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"" -1 370 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 371 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 372 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 373 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 374 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 375 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 376 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 377 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 378 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 379 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 380 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 381 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 382 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 383 "" 1 10 0 0 0 0 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 Fo nt 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 0" -1 257 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 }{PSTYLE "R3 Font 2" -1 258 1 {CSTYLE "" -1 -1 "Gene va" 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 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple Worksheets, 2001" }{TEXT 346 144 "\n(c) John H. Mathews Rus sell W. Howell\nmathews@fullerton.edu howell@westmont.edu\n\nCompl imentary software to accompany the textbook:" }}{PARA 257 "" 0 "" {TEXT 257 82 "COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9" }{TEXT 345 197 "\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 344 1 "\n" }{TEXT 256 36 "CHAPTER 7 TAYLOR and LAURENT SERIES" } {TEXT 313 2 "\n\n" }{TEXT 256 43 "Section 7.4 Singularities, Zeros an d Poles" }{TEXT 312 1 "\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " The point " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 13 " \+ is called a " }{TEXT 354 14 "singular point" }{TEXT -1 5 ", or " } {TEXT 355 11 "singularity" }{TEXT -1 27 ", of the complex function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 6 " if " }{XPPEDIT 18 0 "f; " "6#%\"fG" }{TEXT -1 31 " is not analytic at the point " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 25 ", but every neighborhood " } {XPPEDIT 18 0 "D[R](a)" "6#-&%\"DG6#%\"RG6#%\"aG" }{TEXT -1 4 " of " } {XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 39 " contains at least on e point at which " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 43 " is a nalytic. For example, the function " }{XPPEDIT 18 0 "f(z) = 1/(1-z); " "6#/-%\"fG6#%\"zG*&\"\"\"F),&F)F)F'!\"\"F+" }{TEXT -1 22 " is not \+ analytic at " }{XPPEDIT 18 0 "alpha = 1;" "6#/%&alphaG\"\"\"" }{TEXT -1 43 ", but is analytic for all other values of " }{XPPEDIT 18 0 "z; " "6#%\"zG" }{TEXT -1 19 ". Thus, the point " }{XPPEDIT 18 0 "alpha = 1" "6#/%&alphaG\"\"\"" }{TEXT -1 25 " is a singular point of " } {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 46 " . As another example, con sider the function " }{XPPEDIT 18 0 "g(z) = Log(z);" "6#/-%\"gG6#%\"zG -%$LogG6#F'" }{TEXT -1 31 ". We saw in Section 5.2 that " }{XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT -1 22 " is analytic for all " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 147 " except at the origin and at the points on the negative real axis. Thus, the