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0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Monaco" 1 9 255 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 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 "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 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 257 41 "COMPLEX ANALYSIS: Maple Worksheets, 2001" }{TEXT 287 144 "\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 286 197 "\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 285 1 "\n" }{TEXT 256 36 "CHAPTER 7 TAYLOR and LAURENT SERIES" } {TEXT 279 2 "\n\n" }{TEXT 256 42 "Section 7.2 Taylor Series Represent ations" }{TEXT 280 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 421 " In Sectio n 4.4 we saw that functions defined by power series have derivatives o f all orders (Theorem 4.16). In Section 6.5 we saw that analytic func tions also have derivatives of all orders (Corollary 6.2). It seems na tural, therefore, that there would be some connection between analytic functions and power series. As you might guess, the connection exists via the Taylor and Maclaurin series of analytic functions." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 29 "Definition 7.2: Taylor series " }{TEXT 288 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 2 "If" }{TEXT 299 2 " " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#% \"zG" }{TEXT 289 3 " i" }{TEXT 300 13 "s analytic at" }{TEXT 301 2 " \+ " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG" }{TEXT 290 4 " , " } {TEXT 302 11 "the series " }{TEXT 303 2 " " }{XPPEDIT 18 0 "sum(f^`(k )`*`(`*alpha*`)`*(z - alpha)^k/k!,k=0..infinity)" "6#-%$sumG6$*.)%\"fG %$(k)G\"\"\"%\"(GF*%&alphaGF*%\")GF*),&%\"zGF*F,!\"\"%\"kGF*-%*factori alG6#F2F1/F2;\"\"!%)infinityG" }{TEXT 291 3 " " }{TEXT 304 14 "is ca lled the " }{TEXT 257 13 "Taylor series" }{TEXT 292 4 " for" }{TEXT 305 2 " " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 294 2 " " } {TEXT 306 15 "centered around" }{TEXT 307 2 " " }{XPPEDIT 18 0 "alpha " "6#%&alphaG" }{TEXT 295 4 " . " }{TEXT 308 18 "When the center is" }{TEXT 309 2 " " }{XPPEDIT 18 0 "alpha = 0" "6#/%&alphaG\"\"!" } {TEXT 296 3 " , " }{TEXT 310 26 " the series is called the " }{TEXT 257 17 "Maclaurin series " }{TEXT 293 4 "for " }{TEXT 311 1 " " } {XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 297 2 " ." }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 31 "Theorem 7.4 (Taylor's Theorem)" }{TEXT 312 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 331 9 "Suppose \+ " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 313 41 " is analytic in a domain G, and that " }{XPPEDIT 18 0 "D[R](alpha) = \{`z:`*abs (z - alpha) < R\}" "6#/-&%\"DG6#%\"RG6#%&alphaG<#2*&%#z:G\"\"\"-%$absG 6#,&%\"zGF/F*!