Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 7
Текст из файла (страница 7)
HEITPEPbIBHOCTb7) o6JiaCTb orrpe.n:eJiem:rn: cpyHKUHH tg z -39BC51 KOMITJieKCHa51ITJIOCKOCTb C, KpOMe rnqeK z = n/2, k = 0, ±1, ±2, ... , a cpyHKUHH ctg z BC5I rrJIOCKOCTb C, KpoMe rnqeK z =kn, k = 0, ±1, ±2, ... ;8) cpyHKUHH tg z H ctg z HerrpepbIBHbl Ha o6JiaCTH orrpe.n:eJieHtt5I;9) lim tg z, lim ctg z He cymecrnyIOT;.:~oo:---4-oo10) tg z = tgz , ctg z = ctg z ;6) HaMTH MHO)l(eCTBa rnqeK, B KOTOpbIX cpyHKUHH tg Z, ctg z rrpHHHMaI{)T11) .ueifcTBHTeJibHbie 3HaqeHH5I;12) qHCTO MHHMbie 3HaqeHH5I.3.28 . .D:JI5I cpyHKUHM sh z H ch z ,l(OKa3aTb CJie,UyIOIUHe yrnep)1(.LJ: eHH5I:1) ch (-z) =ch z, sh (-z) =-sh z;222) ch z - sh z = 1;3) ch (z1 + z2) =ch z1 ch z2 + sh z1 sh z2;4) sh (z+ ikTI) = (-1/sh z, ch (z + ikTI)=C-llchz, k=0,±1,±2, ...
;5) sh (z + in/2) = i ch z, ch (z + in/2) = i sh z;6) sh (iz) = i sin z, ch (iz) = cos z;7)rrp.11 z = x + iyRe sh z = sh x cosy, Re ch z = ch x cosy,Im sh z = ch x sin y, Im ch x = sh x sin y;8) rrpH Z = X + iyIsh z I= ~~ch_2_x_--co_s_2 -y, Ich z I= ~ch 2 x-sin 2 y;Ish z I= ~sh 2x + sin 2 y, Ich z I= ~sh 2x + cos 2 y ,I thz I=ch2x - cos2ych2x+ cos2y9) cj:>yHKUHH sh z H ch z ~rnmrIOTCH HerrpepbIBHbIMH Ha KOMrrJieKCHOH ITJIOCKOCTH C;10) lim sh z H lim ch z He cymecrnyIOT;.;:-too:-too40I'lza6a 311) cpyHKUIUI sh z o6paruaeTC5! B HYJib TOJibKO B ToqKax za cpyHKIJJrn ch z - TOJibKO B TOqKax z= ikTI ,= i( 'ITl2 + kTI ), k = 0, ±1 , ±2, ..
. ;12) IIpH Z =X + iylsha)xi ::; lch zl ::; ch x,lshxi ::; sh zl ::; ch x;HaHTH MHO)l(ecrna ToqeK Z, B KOTOpbIX cpyHKUHH sh z, ch znpHHHMaIDT ;::i:eHCTBHTeJibHbie 3HaqeHH5!;6) HaHTH MHO)l(ecrna ToqeK Z, B KOTOpbIX cpyHKUHH sh z, ch znpHHHMaIDT q11CTO MHHMbie 3Haqemrn;13) LJ:OKa3aTb cpopMyrrhI: chz =ch z, shz =sh z .3.29.PaccMOTpHM MHoroqJieHP ( z) = a0 z + a 1z11-! + ... + aII1111,a0:#0 ;a) 5!BJI5!eTc5! JIH P,,(z) HenpepbIBHOH cpyttKUHeH: Ha6) BepHO JIH, qTQ3.30 .
.[.(oKa3aTb,C?P,,(z) = P,,(z) ?qrnlim P,, (z) =oo•: --+oo3.31. IIoKa3aTb, qTo ypaBHetttte P,,(z)TOM KpaTHOCTH.= 0 HMeeT n KOpHeii c yqe-3.32. PaccMorpHM pau110HaJibHYID cpyHKUHID R(z);a) noKa3aTb, qToR(z) HMeeT Ha C He 6oJiee KOHeqHoro q11cJiaToqeK pa3pbIBa H cyruecrnyeT KOHeqHbIH HJIH 6eCKOHeqHbIH npe;::i:eJIlimR( z);.: --+ oo6) Beptto rr11, 'ITO R(z) = R(z)?3.33.
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flyCTbz E { z E C : Iarg z - 2kn In I::; a< 7i I 211 , k = 0, 1, 2, ... , n -1} .3.41. ,IJ:oKa3aTb, qTQ rrptt izl :::; R crrpaBe.LI.JIHBbI ttepaBettcTBajch zi :::; ch R,jsh zj :::; sh R,jcos zi :::; ch R,jsin zj :::; sh R.3.42. ,IJ:oKa3aTb, 'ITO .LI.JIJI MHOfO'!JieHaP,,( z ) = a 0 z11+ a 1z111-+ .. . + a11,a 0 ::t 0,cymecTByeT rnKoe R > 0, 'ITO B o6nacrn { z: izl > R} cpyHKUH.sIP,,(z) I z" paBHOMeptto tterrpepbrntta.3.43. ,IJ:oKa3aTb, 'ITO .LI.JI.sr pauttoHaJibttoii cpyttKUHH R(z) cyruecTsyeT TaKoe R > 0 H TaKoe HaTypaJibHOe m, 'ITO cpyHKUHJI R(z) I zpaBHOMeptto tterrpepbIBHa B o6nacrn { z: lzl > R} .11142I'llaea 33.44.BepHo mr yrnep)J(,UeH11e npe,ub1,uyruei1 3a,uacrn: mm: cpyHKUHHeP,,c:J, meP11 (z) - MHoroqneH CTeneH11n::::1.3.45. IToKa'.3aTb, qTQ nptt !al < 1 pauirnHaJibHa5! cpyHKU115!a-zf (z) = --_- 5!Blli!eTC5! ttenpepbIBHOM B 3aMKeyTOM Kpyre { z: lzl ::::: 1} 111-azOT06pa)KaeT B3a11MHO O.UH03Haqtto 3TOT Kpyr Ha ce65! .3.46.
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