origin and each point on the ne gative real axis is a singularity of " }{XPPEDIT 18 0 "g" "6#%\"gG" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " The point " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" } {TEXT -1 14 " is called an " }{TEXT 356 20 "isolated singularity" } {TEXT -1 24 " of a complex function " }{XPPEDIT 18 0 "f;" "6#%\"fG" } {TEXT -1 6 " if " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 21 " is n ot analytic at " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 33 ", b ut there exists a real number " }{XPPEDIT 18 0 "`R > 0`;" "6#%&R~>~0G " }{TEXT -1 12 " such that " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 47 " is analytic everywhere in the punctured disk " }{XPPEDIT 18 0 "D [R]^`*`*`(a)`;" "6#*&)&%\"DG6#%\"RG%\"*G\"\"\"%$(a)GF*" }{TEXT -1 17 " . The function " }{XPPEDIT 18 0 "f(z) = 1/(1-z)" "6#/-%\"fG6#%\"zG*& \"\"\"F),&F)F)F'!\"\"F+" }{TEXT -1 34 " has an isolated singularity \+ at " }{XPPEDIT 18 0 "alpha = 1" "6#/%&alphaG\"\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " The f unction " }{XPPEDIT 18 0 "g(z) = Log(z)" "6#/-%\"gG6#%\"zG-%$LogG6#F' " }{TEXT -1 33 ", however, has a singularity at " }{XPPEDIT 18 0 "alp ha = 0;" "6#/%&alphaG\"\"!" }{TEXT -1 96 " (or at any point of the ne gative real axis) that is not isolated, because any neighborhood of " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 52 " will contain points on the negative real axis, and " }{XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT -1 121 " is not analytic at those points. Functions with isolated sing ularities have a Laurent series because the punctured disk " } {XPPEDIT 18 0 "D[R]^`*`*`(a)`" "6#*&)&%\"DG6#%\"RG%\"*G\"\"\"%$(a)GF* " }{TEXT -1 28 " is the same as the annulus " }{XPPEDIT 18 0 "A(alpha, 0,R);" "6#-%\"AG6%%&alphaG\"\"!%\"RG" }{TEXT -1 118 ". We now look at this special case of Laurent's theorem in order to classify three typ es of isolated singularities. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 29 "Example for a pole, Page 291." } {TEXT 353 26 " Consider the function " }{XPPEDIT 18 0 "f(z) = cot(z )" "6#/-%\"fG6#%\"zG-%$cotG6#F'" }{TEXT 348 45 " .\nThe leading term i n the series expansion " }{XPPEDIT 18 0 "S(z)" "6#-%\"SG6#%\"zG" } {TEXT 350 6 " is " }{XPPEDIT 18 0 "1/z" "6#*&\"\"\"F$%\"zG!\"\"" } {TEXT 349 6 " and\n" }{XPPEDIT 18 0 "limit(S(z), z=0) = infinity" "6 #/-%&limitG6$-%\"SG6#%\"zG/F*\"\"!%)infinityG" }{TEXT 351 27 " in th e same manner as " }{XPPEDIT 18 0 "limit(cot(z), z=0) = infinity" "6 #/-%&limitG6$-%$cotG6#%\"zG/F*\"\"!%)infinityG" }{TEXT 352 3 " .\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "f:='f': L:='L': s:='s': S:='S': z: ='z': Z:='Z':\nf := z -> cot(z):\nS := series(f(Z), Z=0, 8):\ns := con vert(S, polynom):\nLS := z -> subs(Z=z,s):\n`f(z) ` = f(z);\n`f(z) ` = subs(Z=z,S);\nL[7](z) = LS(z);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 118 "x:='x': y:='y':\nplot(\{f(x),LS(x)\}, x=-3.14 ..3.14, y=-15..15,\n title=`y = cot(x) and y = L7(x)`,\n tickmarks =[7,7]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 45 "Example of a removable singularity, Page 291." } {TEXT 314 16 " The function " }{XPPEDIT 18 0 "f(z) = sin(z)/z" "6#/- %\"fG6#%\"zG*&-%$sinG6#F'\"\"\"F'!