\"\"F/F(" }{TEXT 314 60 " is contained in G. Then the Taylor series converges to " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG " }{TEXT 315 18 " for all z in " }{XPPEDIT 18 0 "D[R](alpha)" "6#- &%\"DG6#%\"RG6#%&alphaG" }{TEXT 316 12 " ; that is " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 332 2 " " }{TEXT 322 2 " " } {XPPEDIT 18 0 "f(z) = sum(f^`(k)`*`(`*alpha*`)`*(z - alpha)^k/k!,k=0.. infinity)" "6#/-%\"fG6#%\"zG-%$sumG6$*.)F%%$(k)G\"\"\"%\"(GF.%&alphaGF .%\")GF.),&F'F.F0!\"\"%\"kGF.-%*factorialG6#F5F4/F5;\"\"!%)infinityG" }{TEXT 317 3 " " }{TEXT 323 8 "for all " }{TEXT 324 1 " " }{XPPEDIT 18 0 "`z`*epsilon*D[R](alpha)" "6#*(%\"zG\"\"\"%(epsilonGF%-&%\"DG6#% \"RG6#%&alphaGF%" }{TEXT 318 3 " .\n" }}{PARA 0 "" 0 "" {TEXT 325 74 " Furthermore, this representation is valid in the largest disk with cen ter " }{TEXT 326 1 " " }{XPPEDIT 18 0 "z = alpha" "6#/%\"zG%&alphaG" } {TEXT 319 3 " t" }{TEXT 327 77 "hat is contained in G, and the conv ergence is uniform on any closed subdisk" }{TEXT 328 2 " " }{XPPEDIT 18 0 "D[r](alpha) = \{`z:`*abs(z - alpha) <= r\}" "6#/-&%\"DG6#%\"rG6# %&alphaG<#1*&%#z:G\"\"\"-%$absG6#,&%\"zGF/F*!\"\"F/F(" }{TEXT 320 3 " \+ " }{TEXT 330 5 "for " }{TEXT 329 1 " " }{XPPEDIT 18 0 "`0 < r` < R " "6#2%&0~<~rG%\"RG" }{TEXT 321 3 " . " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 13 "Corollary 7.3" }{TEXT 333 12 " Suppo se " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 334 57 " is anal ytic in the domain G that contains the point " }{XPPEDIT 18 0 "z = \+ alpha" "6#/%\"zG%&alphaG" }{TEXT 335 9 " . Let " }{XPPEDIT 18 0 "z[0 ]" "6#&%\"zG6#\"\"!" }{TEXT 336 46 " be a singular point of minimum d istance to " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 337 9 " . I f " }{XPPEDIT 18 0 "abs(z[0] - alpha) = R" "6#/-%$absG6#,&&%\"zG6#\" \"!\"\"\"%&alphaG!\"\"%\"RG" }{TEXT 338 9 " , then\n" }{TEXT 256 4 "( i) " }{TEXT 344 33 "\011the Taylor series converges to " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 339 13 " on all of " }{XPPEDIT 18 0 "D[R](alpha)" "6#-&%\"DG6#%\"RG6#%&alphaG" }{TEXT 340 8 " , and \n" }{TEXT 256 5 "(ii) " }{TEXT 345 5 "\011if " }{XPPEDIT 18 0 "S > R " "6#2%\"RG%\"SG" }{TEXT 343 43 ", the Taylor series does not converg e to " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 341 13 " on al l of " }{XPPEDIT 18 0 "D[S](alpha)" "6#-&%\"DG6#%\"SG6#%&alphaG" } {TEXT 342 2 " ." }{TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 22 "Example 7.3, Page 272." }{TEXT 281 14 " Show that " }{XPPEDIT 18 0 "1/(1- z)^2 = sum((n+1)*z^n, n=0..infinity)" "6#/*&\"\"\"F%*$,&F% F%%\"zG!\"\"\"\"#F)-%$sumG6$*&,&%\"nGF%F%F%F%)F(F0F%/F0;\"\"!%)infinit yG" }{TEXT 267 18 " , is valid for " }{XPPEDIT 18 0 "z" "6#%\"zG" } {TEXT 268 6 " in " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 269 2 ": " } {XPPEDIT 18 0 "abs(z) < 1" "6#2-%$absG6#%\"zG\"\"\"" }{TEXT 270 3 " . \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "f:='f': p:='p': P:='P': s:=' s': t:='t': z:='z': Z:='Z':\nf := z -> 1/(1-z)^2:\nt := taylor(f(Z), Z =0, 10):\ns := subs(Z=z,t):\np := z -> subs(Z=z,convert(t, polynom)): \n`f(z) ` = f(z);\n`f(z) ` = s;\nP[9](z) = p(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Or we could use Maple's \"unapply\" procedure." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "f:='f': p:='p': P:='P': s:='s': t :='t': z:='z': Z:='Z':\nf := z -> 1/(1-z)^2:\ns := taylor(f(z), z=0, 1 0):\np:=unapply(convert(taylor(f(z),z=0,10),polynom),z):\n`f(z) ` = f (z);\n`f(z) ` = s;\nP[9](z) = p(z);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 260 54 "Sum up the terms to verify that we have things right.\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "f:='f': n:='n': S:='S': z:='z' :\nf := z -> 1/(1-z)^2:\nS9 := z -> sum((n+1)*z^n, n=0..9):\nS := z -> sum((n+1)*z^n, n=0..infinity):\n`f(z) ` = f(z);\ns[9](z),` = `,Sum((n +1)*z^n, n=0..9) = S9(z);\ns[infinity](z),` = `,Sum((n+1)*z^n, n=0..in finity) = S(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 26 "Example 7.4, Page 273. (a)" }{TEXT 282 12 " Show that " }{XPPEDIT 18 0 "1/(1- z^2) = sum(z^(2*n), n=0..i nfinity)" "6#/*&\"\"\"F%,&F%F%*$%\"zG\"\"#!\"\"F*-%$sumG6$)F(*&F)F%%\" nGF%/F0;\"\"!%)infinityG" }{TEXT 271 17 " , is valid for " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 272 6 " in " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 273 2 ": " }{XPPEDIT 18 0 "abs(z) < 1" "6#2-%$absG6#%\"zG\"\"\" " }{TEXT 274 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "f:='f': p: ='p': P:='P': s:='s': t:='t': z:='z': Z:='Z':\nf := z -> 1/(1-z^2):\nt := taylor(f(Z), Z=0, 20):\ns := subs(Z=z,t):\np := z -> subs(Z=z,conv ert(t, polynom)):\n`f(z) ` = f(z);\n`f(z) ` = s;\nP[18](z) = p(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Or we could use Maple's \"unapp ly\" procedure." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "f:='f': p:='p': P:='P': s:='s': t:='t': z:='z': Z:='Z':\nf := z -> 1/(1-z^2):\ns := t aylor(f(z), z=0, 20):\np:=unapply(convert(taylor(f(z),z=0,20),polynom) ,z):\n`f(z) ` = f(z);\n`f(z) ` = s;\nP[18](z) = p(z);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 261 54 "Sum up the terms to verify that we hav e things right.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "f:='f': n:='n ': s:='s': S:='S': z:='z':\nf := z -> 1/(1-z^2):\nS9 := z -> sum(z^(2* n), n=0..9):\nS := z -> sum(z^(2*n), n=0..infinity):\n`f(z) ` = f(z); \ns[18](z),` = `,Sum(z^(2*n), n=0..9) = S9(z);\ns[infinity](z),` = `,S um(z^(2*n), n=0..infinity) = S(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 256 3 "(b)" }{TEXT 283 14 " \+ Show that " }{XPPEDIT 18 0 "1/(1 + z^2) = sum ((-1)^n*z^(2*n), n=0. .infinity)" "6#/*&\"\"\"F%,&F%F%*$%\"zG\"\"#F%!\"\"-%$sumG6$*&),$F%F*% \"nGF%)F(*&F)F%F1F%F%/F1;\"\"!