Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 3
Текст из файла (страница 3)
z2) - paccTmrnHe Me)f(ny cTepeorpacpttqecKHMHrrpoeKI.(H.SIMH ToqeK Zt H Z2 pacwHpeHHOM KOMrrneKCHOM rrnOCKOCTH ..[{oKa3aTh, qTo crrpaBennHBhI cpopMyJih1:1) d( z,' Z, ) =.II z, - Z2I I2\11+ I z,212) d(z1, 00 )=(l+l.: 1 l f\11 + I Z21122(z,, Z2 "* )00;1•1.24. Hai1TH fMT z E C, ynoBnernop.siIOIUHX yrnoBHIO:1) d(z,O)<R,O<R<l; 2) d(z,=)<R,O<R<l;3) d(z,i)>ll.J2.;1.25. Bh1qttcn11Th:l) ef;;2)~;5)7)7r4) ll2<d(z,l)<l1Ji..3)-Jl=i;4)~-4 + 3i;7rcos--isin-;33.Jl + e;<P, - tr < (/J'~ tr; 8) .Ji+ e;"', - Jr< (jJ 5, Jr;9)~,::=l+i;IO)efl+i;11)~.13KOMI1JIEKCHb!E 411CJIA 1111X CBOHCTBA1.26. IIycTb a = a+ ip.
)J,oKa3aTb,a .ua10Tc~ cpopMynaMH:2z =(/¥ ·Wal-a)(/¥ ·Wal-a)qTo Bee peIUeHH~ ypaBHeHH~1) z=±-al+a- + 1 - - , /J>O;2) z=±---122al+a2--1.27 . .Il:oKa3aTb,. /J<O.2qTo .un~ 11106oro z El~z 2 -l+zl+l~z2-l-zl1.28. IIycThco= e iZrr.Jn=C crrpaBe.UJJHBa cpopMynalz-ll+lz+ll.11 HaTypanbHOek HeKpaTHOn E N . .Il:oKa-3aTb, qTo1 + cok + co2k + ... + co<11 - I)k = O.1.29. IIycTbco - 111060H: KopeHb ttarypanhHOtt cTerreHHn 113 e.uHHH-U:hI, OTJ1MqHbIH OT e,UHHMU:bI.
)J,OKa3aTb, qTo crrpaBeLlJlMBbI cpopMyJihI:1) 1-r2w-r3w' -r ... -r11w"- 1 =nl(w-l);'' n-12) 1 + 4w + 9w + ... + 11 w =n(l -w) + 2n, .(1-wr1.30. I1ycTbCTerreHH n 2: 1ecntt z,, Z2, ... ,P,, (z) = a0 z" +... +a,,_, z +a,, , a0-::j. 0, - MHoroqneHc KOMIIJleKCHbIMM K03cpcp11u:11eHTaMH . .Il:oKa3aTb, qToZ11 - pa3J1HqHbie KOpHH MHOroqJieHa P11(Z) KpaTHOCTHai, a2, ...
, a/II COOTBeTCTBeHHO, TO crrpaBe.UJIMBO pa3JIO)l(eHtte- nm•=/.(.-z,)-J'.,(z)-ao1.31. I1ycn Zi,al"'m, 'L;;=ia,--n.Z2, . .. , Z11 - KOpHH MHoroqneHaBhIIIHCaHHbre c yqeToM KpaTHOCTH. .Il:oKa3aTb cpopMyJihI BHerna,, = ( - 1)11Z1 • Z2• ••• •Z,,.IJwea 1141.32. L(oKa:mTh,'ITO ecmf z 0 EC-KopeHh Kparnocrnaoanre6-paw:1ecKoro Mttoroqnetta cTeneHH 60JihUJe 1 c .ueH:CTBHTeJihHhlMHz0K03cpcp11u11eHTaM11, TO-TaK)l(e Kopetth 3Toro MHoroqnetta 11 eroKpaTHOCTh paBHa CX() .1.33. IIycTh P,, (z) = a0 z" + ... + a 1z + a ao # 0, - MttoroqneHcTenett11 n c .n:ei1:crn11TeJihtth1M11 K03cpqmu11ettTaM11. Dyen., KpoMeToro, Xi, ...
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L(oKa3aTh,1.35. L(oKa3aTh,. '" x1) sm-=I2211Q11 (x)qToHMeeT He 6onee2n KOptteH:.'ITO cnpaBe.umrnhI cne.uy10m11e paBeHcrna:°"'211~•=oC'211 (- l)"-' cos 2( n - k) x;2) I~:o C~1 (- IY-' sin 2(n - k )x = O;3)i I2"cos 211 x=-,,_ 0 C,, cos2(n-k)x;2-",._-"4) I~: 0 c~1 sin 2(n-k)x = O;5) sin211•6) I~:~'7)1x =-c;COS-'11+1 X8) L:~:~101-""~:· 1 CZ-"+ 1( - l)"- 1 sin (2(11 - k)~II.-2211+111 • 1 (-1)"-'I=-,2-ll+Ic;,,., sincos (2(n - k)L1n+I'· -Oc'A-(2(11-k),COS_JI+ 1+ l)x;+ I)x = O;(2( II-k)'+ I) x;+ I)x = 0.1.36.
L(oKa3aTh cne.n:y10mtteyrnep)l(.n:ett115f.a) T,,(x) = cos (11 arc cos x) ecTh anre6paw1ecK11H: MHoroqneHCTeneHH 11. 3TOT MHOfO'lJieH Ha3bJBaeTC5f ,l/H020l/J/eHOJ11 lfe6bztUe6aKOMITJIEKCHhlE 4HCJIA H l1X CBOHCTBA1511epeo20 pooa. I1oKa3aTb, 'ITO cTapI1111i1: K03cpqrnu11eHT y T11 (x)paBeH 2 11 - 1.6) KopHH rrpoH3BO.UHOH ame6pa11qecKoro MHOfO'IJleHa rrpHHa,n:JJe)KaT BbIIIYKJlOH o60JlOlJKe KOpHeH CaMoro MHOfOlJJleHa.1.37. ITycTb P11(z) - ame6pa11qecKHH MHoroqneH cTerreHH n.
KaKHMH .uomKHbI 6bITb ero K03cpcpHUHeHTbI, ecJIH:2) P,, (z) = -P,,Cz).1) P (z) = P (Z);IIII1.38. ITycTb a - .uei1:cTBHTeJlbHoe lJHCJlO. ,[(oKa3aTb, 'ITO ecJJHMHOfOlJJieH P11(Z) CTerreHH n HMeeT 1l ,n:eHCTBHTeJlbHblX KOpHdi, TOMHoroqneH Qn(z) P11 (z + ia) + P,,(z - ia) TaK)Ke HMeeT n ,n:ei1:crn11TeJJbHbIX KOpHeH.=1.39.
,[(oKa3aTb TO)J(,n:ecrna:L:;J zk 1 +I L::. 1z, 1= L1s1«s,,I zk +z., 1 n;:: 2;2) 11 L::J zk 1 -I L::=I z, 1= Lisk<sSnl Z; - z, 1 n;:: 2.21) (n - 2)2,1221,1.40. .lJ:oKa3aTb HepaBeHCTBa:1)IL::= z 1:::;JnL:;Jz !21 11;p-l2)L:;Jz, J:=;;11--;-(L:;Jz1IP)"P,p;::l;1.41. ITycTb p > 0, a q < 0 . .lJ:oKa3aTb, 'ITO crrpaBe,n:nHBbI cne.uy10ume HepaBeHcrna:1)~I z,. Z2 . .... z,, I~ ( ~1 :L:J Zt I")!/" ;1.42 . .UoKa3aTb cne.n:y10m11e paBeHcrna:l1) Iim ( p-+0112)Jim ( -lp-++-L::J z, Ip11)lip=~I z, · ::, · ... -z,, I;L:;J :: II' )"'' = max I z, J;1ISlS11I'Jwea 1163) lim ( -1 L:~J ztp-+-11I'' )"''=min I ztI SI.
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