\"\"" }{TEXT 262 35 " \nhas a remov able singularity at " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT 263 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "f:='f': L:='L': s:= 's': S:='S': z:='z': Z:='Z':\nf := z -> sin(z)/z:\n`f(z) ` = f(z);\nS \+ := series(sin(Z), Z=0, 11)/Z:\n`f(z) ` = subs(Z=z,S);\nS := series(f(Z ), Z=0, 11):\n`f(z) ` = subs(Z=z,S);\ns := convert(S, polynom):\nLS := z -> subs(Z=z,s):\nL[8](z) = LS(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 260 1 "\n" }{TEXT 256 45 "Ex ample of a removable singularity, Page 291." }{TEXT 315 16 " The func tion " }{XPPEDIT 18 0 "f(z) = (cos(z) - 1)/z^2" "6#/-%\"fG6#%\"zG*&,& -%$cosG6#F'\"\"\"F-!\"\"F-*$F'\"\"#F." }{TEXT 261 35 " \nhas a remova ble singularity at " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT 264 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "f:='f': L:='L': s:= 's': S:='S': z:='z': Z:='Z':\nf := z -> (cos(z) -1)/z^2:\n`f(z) ` = f( z);\nS := series((cos(Z) -1), Z=0, 11)/Z^2:\n`f(z) ` = subs(Z=z,S);\nS := series(f(Z), Z=0, 11):\n`f(z) ` = subs(Z=z,S);\ns := convert(S, po lynom):\nLS := z -> subs(Z=z,s):\n`f(z) ` = f(z);\nL[8](z) = LS(z);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 39 "Example of a pole of order 2, Page 291." }{TEXT 326 16 " The function " }{XPPEDIT 18 0 "f(z) = sin(z)/z^3" "6#/-%\"fG6#%\"zG *&-%$sinG6#F'\"\"\"*$F'\"\"$!\"\"" }{TEXT 324 24 " \nhas a pole of or der " }{XPPEDIT 18 0 "n = 2" "6#/%\"nG\"\"#" }{TEXT 327 7 " at " } {XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT 325 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "f:='f': L:='L': s:='s': S:='S': z:='z': Z:=' Z':\nf := z -> sin(z)/z^3:\n`f(z) ` = f(z);\nS := series(sin(Z), Z=0, \+ 13)/Z^3:\n`f(z) ` = subs(Z=z,S);\nS := series(f(Z), Z=0, 13):\n`f(z) ` = subs(Z=z,S);\ns := convert(S, polynom):\nLS := z -> subs(Z=z,s):\nL [8](z) = LS(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 35 "Example of a simple pole, Page 291." } {TEXT 330 16 " The function " }{XPPEDIT 18 0 "f(z) = exp(z)/z" "6#/- %\"fG6#%\"zG*&-%$expG6#F'\"\"\"F'!\"\"" }{TEXT 328 25 " \nhas a simpl e pole at " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT 329 4 " .\n \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "f:='f': L:='L': s:='s': S:=' S': z:='z': Z:='Z':\nf := z -> exp(z)/z:\n`f(z) ` = f(z);\nS := series (exp(Z), Z=0, 8)/Z:\n`f(z) ` = subs(Z=z,S);\nS := series(f(Z), Z=0, 8) :\n`f(z) ` = subs(Z=z,S);\ns := convert(S, polynom):\nLS := z -> subs( Z=z,s):\nL[6](z) = LS(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 265 1 "\n" }{TEXT 256 47 "Examples of an \+ essential singularity, Page 291." }{TEXT 333 16 " The function " } {XPPEDIT 18 0 "f(z) = z^2*sin(1/z)" "6#/-%\"fG6#%\"zG*&F'\"\"#-%$sinG6 #*&\"\"\"F.F'!\"\"F." }{TEXT 331 36 " \nhas an essential singularity \+ at " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT 332 3 " .\n" } {TEXT 316 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "f:='f': L:='L': p:='p': s:='s': z:='z': Z:='Z':\nf := z -> z^2*sin(1/z):\ns := conver t(series(sin(z), z=0, 12), polynom):\nS := subs(z=1/Z,s):\np := z -> s ubs(Z=z, expand(Z^2 * S)):\n`f(z) ` = f(z);\nL[9](z) = p(z);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 66 "There will be infinitely many terms involving negative powers \+ of " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 286 11 " ,\nhence, " } {XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 288 35 " has an essenti al singularity at " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT 287 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 12 "Theorem 7.10" } {TEXT 357 16 " A function " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG " }{TEXT 358 15 " analytic in " }{XPPEDIT 18 0 "D[R](a)" "6#-&%\"DG6 #%\"RG6#%\"aG" }{TEXT 359 39 " has a zero of order k\nat the point \+ " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG" }{TEXT 360 46 " if an d only if its Taylor series given by " }{XPPEDIT 18 0 "f(z) = sum(c[ n]*(z - alpha)^n,n=0..infinity)" "6#/-%\"fG6#%\"zG-%$sumG6$*&&%\"cG6#% \"nG\"\"\"),&F'F0%&alphaG!\"\"F/F0/F/;\"\"!%)infinityG" }{TEXT 361 9 " \nhas " }{XPPEDIT 18 0 "c[0]=0,c[1]=0,`...`,c[k-1]=0" "6&/&%\"cG6# \"\"!F'/&F%6#\"\"\"F'%$...G/&F%6#,&%\"kGF+F+!\"\"F'" }{TEXT 362 9 " \+ and " }{XPPEDIT 18 0 "c[k]<>0" "6#0&%\"cG6#%\"kG\"\"!" }{TEXT 363 2 " ." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 334 1 "\n" }{TEXT 256 23 "Example 7.10, Page 292." }{TEXT 317 14 " Show that " } {XPPEDIT 18 0 "f(z)= z*sin(z^2)" "6#/-%\"fG6#%\"zG*&F'\"\"\"-%$sinG6#* $F'\"\"#F)" }{TEXT 267 23 " has a zero of order " }{XPPEDIT 18 0 "3 " "6#\"\"$" }{TEXT 289 6 " at " }{XPPEDIT 18 0 "z=0" "6#/%\"zG\"\"! " }{TEXT 290 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "f:='f': p: ='p': P:='P': s:='s': z:='z': Z:='Z':\nf := z -> z*sin(z^2):\ns := con vert(series(f(Z), Z=0, 24), polynom):\np := z -> subs(Z=z,s):\n`f(z) ` = f(z);\nP[23](z) = p(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 268 7 "Thus, " }{XPPEDIT 18 0 "f(z) = \+ z*sin(z^2)" "6#/-%\"fG6#%\"zG*&F'\"\"\"-%$sinG6#*$F'\"\"#F)" }{TEXT 269 24 " has a zero of order " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 291 6 " at " }{XPPEDIT 18 0 "z=0" "6#/%\"zG\"\"!" }{TEXT 292 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 12 "Theorem 7.11" }{TEXT 364 13 " Suppose " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 365 18 " is analytic in " }{XPPEDIT 18 0 "D[R](a)" "6#-&%\"DG6#%\"RG 6#%\"aG" }{TEXT 366 11 " . Then " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6# %\"zG" }{TEXT 367 40 " has a zero of order k at the point " } {XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG" }{TEXT 368 50 " if and o nly if it can be expressed in the form " }{XPPEDIT 18 0 "f(z) = (z - \+ alpha)^k*g(z)" "6#/-%\"fG6#%\"zG*&),&F'\"\"\"%&alphaG!\"\"%\"kGF+-%\"g G6#F'F+" }{TEXT 369 11 " , where " }{XPPEDIT 18 0 "g(z)" "6#-%\"gG6# %\"zG" }{TEXT 370 18 " is analytic at " }{XPPEDIT 18 0 "z = alpha" " 6#/%\"zG%&alphaG" }{TEXT 371 7 " and " }{XPPEDIT 18 0 "g(alpha)<>0" "6#0-%\"gG6#%&alphaG\"\"!" }{TEXT 372 2 " ." }{TEXT -1 2 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 13 "Corollary 7.4" }{TEXT 373 7 " \+ If " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 376 5 " and " } {XPPEDIT 18 0 "g(z);" "6#-%\"gG6#%\"zG" }{TEXT 377 18 " are analytic a t " }{XPPEDIT 18 0 "z = alpha;" "6#/%\"zG%&alphaG" }{TEXT 374 27 " a nd have zeros of orders " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT 380 5 " and " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT 381 16 ", respectively, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 382 19 "then t heir product " }{TEXT -1 0 "" }{TEXT 378 1 " " }{XPPEDIT 18 0 "h(z) = \+ f(z)*g(z);" "6#/-%\"hG6#%\"zG*&-%\"fG6#F'\"\"\"-%\"gG6#F'F," }{TEXT 379 23 " has a zero of order " }{XPPEDIT 18 0 "m+n;" "6#,&%\"mG\"\" \"%\"nGF%" }{TEXT 383 16 " at the point " }{XPPEDIT 18 0 "z = alpha " "6#/%\"zG%&alphaG" }{TEXT 375 2 ". " }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 335 1 " \n" }{TEXT 256 23 "Example 7.11, Page 293." }{TEXT 318 14 " Show tha t " }{XPPEDIT 18 0 "f(z)= z^3*sin(z)" "6#/-%\"fG6#%\"zG*&F'\"\"$-%$si nG6#F'\"\"\"" }{TEXT 270 23 " has a zero of order " }{XPPEDIT 18 0 " 4" "6#\"\"%" }{TEXT 294 6 " at " }{XPPEDIT 18 0 "z=0" "6#/%\"zG\"\"! " }{TEXT 293 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "f:='f': p: ='p': P:='P': s:='s': z:='z': Z:='Z':\nf := z -> z^3*sin(z):\ns := con vert(series(f(Z), Z=0, 15), polynom):\np := z -> subs(Z=z,s):\n`f(z) ` = f(z);\nP[14](z) = p(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 271 7 "Thus, " }{XPPEDIT 18 0 "f(z) = \+ z^3*sin(z)" "6#/-%\"fG6#%\"zG*&F'\"\"$-%$sinG6#F'\"\"\"" }{TEXT 272 24 " has a zero of order " }{XPPEDIT 18 0 "4" "6#\"\"%" }{TEXT 296 6 " at " }{XPPEDIT 18 0 "z=0" "6#/%\"zG\"\"!" }{TEXT 297 2 " ." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 12 "Theorem 7.12" }{TEXT -1 16 " \+ A function " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 34 " analytic in the punctured disk " }{XPPEDIT 18 0 "D[R](a)^`*`" "6#)- &%\"DG6#%\"RG6#%\"aG%\"*G" }{TEXT -1 31 " has a pole of order k at " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG" }{TEXT -1 51 " if a nd only if it can be expressed in the form \n" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "f(z) = h(z)/(z - alpha)^k" "6#/ -%\"fG6#%\"zG*&-%\"hG6#F'\"\"\"),&F'F,%&alphaG!\"\"%\"kGF0" }{TEXT -1 7 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "h(z)" "6#-%\"hG6#%\"zG" }{TEXT -1 18 " \+ is analytic at " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG" } {TEXT -1 8 " and " }{XPPEDIT 18 0 "h(alpha)<>0" "6#0-%\"hG6#%&alpha G\"\"!" }{TEXT -1 4 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 13 "C orollary 7.5" }{TEXT -1 8 " If " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6# %\"zG" }{TEXT -1 46 " is analytic and has a zero of order k at " } {XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG" }{TEXT -1 4 " , " }} {PARA 0 "" 0 "" {TEXT -1 6 "then " }{XPPEDIT 18 0 "g(z) = 1/f(z)" "6# /-%\"gG6#%\"zG*&\"\"\"F)-%\"fG6#F'!\"\"" }{TEXT -1 30 " has a pole of order k at " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 13 "Corollary 7.6" } {TEXT -1 8 " If " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 30 " has a pole of order k at " }{XPPEDIT 18 0 "z = alpha" "6#/ %\"zG%&alphaG" }{TEXT -1 10 " , then " }{XPPEDIT 18 0 "h(z) = 1/f(z) " "6#/-%\"hG6#%\"zG*&\"\"\"F)-%\"fG6#F'!\"\"" }{TEXT -1 34 " has a re movable singularity at " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG " }{TEXT -1 5 " . \n" }}{PARA 0 "" 0 "" {TEXT -1 14 "If we define " }{XPPEDIT 18 0 "h(alpha) = 0" "6#/-%\"hG6#%&alphaG\"\"!" }{TEXT -1 10 " , then " }{XPPEDIT 18 0 "h(z)" "6#-%\"hG6#%\"zG" }{TEXT -1 30 " h as a zero of order k at " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alp haG" }{TEXT -1 3 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 1 "\n" } {TEXT 256 23 "Example 7.12, Page 295." }{TEXT 319 34 " Locate the ze ros and poles of " }{XPPEDIT 18 0 "h(z) = tan(z)/z" "6#/-%\"hG6#%\"zG *&-%$tanG6#F'\"\"\"F'!\"\"" }{TEXT 298 30 " ,\nand determine their ord er.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "h:='h': L:='L': s:='s': s :='s': z:='z': Z:='Z':\nh := z -> tan(z)/z:\n`h(z) ` = h(z);\nS := ser ies(h(Z), Z=0, 12):\n`s(z) ` = subs(Z=z,S);\ns := convert(S, polynom): \nLS := z -> subs(Z=z,s):\nL[10](z) = LS(z);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 7 "Thus, " } {XPPEDIT 18 0 "h(z) = tan(z)/z" "6#/-%\"hG6#%\"zG*&-%$tanG6#F'\"\"\"F' !\"\"" }{TEXT 299 34 " has a removable singularity at " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT 300 30 " .\n\nNext consider the po ints " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 337 9 " = ... , " } {XPPEDIT 18 0 "2*pi, -pi, pi, 2*pi" "6&*&\"\"#\"\"\"%#piGF%,$F&!\"\"F& *&F$F%F&F%" }{TEXT 336 6 " , ..." }{TEXT 301 3 " . " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "tan(pi)/pi = tan(Pi)/Pi;\ntan(2*pi)/(2*pi) = tan (2*Pi)/(2*Pi);\ntan(3*pi)/(3*pi) = tan(3*Pi)/(3*Pi);\ntan(4*pi)/(4*pi) = tan(4*Pi)/(4*Pi);\ntan(5*pi)/(5*pi) = tan(5*Pi)/(5*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 6 "T hus " }{XPPEDIT 18 0 "h(z) = tan(z)/z" "6#/-%\"hG6#%\"zG*&-%$tanG6#F' \"\"\"F'!\"\"" }{TEXT 303 23 " has simple zeros at " }{XPPEDIT 18 0 "z = n*pi" "6#/%\"zG*&%\"nG\"\"\"%#piGF'" }{TEXT 275 9 " where " } {XPPEDIT 18 0 "`n = ... -2, -1, 1, 2, ...`" "6#%:n~=~...~-2,~-1,~1,~2, ~...G" }{TEXT 302 31 " . \n\nNext consider the points " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 339 9 " = ..., " }{XPPEDIT 18 0 "pi/2 - 2pi, \+ pi/2 -pi, pi/2, pi/2 + pi, pi/2 +2pi" "6',&*&%#piG\"\"\"\"\"#!\"\"F& *&F'F&F%F&F(,&*&F%F&F'F(F&F%F(*&F%F&F'F(,&*&F%F&F'F(F&F%F&,&*&F%F&F'F( F&*&F'F&F%F&F&" }{TEXT 338 9 " , ... . " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "tan(pi/2)/(pi/2) = limit(h(x),x=Pi/2,left);\ntan(3*pi/2)/(3*p i/2) = limit(h(x),x=3*Pi/2,left);\ntan(5*pi/2)/(5*pi/2) = limit(h(x),x =5*Pi/2,left);\ntan(7*pi/2)/(7*pi/2) = limit(h(x),x=7*Pi/2,left);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 6 "Thus " }{XPPEDIT 18 0 "h(z) = tan(z)/z" "6#/-%\"hG6#%\"zG*&-%$ tanG6#F'\"\"\"F'!\"\"" }{TEXT 304 16 " has poles at " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 341 9 " = ..., " }{XPPEDIT 18 0 "pi/2 - 2pi, \+ pi/2 -pi, pi/2, pi/2 + pi, pi/2 +2pi" "6',&*&%#piG\"\"\"\"\"#!\"\"F& *&F'F&F%F&F(,&*&F%F&F'F(F&F%F(*&F%F&F'F(,&*&F%F&F'F(F&F%F&,&*&F%F&F'F( F&*&F'F&F%F&F&" }{TEXT 340 10 " , ... . \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "s := z -> series(h(z), z=Pi/2, 3):\n`h(z) ` = h(z);\n `h(z) ` = s(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 7 "Hence " }{XPPEDIT 18 0 "h(z) = tan(z)/z " "6#/-%\"hG6#%\"zG*&-%$tanG6#F'\"\"\"F'!\"\"" }{TEXT 305 23 " has si mple poles at " }{XPPEDIT 18 0 "z = (n+1/2)*pi" "6#/%\"zG*&,&%\"nG\" \"\"*&F(F(\"\"#!\"\"F(F(%#piGF(" }{TEXT 278 9 " where " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 306 16 " is an integer." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 321 1 "\n" }{TEXT 256 23 "Example 7.13, Page 296." } {TEXT 320 24 " Locate the poles of " }{XPPEDIT 18 0 "g(z) = 1/(5*z^ 4 + 26*z^2 + 5)" "6#/-%\"gG6#%\"zG*&\"\"\"F),(*&\"\"&F)*$F'\"\"%F)F)*& \"#EF)*$F'\"\"#F)F)F,F)!\"\"" }{TEXT 307 28 " ,\nand specify their ord er.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "g:='g': z:='z':\ng := z -> 1/(5*z^4 + 26*z^2 + 5):\n`g(z) ` = g(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 279 27 "Find the singula rities of " }{XPPEDIT 18 0 "g(z) = 1/(5*z^4 + 26*z^2 + 5)" "6#/-%\"gG 6#%\"zG*&\"\"\"F),(*&\"\"&F)*$F'\"\"%F)F)*&\"#EF)*$F'\"\"#F)F)F,F)!\" \"" }{TEXT 308 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "Zn := so rt([solve(denom(g(z))=0, z)]):\n`For g(z) ` = g(z);\n`The singulariti es are:`;\nz1 := subs(z=Zn[1],z): z[1] = z1;\nz2 := subs(z=Zn[2],z): z [2] = z2;\nz3 := subs(z=Zn[3],z): z[3] = z3;\nz4 := subs(z=Zn[4],z): z [4] = z4;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 280 6 "Thus " }{XPPEDIT 18 0 "g(z) = 1 / (5z^4 + 26z^2 + 5)" "6#/-%\"gG6#%\"zG*&\"\"\"F),(*&\"\"&F)*$F'\"\"%F)F)*&\"#EF)*$F'\" \"#F)F)F,F)!\"\"" }{TEXT 309 26 " , has four simple poles." }}} {EXCHG {PARA 0 "" 0 "" {TEXT 322 1 "\n" }{TEXT 256 23 "Example 7.14, P age 296." }{TEXT 323 34 " Locate the zeros and poles of " } {XPPEDIT 18 0 "g(z) = pi*cot(pi*z)/z^2" "6#/-%\"gG6#%\"zG*(%#piG\"\"\" -%$cotG6#*&F)F*F'F*F**$F'\"\"#!\"\"" }{TEXT 281 30 " ,\nand determine \+ their order.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 297 "g:='g': L:='L': \+ p:='p': s:='s': S:='S': z:='z': Z:='Z':\ng := z -> Pi*cot(Pi*z)/z^2:\n s := series(Pi*cot(Pi*z), z=0, 7)/z^2:\nS := series(g(z), z=0, 7):\np \+ := convert(series(Pi*cot(Pi*Z), Z=0, 7), polynom):\nLS := z -> subs(Z= z,expand(p/z^2)):\n`g(z) ` = g(z);\n`g(z) ` = s;\n`g(z) ` = S;\nL[3](z ) = LS(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 6 "Thus " }{XPPEDIT 18 0 "g(z) = pi*cot(pi*z)/z^2" "6#/-%\"gG6#%\"zG*(%#piG\"\"\"-%$cotG6#*&F)F*F'F*F**$F'\"\"#!\"\"" } {TEXT 283 24 " , has a pole of order " }{XPPEDIT 18 0 "n = 3" "6#/%\" nG\"\"$" }{TEXT 311 6 " at " }{XPPEDIT 18 0 "z = 0" "6#/%\"zG\"\"!" }{TEXT 310 40 " .\n\nNext consider the points z = ... , " }{XPPEDIT 18 0 "-3,-2,-1,1,2,3" "6(,$\"\"$!\"\",$\"\"#F%,$\"\"\"F%F)F'F$" } {TEXT 342 10 " , ... . \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 344 "pi*co t(pi)/(pi)^2 = limit(g(x),x=1,right);\n2*pi*cot(2*pi)/(2*pi)^2 = limit (g(x),x=2,right);\n3*pi*cot(3*pi)/(3*pi)^2 = limit(g(x),x=3,right);\ns := z -> series(g(z), z=1, 5):\n`g(z) ` = g(z),` Expand about the poi nt `,z[0] = 1;\n`g(z) ` = s(z);\ns := z -> series(g(z), z=2, 5):\n`g( z) ` = g(z),` Expand about the point `,z[0] = 2;\n`g(z) ` = s(z);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 6 "Thus " } {XPPEDIT 18 0 "g(z) = pi*cot(pi*z)/z^2" "6#/-%\"gG6#%\"zG*(%#piG\"\" \"-%$cotG6#*&F)F*F'F*F**$F'\"\"#!\"\"" }{TEXT 285 44 " has simple pol es at the points z = ... , " }{XPPEDIT 18 0 "-3,-2,-1,1,2,3" "6(,$\" \"$!\"\",$\"\"#F%,$\"\"\"F%F)F'F$" }{TEXT 343 10 " , ... . ." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 347 19 "End of Section 7.4." }{TEXT -1 0 "" }}}}{MARK "0 0 0" 20 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }