Сборник задач по высшей математике 2 том
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r) sinz = e |
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2i (ei(X+iY) - e-i(X+iY)) = 2i (e iX - Y - |
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= ii (e-Y(cosx + isinx) - |
eY(cos(-x) + isin(-x))) = -~(cosx(e-Y - eY) + |
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zcosx· |
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T.e. u(x,y) = sinxchyj v(x,y) = cosyshy. |
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/I.1tSI. OaHHb/,X rjJYH'IC'U,ufJ. HafJ.mu ux oefJ.cmeume.llibHY'lO "tacm'b u(x, y) |
u .MHUMy'lO |
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"tacm'b v(x,y): |
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7.1.2. |
z. |
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7.1.3. |
iz. |
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7.1.4. |
(Z)2. |
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7.1.5. |
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2z + i. |
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7.1.6. |
z3. |
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7.1.7. |
Rez+ilmz. |
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7.1.8. |
z+z. |
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7.1.9. |
l+i |
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-- . , |
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7.1.10. |
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7.1.11. |
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Z2' |
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z- z' |
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7.1.12. |
eZ. |
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7.1.13. e-iz . |
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7.1.14. |
shz. |
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7.1.15. |
sin(2z). |
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7.1.16. |
cosz. |
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7.1.17. |
ilnz. |
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7.1.18. |
In(z2). |
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7.1.19. |
sh(z + i). |
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7.1.20. |
,lJ;JIf! gaHHoii <PYHKII,HH J(z), rge z = rei'P, HaiiTH IJ(z)1 H ArgJ(z): |
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a) J(z) = Z2 j |
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6) J(z) = t |
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B) J(z) = eZ • |
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o a) MMeeM Z2 = (rei'P)2 = r2 . ei .2'P. |
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TaK KaK |
lei'P I |
1 Mf! JIlo6oro |
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geiicTBHTeJIbHoro cp, |
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a Arg(ei'P) = cp + 27rk, k |
E Z, TO |
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IZ21 = r2, |
Arg(z2) = 2cp + 27rk, |
k E Z. |
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6) TaK KaK ~ = |
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= _1_. =!. ei'P, TO |
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rei'P |
re-''P |
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I~I= ~, |
Arg G) = cp + 27rk, |
k E Z. |
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B) IIoCKOJIbKY eZ = ex+ iy = eX . eiy = er cos 'P . eir sin 'P, |
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leZI=ercos'P, Arg(eZ)=rsincp+27rk, kEZ. |
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/I.1tSI. OaHHb/,X rjJYH'IC'U,UfJ. J(z) HafJ.mu |
IJ(z)1 |
u ArgJ(z): |
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7.1.21. |
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7.1.22. |
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7.1.23. |
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7.1.24. |
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7.1.25. |
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7.1.26.OnpegeJIHTb <PYHKIJ;HIO J(z), rge z = X + iy, eCJIH Re J(z) = X H
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ImJ(z) = -Yo |
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TaK KaK Z = X + iy, a z = X - |
iy, TO Z + z = 2x, z - |
z = 2iy, oTKyga |
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z+z |
z-z |
.z-z |
(1.2) |
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X = -2-' |
Y = ~ |
= -z-2-' |
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CJIe,llpBaTeJIbHO, J( Z) |
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z + Z . z - z |
z + z |
z - z |
- u |
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= x - zy = - 2 - - z· 2i |
= - 2 - - |
- 2 - = |
Z. nTaK, |
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J(Z) = Z. |
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7.1.27. |
Onpe.D:eJIHTb q,YHKU.HIO J(z), r.D:e z = x+iy, eCJIH Re J(z) = eX cosy |
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H 1m J(z) = eX sin y. |
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o I1MeeM: |
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J(z) = eX cosy + iex siny = eX (cos y + i siny) = eX. eiy |
= ex +iy |
= eZ • |
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,npyroit cnoco6: BOCnOJIb3yeMcH paBeHCTBaMH (1.2) H (1.1). Tor.D:a |
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Z+z |
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J(z) = eX cosy + i· eX siny = e-2- cos 2i + i· e-2- sin 2i = |
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z+z (1 (t.z-z |
t.z-Z) |
1(t.z-z |
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t.z-Z)) |
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= e-2- |
'2 |
e |
2'i" + |
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2'i" |
+ i . 2i |
e 2'i" - e- 2'i" |
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1 z+z (Z-Z |
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+ e |
z-z |
_Z-Z) |
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= '2e 2 |
e |
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+ e 2 |
2 - |
e 2 |
= '2e |
2 |
·2e 2 |
= eZ • |
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Onpeoe.!!um'b tPYH'lCqUto J(z), |
zoe z = x + iy, no 3aOaHH'btM Re J(z) = u(x, y) |
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u ImJ(z) = v(x,y): |
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7.1.28.u = -y, v = x. 7.1.29. u = x2 - y2, V = 2xy.
7.1.30. u = |
x , v = |
y . |
x2 |
+ y2 |
x2 + y2 |
7.1.31.u = chycosx, v = -shysinx.
7.1.32. |
RaitTH 3Ha'leHHe q,YHKU.HH Z - |
~ B TO'lKe 3 + 2i. |
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o I1MeeM: |
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-- . |
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3 - 2i |
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3 + 2z - |
3 + 2i = 3 - |
2z - |
(3 + 2i)(3 _ 2i) = 3 - 2z - 9 + 4 |
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= (3 - 13) + i (-2 + 13) = ~~ - |
i· i~. |
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7.1.33.BbI'lHCJIHTb 3Ha'leHHe Z5 B TO'lKe
.;2 ..;2 zO=T+zT'
3anHcaTb OTBeT B anre6paH'leCKoit, TpHroHOMeTpH'leCKoit H nOKa3aTeJIbHoit q,0PMax.
o TaK KaK BbI'lHCJIeHHe 3Ha'leHHH
(V; +iV;r
Henocpe.D:CTBeHHo B anre6paH'leCKoit q,opMe .D:OBOJIbHO TPY.D:OeMKO, 3anHmeM
.'11"
'iHCJIO Zo B nOKa3aTeJIbHoit q,0pMe: Zo = 1· et '4 (CM. pHC. 102); OTCIO.D:a CJIe-
.D:YeT, 'ITO
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i 5'11" |
511'· . 511' |
.;2 ..;2 |
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zo= |
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·e 4 |
4 =cosT+ZSlllT=-T-ZT=-zo. |
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y
x
Puc. 102
7.1.34.Bbl'lHCJIHTb sin(1I" + i).
Q TIOJIb3YflCb q,opMYJIaMH (1.1), nOJIY'IHM
sin(1I"+i) = ii(ei (71+i) _e- i(7I+i)) = ii(e-Hi7l-el-i7l) =
ii (e-1 ei71 - |
e1 e-i7l ) = |
ii (e- 1 (cos 11" + i sin 11") - e1 (cos (-11") + i sin( -11"))) |
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=-i·sh1. |
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=2i(e- 1 .(-1+0)-e1 .(-1+0))=i· e |
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B'bt"tuc.IIum'b |
3Ha"teHWI |
tPYH'lCqUU J (z) 6 mO"t'ICax |
ZI, |
z2' B |
3aoa"tax 7.1.37- |
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7.1.38 om6em 3anUCam'b 6 nO'ICa3ame.ll'bHoit, mpU20HO.Mempu"teC'lCoit U a.ll2e6pau"tec'lCoit tP0p.Max.
7.1.35.J(z) = Z2 - 2z + i, ZI = -2 + 3i, Z2 = 4 - 3i.
7.1.36.J(z) = ~ - 2i, ZI = 1 - i, Z2 = 4.
7.1.37. |
J(z) = |
1;1' ZI |
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= 2 + 2i, Z2 = 2ei I. |
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7 1 38 |
J() - |
7 |
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1 |
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71n2 ( |
11" .. 11") |
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z - |
z , ZI |
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2" - |
t 2""'Z2 - |
V~ |
cos "4 + t sm"4 . |
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7.1.39.J(z) = eZ, ZI = 1 +i, Z2 = In2 -101I"i.
7.1.40.J(z) = chz, ZI = ~i, Z2 = In3 + i~.
7.1.41.J(z) = In(iz), ZI = -1, Z2 = 1.
7.1.42.J(z) = COSZ, ZI = 211" - i, Z2 = 211"i.
p'.IISI. OaHH'btX tPYH'lCqUit J(z), 20e z = x + iy, Haitmu ux oeitcm6Ume.ll'bHY?O "tacm'b u(x,y) u .MHU.MY?O "tacm'b v(x,y):
7.1.43. |
iz 2. |
7.1.44. |
(Z)3 + 2i - 1. |
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7.1.45. |
Re(z2) + i Im( (Z)2). |
7.1.46. |
z+1 |
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z - i' |
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z 'Iz -II- |
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7.1.47. |
7.1.48. |
i cos(z - i). |
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7.1.49. |
sh(iz2 ). |
7.1.50. |
7.1.51. |
tgz. |
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4.!I.R aa'H'H,'btX fPY'H'IC'Il,u'fJ, J(Z), aae z = rei<p, 'Ha'fJ,mu IJ(z)1 u ArgJ(z):
7.1.52. |
oz, r,ll;e 0 E lR. |
7.1.53. z . pe iO , r,ll;e p, (J E lR. |
7.1.54. |
1 |
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n' r,ll;e n E N. |
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z |
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Onpeae.n.um'b fPY'H'IC'Il,u?O J(z), aae z = x + iy, no 3aaa'H'H'bt.M ReJ(z) = u(x,y) u ImJ(z) = v(x,y):
7.1.55. |
x 2 |
- y2 |
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u= |
+ y2)2' |
v= -- |
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(x2 |
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2xy' |
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7.1.56. |
U = x 3 - |
3xy2, |
V = y3 - 3x2y. |
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7.1.57. |
x 2 + y2 + 1 |
x 2 + y2 - 1 |
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U=X |
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+ y2 |
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x2 + y2 |
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x2 |
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BwtUc.!!um'b 3'Ha"l.e'HUe fPY'H'IC'Il,UU J(z) 6 |
mO"l.'lCe Zo: |
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7.1.58. |
J(z) = z + 11, Zo = -2 + i. |
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z- |
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7.1.59. |
J(z) = (Z)2. Imz, Zo = 1- 2i. |
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7.1.60. |
J(z) = 16 , Zo = 1 - |
iV3. |
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z |
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7.1.61.J(z) = 2i sin 2iz, Zo = -l.
7.1.62.J(z) = sh(z + i), Zo = 2 - i.
KOHTponbHble Bonpocbl M 60nee cnmKHble 3aW-HMH
7.1.63. MorYT JIH Y ,ll;BYX pa3JIHqHbIX cPYHKII.Hit KOMnJIeKCHOro nepeMeH-
Horo 6bITb
a) pa3JIHqHbIe ,ll;eitcTBHTeJIbHbIe qacTH H O,ll;HHaKOBble MHHMble qaCTH;
6) O,ll;HHaKOBbIe ,ll;eitcTBHTeJIbHble qaCTH H pa3JIHqHble MHHMble qaCTH;
B) O,ll;HHaKOBbIe ,ll;eitcTBHTeJIbHbIe qaCTH H O,ll;HHaKOBble MHHMble qaCTH;
r) pa3JIHqHble ,ll;eitcTBHTeJIbHble qaCTH H pa3JIHqHble MHHMble qaCTH?
7.1.64. BepHO JIH, 'ITOeZ '"0 npH JIl060M z E C? 7.1.65. PemHTb YPaBHeHHe sinz = O.
7.1.66. CYIII.eCTBYIOT JIH TaKHe TOqKH z E C, 'ITOIcos zl > I? 7.1.67. BepHo JIH, 'ITOcPYHKII.HH sin z He OrpaHHqeHa Ha C?
7.1.68. HaitTH Bce TaKHe TOqKH z E C, B KOTOPbIX 3HaqeHHH cPYHKII.HH eZ qHCTO MHHMble.
443
7.1.69.B03MOlKHO JIH O,!l;H03HaqHO 3a,r:J;aTb <PYHKll,HIO f(z), eCJIH H3BeCTHO, 'ITO
a) If(z)1 = Izl, Ref(z) = Rez (npH Rez > 0);
6) Argf(z) = Argz, Ref(z) = Rez (npH Rez > O)?
§ 2. AHAnLt1TLt1'-1ECKLt1E«1»YHKLI,Lt1Lt1
TIYCTb <PYHKI1,HH J(z) onpe,ll;eJIeHa B HeKoTopoil: OKpeCTHOCTH TOqKH Zo KOM-
nJIeKcHoil: nJIOCKOCTH.
~ IIpoU360ihl,o(J, !'(zo) <PYHKI1,HH J(z) B TOqKe Zo H83bIBaeTCH npe,ll;eJI
r |
J(zo + ~z) - |
J(zo) - |
J'( |
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l>.!~o |
~z |
- |
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Zo, |
eCJIH OH CYII1,eCTByeT H KOHeqeH.
~ECJIH CYII1,eCTByeT npOH3BO,ll;HaH !'(zo), TO <PYHKI1,HH J(z) Ha3bIBaeTCH aurjJrjJe-
pe'HtjupyeMO(J, B TOqKe ZOo ~
MHOlKeCTBO D TOqeK pacmHpeHHoil: KOMllJIeKcHoil: nJIOCKOCTH IC U {oo} Ha3bl-
BaeTCH 06J1aCm'b1O, eCJIH
1)MHOlKeCTBO D OTKPblTO, T. e. ,!l;JIH KalK,ll;Oil: TOqKH, npHHa,!l;JIelKaII1,eil: D, cyII1,e- cTByeT oKpecTHocTb 3TOil: TOqKH, npHHa,!l;JIelKaII1,aH D;
2)MHOlKeCTBO D CBH3HO, T. e. JII06ble ,ll;Be TOqKH H3 D MOlKHO COe,ll;HHHTb HenpepblBHoil: KPHBOil:, Bce TOqKH KOTOPOil: npHHa,!l;JIelKaT D.
~<l>YHKI1,HH J(z) Ha3bIBaeTCH a'Ha./Iumu'l.eclGo(J, 6 o6J1acmu D, eCJIH OHa ,ll;H<p<pe-
peHI1,HpyeMa B KalK,ll;Oil: TOqKe 3TOil: 06JIacTH. |
~ |
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<l>YHKI1,HH J(z) Ha3blBaeTCH a'HaJlumu'l.ecIGo(J, 6 |
mO'l.IGe zo, eCJIH OHa ,ll;H<p<pepeH- |
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I1,HpyeMa B KalK,ll;Oil: TOqKe HeKoTopoil: OKpeCTHOCTH TOqKH ZOo |
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TeopeMa 7.1. .oIlR Toro, 'Ho6blcl>YHK~~R |
J(z) = u(x, y) + iv(x, y) (z = x + iy) |
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6bllla A~cI>cI>epeH~~pyeMa |
B TO'lKeZo = x() + iyo, AOCTaTO'lHO, 'lTo6bl |
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1) 'laCTHblenpO~3BOAHble |
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au(x, y) |
au(x,y) |
av(x,y) |
av(x, y) |
ax |
ay |
ax |
ay |
cy~ecTBoBall~ ~ 6blll~ HenpepblBHbl B HeKoTopoili OKpeCTHOCT~ TO'lK~ ,(xo, yo) (KaK cl>YHK~~~ AByX AeilicTB~TenbHblx nepeMeHHblX x ~ y);
2) B TO'lKe(xo, Yo) 6blll~ BblnOIlHeHbl ycnoBVl1I KOWVI-PVlMaHa
au(x,y) |
av(x,y) |
au(x, y) |
av(x, y) |
ax |
=-:,....:..::..:. |
ay |
(2.1) |
ay |
ax |
3aMeTHM, qTO yCJIOBHH KomH-PHMaHa HBJIHIOTCH Heo6xo,ll;HMblMH ,!l;JIH ,ll;H<p<pepeHI1,HpOBaHHH <PYHKI1,HH J(z) B TOqKe Zo = xo + iyo.
444
B rrOJIHpHhIX KOOp,o;HHaTax (r, rp) yCJIOBHH KOIlIH-PHMaHa 3arrHChIBaIOTCH CJIe- !1YIOIII;HM 06pa30M:
au(r, rp) |
1 av(r, rp) |
av(r,rp) |
1 au(r, rp) |
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-a~r:"":"":'" |
=;: arp |
or |
= -;: arp |
(2.2) |
ECJIH cYIII;ecTByeT rrpoH3Bo,o;HaH J' (z), TO ee MO)KHO 3arrHcaTb O,o;HHM H3 CJIe.o;y- IOIII;HX crroc060B:
HJIH
J'(z) = !. (au + iaV) = ! |
(av _ iaU) . |
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z |
or or |
z |
arp arp |
,[VIH rrpoH3Bo,o;HhIX OT <PYHKII;Hit KOMrrJIeKcHoro rrepeMeHHoro HMeIOT MeCTO rrpaBHJIa, aH8JIOrH'IHhIecooTBeTcTBYIOIII;HM rrpaBHJIaM ,o;JIH rrpoH3Bo,o;HhIX OT <PYHKII;Hit ,o;eitcTBHTeJIbHOrO rrepeMeHHoro. A HMeHHO: eCJIH B TO'lKe z CYIII;eCTBYIOT rrpOH3Bo,o;HhIe J'(z) H g'(z), TO CYIII;eCTBYIOT H rrpOH3Bo,o;HhIe (C. J(z))', (J(z) ± g(z))', (J(z)· g(z))', (J(z)/g(z))', rrpH'IeMBbIIIOJIHHIOTCH CJIe.o;yIOIII;He paBeHCTBa:
(C· J(z))' = C· J'(z)', r,o;e C E C,
(J(z) ± g(z))' = J' (z) ± g' (z),
(J(z). g(z))' = J'(z)· g(z) + J(z)· g'(z),
( J(Z))' |
J'(z)· g(z) - J(z)· g'(z) (rrpH g(z) #- 0). |
g(z) |
l(z) |
ECJIH <PYHKII;HH J(z) - aH8JIHTH'IeCKaHB 06JIacTH D, TO ee ,o;eitcTBHTeJIbHaH |
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'1acTb u( x, Y) H MHHMaH |
'1aCTb v( x, Y) HBJIHIOTCH <PYHKII;HHMH, ZapMOH.U"'I.eCICUMU |
B D. 3TO 3Ha'lHT,'ITOY K~,o;Oit H3 <PYHKII;Hit u(x, Y) H V (x, Y) CYIII;eCTBYIOT HerrpephIBHhIe B D '1acTHhIerrpOH3Bo,o;Hble 2-ro rropH,o;Ka, H ,o;JIH Ka)K,o;oit H3 HHX BepHo ypa6H.eH.Ue JIanllaca
r,o;e 6. - orrepaTop JIarrJIaca (CM. c. 243). ECJIH <PYHKII;HH u(x, y) (<pYHKII;HH v(x, y))
HBJIHeTCH rapMOHH'IeCKoit B HeKoTopoit |
06JIacTH D (Bo06III;e rOBopH, O,o;HOCBH3- |
HOit5 ), TO CYIII;ecTByeT aH8JIHTH'IeCKaHB |
D <PYHKII;HH J(z) C ,o;eitcTBHTeJIbHoit '1a- |
CTbIO u(x, y) (COOTBeTCTBeHHO, C MHHMOit '1aCTbIOv(x, y)), orrpe,o;eJIHeMaH C TO'lHO-
CTb~ ,0;0 rrOCTOHHHoro CJIaraeMoro.
7.2.1. HaitTH TO'lKH,B KOTOPhIX Cyrn;ecTByeT npOH3BO,n:HaJI <PYHKll,HH eZ ,
H BhI'IHCJIHTb9Ty npOH3BO,n:HYIO.
5To eCTb OrpaHHQeHHoit 3aMKHYToit HecaMonepeceKalOIIIeitcH JIHHHeit. 06JIacTH, onUCbI-
BaeMbIe B npuBoAuMbIX AaJIee 3a.D,aQax, HBJIHIOTCH OAHOCBH3HbIMU.
445
a TaK KaK (CM. 3a,u.a'lY7.1.1 B) eZ = eX cos y + i . eX siny, TO
u(x, y) = eX COS y, v(x, y) = eX sin y.
HaiI:.D:eM 'IaCTHblerrpOH3BO.D:Hble g~, g~, g~, g~ H BbUICHHM, B OKpeCTHO-
CTj(X KaKHX TO'leKOHH cym;eCTBYIOT H HerrpepbIBHbI, a TaK:lKe B KaKHX TO'lKaX rrJIOCKOCTH BbIIIOJIHj(IOTCj( YCJIOBHj( KOIlIH-PHMaHa (2.1):
au = .Q. (eX cos y) = eX cos y |
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av = .Q. (eX sin y) = eX cos y |
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ax |
ax |
ay |
ay |
, |
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T. e. g~ = g~ .D:JIj( JIlO6bIX .D:eiI:cTBHTeJIbHbIX x |
H y, H 3TH 'IaCTHblerrpOH3BO.D:- |
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Hble HerrpepbIBHbI BO Bceil: rrJIOCKOCTH ]R2; KpOMe Toro, |
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au = .Q. (eX cos y)= _ex sin y |
av = .Q. (eX sin y) = eX siny |
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ay |
ay |
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ax |
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T. e. g~ = - g~ .D:J!j('JII06bIX.D:eil:cTBHTeJIbHbIX x H y, H |
3TH 'IaCTHblerrpOH3- |
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BO.D:Hble HerrpepbIBHbI BO Bceil: rrJIOCKOCTH ]R2.
TaK KaK YCJIOBHj( KOIlIH-PHMaHa (2.1) BbIIIOJIHj(IOTCj(.D:J!j( JIlO6oil: rrapbI
.D:eil:cTBHTeJIbHbIX'IHCeJI(x, y), H 'IaCTHblerrpOH3BO.D:Hble g~, g~, g~, g~ CY- m;eCTBYIOT H HerrpepbIBHbI B OKpeCTHOCTH JIlO6oil: TO'lKH(x, y), TO rrpOH3BO.D:-
HM f' (z) |
cym;ecTByeT B JIlO6oil: TO'lKez = x +iy KOMrrJIeKcHoil: rrJIOCKOCTH Co |
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HaiI:.D:eM 3Ty rrpOH3BO.D:HYIO: |
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l'(z) |
= g~ + i |
g~ = eX cos y + i . eX sin y = eX (cos y + i sin y) = eZ. |
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IhaK, f'(z) = (e z )' |
= eZ |
(Vz E C). |
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7.2.2. |
YKa3aTb 06JIacTb .D:H<p<pepeHIJ;HpyeMocTH <PyHKIJ;HH J(z) = Z H BbI- |
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'IHCJIHTbrrpOH3BO.D:HYIO. |
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a TaK KaK Z = X - |
iy, TO u(x,y) = x, v(x,y) = -y H |
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au = ax = 1 |
av = a( -y) = -1 |
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ax |
ax |
'ay |
ay |
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.D:JIj( JIlO6bIX .D:eil:cTBHTeJIbHbIX x |
H |
y. CJIe.D:OBaTeJIbHO, g~ = 1 =I -1 = g~, |
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H rrepBoe H3 .D:ByX YCJIOBHiI: KOllH-PHMaHa (2.1) He BbIIIOJIHj(eTCj( HH .D:J!j(
KaKOil: rrapbI .D:eil:cTBHTeJIbHbIX 'IHCeJI(x, y). 3Ha'lHT,<PYHKIJ;Hj( |
J(z) = Z He |
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.D:H<p<pepeHIJ;HpyeMa HH B KaKoil: TO'lKez E Co |
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7.2.3. |
Hail:TH TO'lKH,B KOTOPbIX cym;ecTByeT rrpOH3BO.D:HM |
<PyHKIJ;HH i, |
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H BbI'IHCJIHTb3Ty rrpOH3BO.D:HYIO. |
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a Cnoco6 1. TaK KaK |
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J(z) = 1 = _1_ = |
x - iy |
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x |
+ iy |
(x + iy)(x - iy) |
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TO |
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u(x,y)= |
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v(x,y) = - 2 |
y |
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2' |
2· |
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+y |
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+y |
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446
lIafi,ll;eM qaCTHble rrpOH3BO,ll;Hble ~~, ~~, ~~, ~~ H BbIHCHHM, B OKpeCTHo-
CTHX KaKHX TOqeK OHH Cyrn;eCTBYIOT H HerrpepbIBHbI, a TaIOKe B KaKHX TOqKax BbIII031HHIOTCH yC310BHH KorrIH-PHMaHa (2.1):
aX.(X2+y2)_.Q.(x2+y2)·x |
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~~ = tx ( X2: y2 ) = "","a,:;;,x---(-X-2-+---""'-~~;)""'2---- |
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(x 2 + y2) - 2x . X |
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(x2 + y2)2 |
ay . (x2 + y2) _ |
.Q.(X2 + y2) . Y |
ay |
ay |
~--------~--~~-------=
(x2 + y2)2
(x 2 + y2) - 2y . Y
(x2 + y2)2
T. e. ~~ = ~~ ,ll;31H 31I060fi rrapbI ,ll;efiCTBHTe31bHbIX qHCe31 (x, y), eC31H x 2+y2 =I
8TH qaCTHble rrpOH3BO,ll;Hble cyrn;eCTBYIOT H HerrpepbIBHbI B OKpeCTHOCTH KaJK,ll;ofi TOqKH rr310CKOCTH 1R?, 3a HCK31lOqeHHeM TOqKH (0,0) . .naJIee,
2xy
av a( |
y) |
ax = ax |
- x 2 + y2 |
T. e. ~~ = - ~~ AJIH 31I060fi napbI ,ll;efiCTBHTe31bHbIX qHCe31 (x, y), eC31H
8TH qaCTHble npOH3BO,ll;Hble cyrn;eCTBYIOT H HerrpepbIBHbI B OKpeCTHOCTH KaJK,ll;ofi TOqKH H3 ]R2, 3a HCK31lOqeHHeM TOqKH
TaK KaK yC310BHH KorrIH-PHMaHa (2.1) BbIII031HHIOTCH AJIH 31I060fi rrapbI
,ll;eiI:cTBHTe31bHbIX qHCe31 (x, y), KpOMe napbI (0,0), H qacTHble rrpOH3BO,ll;Hble
au au av av
ax' ay' ax' ay cyrn;ecTBylOT H HerrpepbIBHbI B oKpecTHocTH 31I060fi TOqKH
113 ]R2 \ {(0, O)}, TO npOH3BO,ll;HaH l'(z) cyrn;ecTByeT B 3110601\ TOqKe z = x +i Y
KOMrr31eKcHol\ rr310CKOCTH C, 3a HCK31lOqeHHeM TOqKH z = O.
447
Hail:.D:eM 9TY npOH3BO.D:HYIO:
y2 _ X2 + i . 2xy |
= |
(X2 + y2)2 |
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1 |
1 |
(x + iy)2 |
- Z2· |
1)' |
1 |
l1TaK, J'(z) = (Z |
= - z2 Vz E C, z #- O. |
KaK BH.D:HO H3 pemeHHH, HaXO)l{.D:eHHe npOH3BO.D:Hoil: TaKHM cnoc06oM He BnOJIHe O'"leBH.D:HO. B .D:aHHOM npHMepe 60JIee IJ;eJIeC006pa3Ho BbI'"IHCJIeHHe npOH3BO.D:Hoil: B nOJIHpHbIX KOOp.D:HHaTax.
TaK KaK
J() |
1 |
1 |
1 |
-i,n |
1 ( |
.. ) |
1 |
. ( |
1. |
) |
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z =-=-,-=-·e |
Y=-cos<p-zsm<p =-cos<p+z |
--sm<p, |
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ret'" |
r |
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u(r,<p) = ¥cos<p, |
v(r,<p) = -¥sin<p. |
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Hail:.D:eM '"IaCTHblenpOH3BO.D:Hb1e ~~ H ~~: |
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-OU |
= -0 (1-cos<p) = --cos<p,1 |
-OV |
= -0 ( |
--1sm<p. ) |
= -1sm<p.. |
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or |
or |
r |
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r2 |
or |
or |
r |
r2 |
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BbI'"IHCJIeHHe'"IacTHbIXnpOH3BO.D:H bIX Z~ H Z~, a TaK:>Ke npoBepKY YCJIo- |
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BHiI: KomH-PHMaHa (2.2) npe.D:OCTaBJIHeM '"IHTaTeJIIO. |
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HaiI:.D:eM npOH3BO.D:H YIO <PYHKIJ;HH J(z) = |
i: |
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f'(z) |
= r. (ou + iOV) = r. |
(_-.1 cos<p + i·-.1 sin<p) = |
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z |
~ |
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z |
~ |
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1 ( |
cos <p - |
.. ) |
1 |
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-i", |
= - , |
1 |
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= - - |
zsm <p = - - , -- . e |
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,= -- |
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zr |
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ret'" . r |
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ret'" . ret'" |
Z2 . |
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IIOJIY'"IHJIHTOT)I{e pe3YJIbTaT, '"ITOH npH pemeHHH nepBbIM cnoco6oM. |
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7.2.4. |
YKa3aTb 06JIaCTb |
.D:H<p<pepeHIJ;HPyeMocTH <PYHKIJ;HH |
J(z) |
= zn |
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(n E N) H Hail:TH J'(z). |
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Q TaK KaK |
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J(z) = zn = (rei",)n = rnein", = rn(cosn<p + isinn<p), |
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TO |
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u(r, <p) = rn cosn<p, |
v(r, <p) = rn sin n<p. |
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Hail:.D:eM '"IacTHblenpOH3BO.D:Hb1e ~~, g~, Z~, Z~ H BbIHCHHM, r.D:e OHH cy-
m;eCTBYIOT H HenpepbIBHbI, a TaK:>Ke B KaKHX TO'"lKaxBbIllOJIHHIOTCH YCJIOBHH KOIIIH-PHMaHa (2.2):
OU = ~(rncosn<p) =nrn-1cosn<p, |
OV = ~(rnsinn<p) =nrncosn<p, |
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or or |
o<p o<p |
, |
448
T. e. ~~ = |
~g~ )I.JlfI JIlo6oil: rrapbI |
qHCeJI (r, <p), |
r > OJ 3TH |
qaCTHble rrpOH3- |
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BOp;Hble CymecTByIOT H HerrpepbIBHbI B JII06oil: TOqKe (r, <p) rrpH r |
> O. .naJIee, |
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av = ~(rn sinn<p) = nrn- 1sinn<p, |
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or |
or |
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T. e. ~~ = |
- ~g~ )I.JlfI JII06oil: rrapbI qHCeJI (r, <p), |
r > 0; 3TH qacTHble rrpOH3- |
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Bop;HbIe cymeCTBYIOT H HerrpepbIBHbI B JII06oil: TOqKe (r, <p) rrpH r |
> O. |
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TaK KaK yCJIOBHfI KomH-PHMaHa (2.2) BbIIIOJIHflIOTCfI )I.JlfI JII06oil: rrapbI |
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t.lHceJI |
(r,<p) (rrpH r |
> 0) H |
qaCTHbIe rrpOH3BO,lJ;Hble ~~, g~, |
~~, g~ cyme- |
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CTBYIOT H HerrpepbIBHbI B oKpecTHocTH KroK,lJ;Oil: TOqKH (r,<p) |
(rrpH r > 0), |
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TO rrpOH3BO,lJ;HafI f'(z) cymecTByeT B JII06oil: TOqKe z = rei'P |
KOMrrJIeKcHoil: |
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ITJIOCKOCTH C (3a HCKJIIOqeHHeM, MOlKeT 6bITb, TOqKH Z = 0). |
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Hail:,lJ;eM 3Ty rrpOH3BOP;HYIO: |
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f'(z) = (zn), = r. (au +i av ) = -2:,-(nrn- 1 cosn<p + inrn- 1 sinn<p) |
= |
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z |
or |
or |
re''P |
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n-l |
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~'P. |
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= ~(cosn<p + isinn<p) |
= nrn- 1 . ~ = nrn- 1 • |
e,(n-l)'P = |
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e''P |
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e''P |
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= nZn- |
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)n-l |
1 |
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= n· ( re''P |
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IfTaK, |
f'(z) = (zn), |
= nzn-l Vz |
E C (z =I- 0). TIpoBepKY YCJIOBHiI: KOmH- |
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PHMaHa (2.1) B TOqKe Z = |
0 H BbIqHCJIeHHe rrpOH3BO,lJ;Hoil: <PYHKll,HH B 3TOil: |
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TOqKe rrpep;OCTaBJIfleM qHTaTeJIIO. |
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,4.M1 oaHHoti tPYH'IC'I.4UU J (z) |
Y'lCa3am'b mO~'lCU, 6 'lComOp'btX cyw,eCm6yem |
npo- |
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U3600Ha.R f'(z), U Hatimu npOU3600HY'lO 6 3mUX l1W~'lCax: |
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7.2.5. |
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J(z) = iz. |
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7.2.6. |
J(z) = Z + 2i. |
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7.2.7. |
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J(z)=iz 2 -3z+1. |
7.2.8. |
J(z) = zRez. |
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7.2.9. |
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J(z) = Z6. |
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7.2.10. |
1 |
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J(z) = 3' |
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7.2.11. |
J(z) = In(z2). |
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7.2.12. |
z |
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J(z) = Ln(z2). |
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7.2.13. |
J(z) = chz. |
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7.2.14. |
J(z) = sini. |
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7.2.15. |
J(z) = sin(z + 2i). |
7.2.16. |
J(z) = cos(iz). |
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7.2.17. |
Hail:TH MHO:>KeCTBO TOqeK, B KOTOPbIX <PYHKll,HfI v(x, y) = 2xy - 3 |
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y,lJ;OBJIeTBOpfleT yCJIOBHIO 6.v = O. Orrpe,lJ;eJIHTb, |
cymecTByeT JIH |
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aHaJIHTHqeCKafI B HeKoTopoil: 06JIaCTH D <PYHKll,HfI J(z) (z=x+iy), |
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P;JIfI KOTOPOil: 1m J = V. ECJIH TaKafI <PYHKll,HfI J(z) cymecTByeT, TO |
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Hail:TH ee. |
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Q 1. Hail:p;eM qaCTHble rrpOH3Bop;Hble: |
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g~ = 2y, |
g~ = 2x, ~:~ = 0, |
~~~ = O. |
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IS CooPHHK 311JUl~ no Bb,cweA M8TCM8THKe. 2 KYPc |
449 |