%)infinityG" }{TEXT 275 19 " , is vali d for " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT 276 6 " in " }{XPPEDIT 18 0 "D" "6#%\"DG" }{TEXT 277 2 ": " }{XPPEDIT 18 0 "abs(z) < 1" "6#2- %$absG6#%\"zG\"\"\"" }{TEXT 278 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "f:='f': p:='p': P:='P': s:='s': t:='t': z:='z': Z:='Z':\nf := z -> 1/(1+z^2):\nt := taylor(f(Z), Z=0, 20):\ns := subs(Z=z,t):\np := z -> subs(Z=z,convert(t, polynom)):\n`f(z) ` = f(z);\n`f(z) ` = s; \nP[18](z) = p(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Or we could use Maple's \"unapply\" procedure." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "f:='f': p:='p': P:='P': s:='s': t:='t': z:='z': Z:='Z':\nf := z - > 1/(1+z^2):\ns := taylor(f(z), z=0, 20):\np:=unapply(convert(taylor(f (z),z=0,20),polynom),z):\n`f(z) ` = f(z);\n`f(z) ` = s;\nP[18](z) = \+ p(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 262 54 "Sum up the terms to verify that we have things righ t.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "f:='f': n:='n': s:='s': S: ='S': z:='z':\nf := z -> 1/(1+z^2):\nS9 := z -> sum((-1)^n * z^(2*n), \+ n=0..9):\nS := z -> sum((-1)^n * z^(2*n), n=0..infinity):\n`f(z) ` = f (z);\ns[18](z),` = `,Sum((-1)^n * z^(2*n), n=0..9) = S9(z);\ns[infinit y](z),` = `,Sum((-1)^n * z^(2*n), n=0..infinity) = S(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 41 "Theorem 7.5 (Uniqueness of power seri es)" }{TEXT 346 30 " Suppose that in some disk " }{XPPEDIT 18 0 "D[ r](alpha)" "6#-&%\"DG6#%\"rG6#%&alphaG" }{TEXT 347 9 " we have" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 352 5 " " } {XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 349 3 " = " }{XPPEDIT 18 0 "sum(a[n]*(z-alpha)^n,n=0..infinity) = sum(b[n]*(z-alpha)^n,n=0.. infinity) " "6#/-%$sumG6$*&&%\"aG6#%\"nG\"\"\"),&%\"zGF,%&alphaG!\"\"F +F,/F+;\"\"!%)infinityG-F%6$*&&%\"bG6#F+F,),&F/F,F0F1F+F,/F+;F4F5" } {TEXT 348 12 " . Then " }{XPPEDIT 18 0 "a[n] = b[n]" "6#/&%\"aG6#% \"nG&%\"bG6#F'" }{TEXT 350 9 " for " }{XPPEDIT 18 0 "n = 0,1,2,`.. .`" "6&/%\"nG\"\"!\"\"\"\"\"#%$...G" }{TEXT 351 4 " . " }{TEXT -1 0 " " }}}{EXCHG {PARA 257 "" 0 "" {TEXT 263 1 "\n" }{TEXT 256 22 "Example \+ 7.5, Page 274." }{TEXT 284 33 " Find the Maclaurin series of " } {XPPEDIT 18 0 "f(z) = sin(z)^3" "6#/-%\"fG6#%\"zG*$-%$sinG6#F'\"\"$" } {TEXT 264 3 " .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "f:='f': p:='p ': P:='P': s:='s': t:='t': z:='z': Z:='Z':\nf := z -> sin(z)^3:\nt := \+ taylor(f(Z), Z=0, 14):\ns := subs(Z=z,t):\np := z -> subs(Z=z,convert( t, polynom)):\n`f(z) ` = f(z);\n`f(z) ` = s;\nP[13](z) = p(z);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 265 54 "Sum up the terms to verify that we have things right.\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 549 "An:='An': Bn:='Bn': f:='f': n: ='n': p:='p': q:='q': s:='s': z:='z':\nf := z -> sin(z)^3:\nAn := (-1) ^n * 3/(4*(2*n+1)!):\nBn := - (-1)^n * 3 * 9^n/(4*(2*n+1)!):\n`f(z) ` \+ = f(z);\np13 := sum(An* z^(2*n+1), n=0..6):\nq13 := sum(Bn* z^(2*n+1), n=0..6):\nSum(An* z^(2*n+1), n=0..6) = p13;\nSum(Bn* z^(2*n+1), n=0.. 6) = q13;\n`Partial sum `, s[13](z) = \n sum(An* z^(2*n+1), n=0..6) \+ + \n sum(Bn* z^(2*n+1), n=0..6);\ns[infinity](z),` = `,\nSum(An* z^(2 *n+1)+Bn* z^(2*n+1), n=0..infinity) =\n sum(An* z^(2*n+1), n=0..infin ity) +\n sum(Bn* z^(2*n+1), n=0..infinity);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "Theorem 7.6" }{TEXT 353 8 " Let " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT 354 7 " and " }{XPPEDIT 18 0 "g(z)" "6#-%\"gG6#%\"zG" }{TEXT 355 40 " hav e the power series representations\n" }{XPPEDIT 18 0 "f(z) = sum(a[n]* (z-alpha)^n,n=0..infinity)" "6#/-%\"fG6#%\"zG-%$sumG6$*&&%\"aG6#%\"nG \"\"\"),&F'F0%&alphaG!\"\"F/F0/F/;\"\"!%)infinityG" }{TEXT 356 9 " f or " }{XPPEDIT 18 0 "`z`*epsilon*D[r[1]](alpha)" "6#*(%\"zG\"\"\"%(e psilonGF%-&%\"DG6#&%\"rG6#F%6#%&alphaGF%" }{TEXT 358 9 " and " } {XPPEDIT 18 0 "g(z) = sum(b[n]*(z-alpha)^n,n=0..infinity)" "6#/-%\"gG6 #%\"zG-%$sumG6$*&&%\"bG6#%\"nG\"\"\"),&F'F0%&alphaG!\"\"F/F0/F/;\"\"!% )infinityG" }{TEXT 357 9 " for " }{XPPEDIT 18 0 "`z`*epsilon*D[r[2 ]](alpha)" "6#*(%\"zG\"\"\"%(epsilonGF%-&%\"DG6#&%\"rG6#\"\"#6#%&alpha GF%" }{TEXT 359 7 " .\nIf " }{XPPEDIT 18 0 "r = min(r[1],r[2])" "6#/% \"rG-%$minG6$&F$6#\"\"\"&F$6#\"\"#" }{TEXT 360 9 " , and " } {XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT 361 32 " is any complex const ant, then\n" }{TEXT 256 3 "(i)" }{TEXT 368 4 " " }{XPPEDIT 18 0 "be ta*f(z) = sum(beta*a[n]*(z-alpha)^n,n=0..infinity)" "6#/*&%%betaG\"\" \"-%\"fG6#%\"zGF&-%$sumG6$*(F%F&&%\"aG6#%\"nGF&),&F*F&%&alphaG!\"\"F2F &/F2;\"\"!%)infinityG" }{TEXT 362 9 " for " }{XPPEDIT 18 0 "`z`*ep silon*D[r[1]](alpha)" "6#*(%\"zG\"\"\"%(epsilonGF%-&%\"DG6#&%\"rG6#F%6 #%&alphaGF%" }{TEXT 363 4 " , \n" }{TEXT 256 4 "(ii)" }{TEXT 369 3 " \+ " }{XPPEDIT 18 0 "f(z) + g(z) = sum((a[n] + b[n])*(z-alpha)^n,n=0..in finity)" "6#/,&-%\"fG6#%\"zG\"\"\"-%\"gG6#F(F)-%$sumG6$*&,&&%\"aG6#%\" nGF)&%\"bG6#F5F)F)),&F(F)%&alphaG!\"\"F5F)/F5;\"\"!%)infinityG" } {TEXT 364 9 " for " }{XPPEDIT 18 0 "`z`*epsilon*D[r](alpha)" "6#*( %\"zG\"\"\"%(epsilonGF%-&%\"DG6#%\"rG6#%&alphaGF%" }{TEXT 365 8 " , an d \n" }{TEXT 256 5 "(iii)" }{TEXT 370 3 " \011" }{XPPEDIT 18 0 "f(z)* g(z) = sum(c[n]*(z-alpha)^n,n=0..infinity)" "6#/*&-%\"fG6#%\"zG\"\"\"- %\"gG6#F(F)-%$sumG6$*&&%\"cG6#%\"nGF)),&F(F)%&alphaG!\"\"F4F)/F4;\"\"! %)infinityG" }{TEXT 366 9 " for " }{XPPEDIT 18 0 "`z`*epsilon*D[r] (alpha)" "6#*(%\"zG\"\"\"%(epsilonGF%-&%\"DG6#%\"rG6#%&alphaGF%" } {TEXT 367 12 " , where " }{XPPEDIT 18 0 "c[n] = sum(a[k]*b[n-k],k=0 ..n)" "6#/&%\"cG6#%\"nG-%$sumG6$*&&%\"aG6#%\"kG\"\"\"&%\"bG6#,&F'F0F/! \"\"F0/F/;\"\"!F'" }{TEXT 371 4 " . \n" }}{PARA 0 "" 0 "" {TEXT 375 31 "Identity (iii) is known as the " }{TEXT 257 14 "Cauchy product" } {TEXT 372 20 " of the series for " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#% \"zG" }{TEXT 373 7 " and " }{XPPEDIT 18 0 "g(z)" "6#-%\"gG6#%\"zG" } {TEXT 374 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 266 19 "End of Section 7.2." }}}}{MARK "0 0 0 " 33 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }