPPEDIT 18 0 "3 + i" "6 #,&\"\"$\"\"\"%\"iGF%" }{TEXT 291 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 540 "dz :='dz':f:='f':F:='F':g:='g ':t:='t':T:='T':z:='z':Z:='Z':z1:='z1':\nf := z -> z:\n`f(z) ` = f(z); \nz := t -> t^2 + 2*t + 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), `d t`;\nInt(f(z),z=C..``) = Int(f(z(t))*z1(t),t=-1..1);\nInt(f(z),z=C..`` ) = Int(evalc(f(z(t))*z1(t)),t=-1..1);\n`The anti-derivative is:`;\ng \+ := t -> simplify(subs(T=t,int(f(z(T))*z1(T), T))):\n`g(t) ` = evalc(g (t));\ng1 := g(1):\ng0 := g(-1):\n`g(1) - g(-1) ` = expand(g1 - g0); \nInt(f(z),z=C..``) = expand(g1 - g0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&f(z)~G%\"zG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*C:~~z(t)~G, (*$)%\"tG\"\"#\"\"\"F**&F)F*F(F*F**&^#F*F*F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)f(z(t))~G,(*$)%\"tG\"\"#\"\"\"F**&F)F*F(F*F**&^#F*F* F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0dz~=~z~'(t)~dt~G,&%\"tG\" \"#^$F'\"\"\"F)%#dtG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$%\"z G/F';%\"CG%!G-F%6$*&,(*$)%\"tG\"\"#\"\"\"F4*&F3F4F2F4F4*&^#F4F4F2F4F4F 4,&F2F3^$F3F4F4F4/F2;!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$In tG6$%\"zG/F';%\"CG%!G-F%6$,**$)%\"tG\"\"$\"\"\"\"\"#*&\"\"'F3)F1F4F3F3 *&F2F3F1F3F3*&^#F3F3,&*$F7F3F2*&\"\"%F3F1F3F3F3F3/F1;!\"\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8The~anti-derivative~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'g(t)~~G,**$)%\"tG\"\"%\"\"\"#F*\"\"#*&F,F*)F(\"\" $F*F**&#F/F,F*)F(F,F*F**&^#F*F*,&*$F.F*F**&F,F*F2F*F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%/g(1)~-~g(-1)~~G^$\"\"%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$%\"zG/F';%\"CG%!G^$\"\"%\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 50 "Which is the same as the value for the other path." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 328 1 "\n" }{TEXT 256 12 "Example 6.*," }{TEXT 336 42 " Similar to the above example, but using " }{XPPEDIT 18 0 "co njugate(z)" "6#-%*conjugateG6#%\"zG" }{TEXT 294 38 " .\nEvaluate the c ontour integrals of " }{XPPEDIT 18 0 "f(z) = conjugate(z)" "6#/-%\"fG 6#%\"zG-%*conjugateG6#F'" }{TEXT 293 49 " over a line segment and por tion of a parabola.\n" }{TEXT 256 3 "(a)" }{TEXT 337 47 " Use the str aight line connecting the points " }{XPPEDIT 18 0 "-1 - i" "6#,&\"\" \"!\"\"%\"iGF%" }{TEXT 295 7 " and " }{XPPEDIT 18 0 "3 + i" "6#,&\" \"$\"\"\"%\"iGF%" }{TEXT 296 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 563 "dz :='dz':f:='f':F:='F':g:='g':t:='t':T:='T':z:='z':Z:='Z':z1:= 'z1':\nf := z -> conjugate(z):\n`f(z) ` = f(z);\nz := t -> 2*t + 1 + I *t:\n`C: z(t) ` = z(t);\n`f(z(t)) ` = evalc(f(z(t)));\nz1 := t -> sub s(T=t,diff(z(T), T)):\n`dz = z '(t) dt ` = z1(t), `dt`;\nInt(f(z),z=C. .``) = Int(f(z(t))*z1(t),t=-1..1);\nInt(f(z),z=C..``) = Int(evalc(f(z( t))*z1(t)),t=-1..1);\n`The anti-derivative is:`;\ng := t -> simplify(s ubs(T=t,int(evalc(f(z(T))*z1(T)), T))):\n`g(t) ` = evalc(g(t));\ng1 : = g(1):\ng0 := g(-1):\n`g(1) - g(-1) ` = expand(g1 - g0);\nInt(f(z),z =C..``) = expand(g1 - g0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&f(z)~ G-%*conjugateG6#%\"zG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*C:~~z(t)~G ,(%\"tG\"\"#\"\"\"F(*&^#F(F(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%)f(z(t))~G,(\"\"\"F&*&\"\"#F&%\"tGF&F&*&^#!\"\"F&F)F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0dz~=~z~'(t)~dt~G^$\"\"#\"\"\"%#dtG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%*conjugateG6#%\"zG/F*;%\"CG%!G-F %6$*&^$\"\"#\"\"\"F4,&F4F4-F(6#,&%\"tGF3*&^#F4F4F9F4F4F4F4/F9;!\"\"F4 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%*conjugateG6#%\"zG/F*; %\"CG%!G-F%6$,&^$\"\"#\"\"\"F4*&\"\"&F4%\"tGF4F4/F7;!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8The~anti-derivative~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'g(t)~~G,(%\"tG\"\"#*&^#\"\"\"F*F&F*F**&#\"\"&F'F* )F&F'F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%/g(1)~-~g(-1)~~G^$\"\"% \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%*conjugateG6#%\"z G/F*;%\"CG%!G^$\"\"%\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 256 3 "(b)" }{TEXT 338 55 " Use the port ion of a parabola connecting the points " }{XPPEDIT 18 0 "-1 - i" "6# ,&\"\"\"!\"\"%\"iGF%" }{TEXT 298 7 " and " }{XPPEDIT 18 0 "3 + i" "6 #,&\"\"$\"\"\"%\"iGF%" }{TEXT 297 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 573 "dz :='dz':f:='f':F:='F':g:='g':t:='t':T:='T':z:='z': Z:='Z':z1:='z1':\nf := z -> conjugate(z):\n`f(z) ` = f(z);\nz := t -> \+ t^2 + 2*t + I*t:\n`C: z(t) ` = z(t);\n`f(z(t)) ` = evalc(f(z(t)));\nz 1 := t -> subs(T=t,diff(z(T), T)):\n`dz = z '(t) dt ` = z1(t), `dt`;\n Int(f(z),z=C..``) = Int(f(z(t))*z1(t),t=-1..1);\nInt(f(z),z=C..``) = I nt(expand(evalc(f(z(t))*z1(t))),t=-1..1);\n`The anti-derivative is:`; \ng := t -> simplify(subs(T=t,int(evalc(f(z(T))*z1(T)), T))):\n`g(t) \+ ` = evalc(g(t));\ng1 := g(1):\ng0 := g(-1):\n`g(1) - g(-1) ` = expand (g1 - g0);\nInt(f(z),z=C..``) = expand(g1 - g0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&f(z)~G-%*conjugateG6#%\"zG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*C:~~z(t)~G,(*$)%\"tG\"\"#\"\"\"F**&F)F*F(F*F**&^#F*F *F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)f(z(t))~G,(*$)%\"tG\"\"# \"\"\"F**&F)F*F(F*F**&^#!\"\"F*F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0dz~=~z~'(t)~dt~G,&%\"tG\"\"#^$F'\"\"\"F)%#dtG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$-%*conjugateG6#%\"zG/F*;%\"CG%!G-F%6$*&-F(6 #,(*$)%\"tG\"\"#\"\"\"F9*&F8F9F7F9F9*&^#F9F9F7F9F9F9,&F7F8^$F8F9F9F9/F 7;!\"\"F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%*conjugateG6# %\"zG/F*;%\"CG%!G-F%6$,**$)%\"tG\"\"$\"\"\"\"\"#*&\"\"'F6)F4F7F6F6*&\" \"&F6F4F6F6*&^#!\"\"F6F:F6F6/F4;F?F6" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%8The~anti-derivative~is:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'g(t )~~G,**$)%\"tG\"\"%\"\"\"#F*\"\"#*&F,F*)F(\"\"$F*F**&#\"\"&F,F*)F(F,F* F**&^##!\"\"F/F*F.F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%/g(1)~-~g( -1)~~G^$\"\"%#!\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$- %*conjugateG6#%\"zG/F*;%\"CG%!G^$\"\"%#!\"#\"\"$" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 53 "Which is \+ different from the value for the other path." }}}{EXCHG {PARA 0 "" 0 " " {TEXT 327 1 "\n" }{TEXT 256 23 "Example 6.10, Page 215." }{TEXT 339 1 "\n" }{TEXT 256 3 "(a)" }{TEXT 340 13 " Show that " }{XPPEDIT 18 0 "int(conjugate(z), z=C[1]..`.`) = - i*pi" "6#/-%$intG6$-%*conjugate G6#%\"zG/F*;&%\"CG6#\"\"\"%\".G,$*&%\"iGF0%#piGF0!\"\"" }{TEXT 299 12 " , where " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" }{TEXT 300 33 " is the semicircular path from " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\" \"" }{TEXT 301 6 " to " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 302 27 " in the upper half plane.\n" }{TEXT 256 3 "(b)" }{TEXT 341 13 " Show that " }{XPPEDIT 18 0 "int(conjugate(z), z=C[2]..`.`) = - 4*i" "6# /-%$intG6$-%*conjugateG6#%\"zG/F*;&%\"CG6#\"\"#%\".G,$*&\"\"%\"\"\"%\" iGF5!\"\"" }{TEXT 303 12 " , where " }{XPPEDIT 18 0 "C[2]" "6#&%\"C G6#\"\"#" }{TEXT 304 30 " is the polygonal path from " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT 305 6 " to " }{XPPEDIT 18 0 "-1+i" " 6#,&\"\"\"!\"\"%\"iGF$" }{TEXT 306 6 " to " }{XPPEDIT 18 0 "1 + i" " 6#,&\"\"\"F$%\"iGF$" }{TEXT 307 6 " to " }{XPPEDIT 18 0 "1" "6#\"\" \"" }{TEXT 308 2 " ." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 3 "(a)" }{TEXT 342 34 " Use the semicircular path from " } {XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT 309 6 " to " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 310 27 " in the upper half plane.\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 577 "dz :='dz':f:='f':F:='F':g:='g':t:= 't':T:='T':z:='z':Z:='Z':z1:='z1':\nf := z -> conjugate(z):\n`f(z) ` = f(z);\nz := t -> - cos(t) + I*sin(t):\n`C: z(t) ` = z(t);\n`f(z(t)) \+ ` = evalc(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..pi); \nInt(f(z),z=C..``) = Int(expand(evalc(f(z(t))*z1(t))),t=0..pi);\n`The anti-derivative is:`;\ng := t -> simplify(subs(T=t,int(evalc(f(z(T))* z1(T)), T))):\n`g(t) ` = evalc(g(t));\ng1 := g(Pi):\ng0 := g(0):\n`g( Pi) - g(0) ` = expand(g1 - g0);\nInt(f(z),z=C..``) = expand(g1 - g0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&f(z)~G-%*conjugateG6#%\"zG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%*C:~~z(t)~G,&-%$cosG6#%\"tG!\"\"*&^# \"\"\"F--%$sinGF(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)f(z(t))~G, &-%$cosG6#%\"tG!\"\"*&^#F*\"\"\"-%$sinGF(F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0dz~=~z~'(t)~dt~G,&-%$sinG6#%\"tG\"\"\"*&^#F*F*-%$cos GF(F*F*%#dtG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%*conjugate G6#%\"zG/F*;%\"CG%!G-F%6$*&-F(6#,&-%$cosG6#%\"tG!\"\"*&^#\"\"\"F<-%$si nGF7F " 0 "" {MPLTEXT 1 0 587 "t:='t':x0:='x0':x1:='x1':y0:=' y0':y1:='y1':z:='z':\nz0:='z0':z1:='z1':Z1:='Z1':Z2:='Z2':Z3:='Z3':\nz 0 := - 1:\nz1 := - 1 + I:\nx0 := Re(z0): y0 := Im(z0): x1 := Re(z1): y 1 := Im(z1):\nZ1 := x0 + (x1-x0)*t + I*(y0 + (y1-y0)*t):\nz0 := - 1 + \+ I:\nz1 := 1 + I:\nx0 := Re(z0): y0 := Im(z0): x1 := Re(z1): y1 := Im(z 1):\nZ2 := x0 + (x1-x0)*t + I*(y0 + (y1-y0)*t):\nz0 := 1 + I:\nz1 := 1 :\nx0 := Re(z0): y0 := Im(z0): x1 := Re(z1): y1 := Im(z1):\nZ3 := x0 + (x1-x0)*t + I*(y0 + (y1-y0)*t):\nC[1], ` is given by `,z[1](t) = Z1; \nC[2], ` is given by `, z[2](t) = Z2;\nC[3], ` is given by `, z[3]( t) = Z3; ` `;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%&%\"CG6#\"\"\"%/~is~g iven~by~~G/-&%\"zGF%6#%\"tG,&!\"\"F&*&^#F&F&F-F&F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%&%\"CG6#\"\"#%/~is~given~by~~G/-&%\"zGF%6#%\"tG,&F-F& ^$!\"\"\"\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%&%\"CG6#\"\"$%/~is~ given~by~~G/-&%\"zGF%6#%\"tG,&\"\"\"F/*&^#F/F/,&F/F/F-!\"\"F/F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 492 "f:='f': F:='F': g:='g': t:='t': z:='z':\nf := z -> c onjugate(z):\nF(z[1](t)) = evalc(f(Z1)),` and `,\ndiff(z[1](t),t) = \+ evalc(diff(Z1,t));\nw1 := evalc(f(Z1)*diff(Z1,t)):\nF(z[1](t))*diff(z[ 1](t),t) = w1; ` `;\nF(z[2](t)) = evalc(f(Z2)),` and `,\ndiff(z[2](t ),t) = evalc(diff(Z2,t));\nw2 := evalc(f(Z2)*diff(Z2,t)):\nF(z[2](t))* diff(z[2](t),t) = w2; ` `;\nF(z[3](t)) = evalc(f(Z3)),` and `,\ndiff (z[3](t),t) = evalc(diff(Z3,t));\nw3 := evalc(f(Z3)*diff(Z3,t)):\nF(z[ 3](t))*diff(z[3](t),t) = w3; ` `;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/ -%\"FG6#-&%\"zG6#\"\"\"6#%\"tG,&!\"\"F+*&^#F/F+F-F+F+%(~~and~~G/-%%dif fG6$F'F-^#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"FG6#-&%\"zG6#\" \"\"6#%\"tGF,-%%diffG6$F(F.F,,&F.F,^#!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/-%\"FG6#-&%\" zG6#\"\"#6#%\"tG,&^$!\"\"F0\"\"\"*&F+F1F-F1F1%(~~and~~G/-%%diffG6$F'F- F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"FG6#-&%\"zG6#\"\"#6#%\"tG \"\"\"-%%diffG6$F(F.F/,&^$!\"#F5F/*&\"\"%F/F.F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/-%\"FG6#-&%\" zG6#\"\"$6#%\"tG,&\"\"\"F/*&^#F/F/,&!\"\"F/F-F/F/F/%(~~and~~G/-%%diffG 6$F'F-^#F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"FG6#-&%\"zG6#\"\" $6#%\"tG\"\"\"-%%diffG6$F(F.F/,&^$!\"\"F5F/F.F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 242 "s1 := int(w1,t):\nInt(F(z[1](t))*diff(z[1](t),t),t) = s1;\ns2 := int(w2, t):\nInt(F(z[2](t))*diff(z[2](t),t),t) = s2;\ns3 := int(w3,t):\nInt(F( z[3](t))*diff(z[3](t),t),t) = s3;\ns := int(w1,t) + int(w2,t) +int(w3, t):\nInt(F(z(t))*diff(z(t),t),t) = s;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%\"FG6#-&%\"zG6#\"\"\"6#%\"tGF/-%%diffG6$F+F1F/F1,&*$) F1\"\"#F/#F/F8*&^#!\"\"F/F1F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% $IntG6$*&-%\"FG6#-&%\"zG6#\"\"#6#%\"tG\"\"\"-%%diffG6$F+F1F2F1,&*&^$! \"#F9F2F1F2F2*&F/F2)F1F/F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$In tG6$*&-%\"FG6#-&%\"zG6#\"\"$6#%\"tG\"\"\"-%%diffG6$F+F1F2F1,&*&^$!\"\" F9F2F1F2F2*&#F2\"\"#F2)F1F " 0 "" {MPLTEXT 1 0 479 "z1:='z1': z2:='z2': z3:='z3': \nS1 := Int(w1,t=0..1): s1 := int(w1,t= 0..1):\nInt(F(z[1](t))*diff(z[1](t),t),t=0..1),` = `, S1 = s1;\nS2 := \+ Int(w2,t=0..1): s2 := int(w2,t=0..1):\nInt(F(z[2](t))*diff(z[2](t),t), t=0..1),` = `, S2 = s2;\nS3 := Int(w3,t=0..1): s3 := int(w3,t=0..1):\n Int(F(z[3](t))*diff(z[3](t),t),t=0..1),` = `, S3 = s3;\nS := Int(w1+w2 +w3,t=0..1):\ns := int(w1,t=0..1) + int(w2,t=0..1) + int(w3,t=0..1):\n Int(F(z(t))*diff(z(t),t),t=0..1),` = `, S = s;\nInt(f(z),z=C..``) = s; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%-%$IntG6$*&-%\"FG6#-&%\"zG6#\"\"\" 6#%\"tGF.-%%diffG6$F*F0F./F0;\"\"!F.%$~=~G/-F$6$,&F0F.^#!\"\"F.F4^$#F. \"\"#F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%-%$IntG6$*&-%\"FG6#-&%\"zG6 #\"\"#6#%\"tG\"\"\"-%%diffG6$F*F0F1/F0;\"\"!F1%$~=~G/-F$6$,&^$!\"#F>F1 *&\"\"%F1F0F1F1F5^#F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%-%$IntG6$*&-% \"FG6#-&%\"zG6#\"\"$6#%\"tG\"\"\"-%%diffG6$F*F0F1/F0;\"\"!F1%$~=~G/-F$ 6$,&^$!\"\"F>F1F0F1F5^$#F>\"\"#F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%- %$IntG6$*&-%\"FG6#-%\"zG6#%\"tG\"\"\"-%%diffG6$F*F-F./F-;\"\"!F.%$~=~G /-F$6$,&F-\"\"'^$!\"$!\"%F.F2^#F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$IntG6$-%*conjugateG6#%\"zG/F*;%\"CG%!G^#!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 377 53 "Which is different from the value for the other pat h." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 " We now \+ give a few important inequalities relating to complex integrals. " }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 43 "Theorem 6.2 (Integral triangl e inequality)" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "f(t) = u(t)+i*v(t)" "6#/-%\"fG6#%\"tG,&-%\"uG6#F'\"\"\"*&%\"iGF,-%\"vG6#F'F,F," }{TEXT -1 49 " is a continuous function of the real parameter " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 8 ", then " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 379 7 " " }{XPPEDIT 18 0 "abs(int(f(t),t = a .. b));" "6#-%$absG6#-%$intG6$-%\"fG6#%\"tG/F,;%\"aG%\"bG" } {XPPEDIT 18 0 "`` <= ``;" "6#1%!GF$" }{XPPEDIT 18 0 "int(abs(f(t)),t = a .. b);" "6#-%$intG6$-%$absG6#-%\"fG6#%\"tG/F,;%\"aG%\"bG" }{TEXT 378 3 " . " }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 256 28 "Theorem 6.3 (ML inequality)" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "f(z) = u(x,y)+i*v(x,y);" "6#/-%\"fG6#%\"zG ,&-%\"uG6$%\"xG%\"yG\"\"\"*&%\"iGF.-%\"vG6$F,F-F.F." }{TEXT -1 42 " i s a continuous function on the contour " }{XPPEDIT 18 0 "C;" "6#%\"CG " }{TEXT -1 9 ", then " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 380 6 " " }{XPPEDIT 18 0 "abs(int(f(z),z = C .. ` `) );" "6#-%$absG6#-%$intG6$-%\"fG6#%\"zG/F,;%\"CG%\"~G" }{XPPEDIT 18 0 " `` <= ``;" "6#1%!GF$" }{XPPEDIT 18 0 "M*L;" "6#*&%\"MG\"\"\"%\"LGF%" } {TEXT 381 8 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "L;" "6#%\"LG" }{TEXT -1 32 " \+ is the length of the contour " }{XPPEDIT 18 0 "C;" "6#%\"CG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "M;" "6#%\"MG" }{TEXT -1 37 " is an upp er bound for the modulus " }{XPPEDIT 18 0 "abs(f(z));" "6#-%$absG6#-% \"fG6#%\"zG" }{TEXT -1 6 " on " }{XPPEDIT 18 0 "C;" "6#%\"CG" } {TEXT -1 13 ", that is " }{XPPEDIT 18 0 "abs(f(z)) <= M;" "6#1-%$ab sG6#-%\"fG6#%\"zG%\"MG" }{TEXT -1 13 " for all " }{XPPEDIT 18 0 "z *epsilon*C;" "6#*(%\"zG\"\"\"%(epsilonGF%%\"CGF%" }{TEXT -1 3 ". " }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 283 19 "End of Section 6.2." }}}}{MARK "0 0 0" 26 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }