Παν. Κρήτης

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Μιχάλης Παπαδημητράκης Μιγαδική Ανάλυση Τμήμα Μαθηματικών Πανεπιστήμιο Κρήτης

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Transcript of Παν. Κρήτης

  • 1 . 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 bC, Riemann. . . . . . . . . . . . 5

    2 C. 112.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    3 . 313.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 . . . . . . . . . . . . . . . . . . . . . 373.4 . . . . . . . . . . . . . . . . . . . . 39

    4 414.1 . . . . . . . . . . . 414.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 . . . . . . . 494.4 . . . . . . . . . . . . . . . 52

    5 . 575.1 . . . . . . . . . . . . . . . . . . . . . . 575.2 Cauchy - Riemann. . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3 . . . . . . . . . . . . . . . . . . . . . . . . . 665.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.6 n- n- . . . . . . . . . . . . . . . . . . . . . . . . 80

    6 Cauchy. 906.1 Cauchy . . . . . . . . . . . . . . . . . . . . . . . . 906.2 Cauchy . . . . . 966.3 . . . . . . . . . . . . . . . . . . . . . 1026.4 Cauchy

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

    7 Cauchy. 1157.1 . . . 1157.2 Cauchy. . . . . . . . . . . . . . . . . . . . . . . . . . 1187.3 ( ) . . . . . . . . . . . . . . . . . . 124

    i

  • 8 Taylor Laurent. 1268.1 . . . . . . . . . . . . . . . . . . . . . . . . 1268.2 . . . . . . . . . . . . 1328.3 . . . . . . . . . . . . . . . . . . . . . . . . 1358.4 Taylor Laurent. . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1548.7 . . . . . . . . . . . . 158

    ii

  • iii

  • 1

    .

    1.1 .

    C . z

    z = (x; y) = x+ iy x; y 2 R:

    x; y; u; v; ; ; a; b; ; , , z; w , , : z = x+ iy, w = u+ iv.

    C , ,

    (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2);

    (x1 + iy1) (x2 + iy2) = (x1x2 y1y2) + i(x1y2 + x2y1): i :

    i2 = 1:

    , i z2 + 1 = 0. , , R.

    C R. z < w z w z; w < R.

    z = x+ iy

    z = x iy:

    z = x+ iy , ,

    Re z = x; Im z = y:

    , z = x+ iy

    jzj =px2 + y2:

    :

    z = z; z1 z2 = z1 z2; z1z2 = z1 z2; z1/z2 = z1/z2;

    Re z = (z + z)/2; Im z = (z z)/(2i);Re (z1 + z2) = Re z1 + Re z2; Im (z1 + z2) = Im z1 + Im z2;

    1

  • jzj 0; jzj = 0 z = 0;jzj2 = zz; jzj = jzj;

    jz1z2j = jz1jjz2j; jz1/z2j = jz1j/jz2j; .

    :

    jRe zj jzj; jIm zj jzj; jzj jRe zj+ jIm zj;

    jz1 + z2j2 = jz1j2 + 2Re (z1z2) + jz2j2: jx1+x2j2 = jx1j2+2x1x2+ jx2j2 x1; x2 2 R.

    , :jz1j jz2j jz1 z2j jz1j+ jz2j:, z = x+ iy 6= 0.

    r = jzj > 0;

    0 2 (; ]

    cos 0 = xr ; sin 0 =

    yr :

    cos = xr ; sin =yr

    = 0 + k2 k 2 Z

    . 0 z = 0 + k2 k 2 Z z. z , ( ) 2. z , (; ]. Arg z 0 z arg z z:

    Arg z = 0; arg z = 0 + k2 = Arg z + k2 k 2 Z:

    , z 6= 0 2 R, arg z

    z = r(cos + i sin );

    r = jzj > 0. z = r(cos + i sin ) z., z 6= 0 .

    z = 0 . , z = 0 -.

    2

  • 1.1.1. z = x > 0 Arg z = 0 arg z = k2 k 2 Z. z = x < 0 Arg z = arg z = + k2 k 2 Z. z = iy y > 0 Arg z = 2 arg z =

    2 + k2 k 2 Z.

    z = iy y < 0 Arg z = 2 arg z = 2 + k2 k 2 Z. z = x + iy (, x; y > 0), Arg z (0; 2 ). z = x + iy (, x < 0, y > 0), Arg z (2 ; ). z = x+ iy (, x < 0, y < 0), Arg z (;2 ). z = x + iy (, x > 0, y < 0), Arg z (2 ; 0).

    (cos 1 + i sin 1)(cos 2 + i sin 2) = cos(1 + 2) + i sin(1 + 2)

    deMoivre:

    (cos + i sin )n = cos(n) + i sin(n)

    n 2 Z., z1; z2 6= 0

    arg(z1z2) = arg z1 + arg z2 (1.1)

    : 1 arg z1 2 arg z2, = 1 + 2 arg(z1z2) , , arg(z1z2) 1 arg z1 2 arg z2 = 1 + 2.

    , (1.1) , Arg (z1z2) = Arg z1 + Arg z2, z1; z2 6= 0.

    .

    1.1.1. Re (iz) = Im z, Im (iz) = Re z, Re (z z) = 0, Im (z + z) = 0.1.1.2. z z = z.

    1.1.3. jzj = z; jzj = iz;1.1.4. z1; z2 6= 0. jz1 + z2j = jz1j+ jz2j t > 0 z2 = tz1.

    1.1.5. z2 + z + 1 = 0, .

    1.1.6. jRe zj+ jIm zj p2jzj.1.1.7. jz3j 6= jz4j,

    z1+z2z3+z4

    jz1j+jz2jjjz3jjz4jj .1.1.8. jzj < 1, jIm (1 z + z2)j < 3.1.1.9. jz44z2+3j jzj = 2; jzj 2;

    3

  • 1.1.10. jz wj2 (1 + jzj2)(1 + jwj2).1.1.11. [] jz wj = j1 wzj jzj = 1 jwj = 1.[] jz wj < j1 wzj jzj; jwj < 1 jzj; jwj > 1.1.1.12. [] jz wj = jz wj Im z = 0 Imw = 0.[] jz wj < jz wj Im z; Imw < 0 Im z; Imw > 0.1.1.13. w z+z = w ; z z = w.1.1.14. (p3i),(1p3i);1.1.15. z 4 4 arg z;1.1.16. z arg(z2); arg(z3); arg(z4);

    1.1.17. J R arg z z Re z > 0. J ; Im z < 0,Im z > 0, Re z < 0 Re z > 0.

    1.1.18. :[] Arg (z1z2) = Arg z1 + Arg z2 < Arg z1 + Arg z2 .[] Arg (z1z2) = Arg z1 + Arg z2 + 2 2 < Arg z1 + Arg z2 .[] Arg (z1z2) = Arg z1 + Arg z2 2 < Arg z1 + Arg z2 2.1.1.19. z1; z2 6= 0. arg z1z2 = arg z1 arg z2 .1.1.20. de Moivre cos(3) = 4(cos )3 3 cos sin(3) =3 sin 4(sin )3.

    1.1.21. 1 + z + z2 + + zn =(

    zn+11z1 ; z 6= 1

    n+ 1; z = 1

    1+ r cos + r2 cos(2)+ + rn cos(n) r sin + r2 sin(2)+ + rn sin(n) r 0.1.1.22. l , z, z, z l. l; z, z, z, iz z, iz.1.1.23. fz : jzj < 1; jz ij < 1g, fz : jzj < 1; Im z > 0; jz ij > 1g.1.1.24. z :z + z = 1, i(z z) 2, 2i(z z) + jzj2 + 1 0, (2 i)z + (2 + i)z = 2, jz2 1j = 1,jzj+ Re z 1.1.1.25. z; w 6= 0, z, w Re (wz) = 0.1.1.26. [] ax + by = c, a; b 0. Re (wz) = c, () w 0 ( c ).[] w 6= 0, Re (wz) 0 Re (wz) 0 w w;[] w 6= 0, w Re (wz) = c. c (1;+1);[] w 6= 0 c1 c2, z c1 Re (wz) c2.

    4

  • [] w;w0 6= 0, Re (wz) = c, Re (w0z) = c0 t 2 R w0 = tw, c0 = tc.[] w;w0 6= 0, Re (wz) = c, Re (w0z) = c0 t 2 R w0 = tw.[] w;w0 6= 0, Re (wz) = c - Re (w0z) c0. ( ) ;

    1.1.27. > 0, z z1z+1

    = . - (0;+1);1.1.28. , z Re (z2) = . (1;+1); Im (z2) = .1.1.29. [] z1, z2, z3 z1z2z1z3 .[] z1, z2, z3, z4 z1z2z1z3

    z4z3z4z2 .

    1.1.30. w = 1+ip3

    2 . J R arg z z Re (wz) = 3. J ; Re (wz) = 3 Re (wz) = 3. Re (wz) = 3;1.1.31. J R arg z z jz 1j = 12 . J ; jz ij = 12 , jz + ij = 12 , jz + 1j = 12 .1.1.32. Re (zn) 0 n 2 N, z - .

    1.2 bC, Riemann. C , , 1,

    ,

    bC = C [ f1g: bC C.

    1 - z ( 0, ) . 1 - R. x 0 : , - +1, , 1. , . (, ) - . , , .

    , bC :1. : z +1 =1 1+ z =1.2. : 1 =1.

    5

  • 3. : z 1 =1 1 z =1.4. : z 1 =1 1 z =1 z 6= 0. ,1 1 =1.5. : 11 = 0

    10 =1.

    6. : z1 = 0 1z =1.

    7. :1 =1.8. : j1j = +1.

    1+1; 11; 0 1; 1 0; 11 ; 00 .

    10 =1. R 10 , x > 0 1x > 0, 1x +1, x < 0 1x < 0, 1x 1. , C R. C 1z z , jzj ( > 0), 1z , j1z j = 1jzj , , , 1z 1. 10 =1.

    b

    b

    b

    b

    z = x+ iy = (x; y; 0)

    A = (; ; )

    N = (0; 0; 1)

    O = (0; 0; 0)

    C = R2

    R3 S2

    1

    R3 C = R2 ( 1). A = (; ; ) R3 z = x + iy = (x; y) = (x; y; 0) C = R2. , - x- - y-, - C = R2.

    ,

    S2 = f(; ; ) 2 R3 j 2 + 2 + 2 = 1g

    R3. S2

    N = (0; 0; 1):

    z = x+iy C Nz N z. S2 N . .

    6

  • A = (; ; ) Nz S2. A = (; ; ) :

    !AN = t !zN t 2 R:

    , :

    0 = t(x 0) 0 = t(y 0) 1 = t(0 1)

    (1.2)

    A = (; ; ) S2 , :

    2 + 2 + 2 = 1: (1.3)

    A = (; ; ) Nz S2 (; ; ; t) (1.2) (1.3). , (1.2)

    = tx; = ty; = 1 t (1.4) (1.3)

    t2(x2 + y2 + 1) = 2t:

    t = 0 t = 2

    x2+y2+1:

    t = 0 (1.4)

    A = N = (0; 0; 1)

    t = 2x2+y2+1

    (1.4)

    A =

    2xx2+y2+1

    ; 2yx2+y2+1

    ; x2+y21

    x2+y2+1

    : (1.5)

    , , (1.2) (1.3) A, (1.5), , N , Nz S2.

    C 3 z = x+ iy 7! A = (; ; ) = 2xx2+y2+1

    ; 2yx2+y2+1

    ; x2+y21

    x2+y2+1

    2 S2 n fNg C S2 n fNg.

    -- S2 n fNg

    S2 n fNg 3 A = (; ; ) 7! z = x+ iy = 1 + i 1 2 C:

    C S2 n fNg , , S2 n fNg C.

    C ! S2 n fNg:

    7

  • 1.2.1. z = 0 S = (0; 0;1) S2 . z = x + iy , jzj > 1, A = (; ; ) S2, 0 < < 1, . ( = 1 N .) z = x + iy , jzj < 1, A = (; ; ) S2, 1 < 0, ., z = x+iy , jzj = 1, A = (; ; ) S2, = 0, . , z A.

    , . R2 R3 ( -

    ) : - ., R2 :

    0 jx x0j; jy y0j p(x x0)2 + (y y0)2 jx x0j+ jy y0j: (1.6)

    (x; y) (x0; y0), p(x x0)2 + (y y0)2 0

    (1.6) jxx0j, jyy0j 0 , , x x0 y y0. , x x0 y y0, jx x0j, jy y0j 0, (1.6)

    p(x x0)2 + (y y0)2 0 , , (x; y)

    (x0; y0). R3.

    , z = x+ iy z0 = x0+ iy0, x! x0 y ! y0, , ,

    2xx2+y2+1

    ! 2x0x02+y02+1

    ; 2yx2+y2+1

    ! 2y0x02+y02+1

    ; x2+y21

    x2+y2+1! x02+y021

    x02+y02+1

    , , A = (; ; ) z = x + iy A0 = (0; 0; 0) z0 = x0 + iy0., A = (; ; ) S2 n fNg A0 = (0; 0; 0) S2 n fNg ( A0 N , , 0 6= 1), ! 0 ! 0 ! 0, ,

    1 ! 010 ;

    1 ! 010 ;

    z = x+ iy A = (; ; ) z0 = x0+ iy0 A = (0; 0; 0).

    N . , z = x+ iy ( 0) +1, jzj ! +1,

    2xx2+y2+1

    ! 0; 2yx2+y2+1

    ! 0; x2+y21x2+y2+1

    ! 1: (1.7)

    ,

    0 2xx2+y2+1

    = 2jxjjzj2+1 2jzjjzj2+1 ! 0; 0 2yx2+y2+1 = 2jyjjzj2+1 2jzjjzj2+1 ! 0;8

  • 0 x2+y21x2+y2+1

    1 = 2x2+y2+1

    = 2jzj2+1 ! 0 jzj ! +1. (1.7) A = (; ; ) z = x+ iy N = (0; 0; 1)., A = (; ; ) S2 nfNg N = (0; 0; 1), ! 0 ! 0 ! 1,

    x2 + y2 = 2

    (1)2 +2

    (1)2 =2+2

    (1)2 =12(1)2 =

    1+1 ! +1

    , , z = x+ iy A = (; ; ) +1.

    .

    . z !1

    jzj ! +1. , -

    z 1 A = (; ; ) N = (0; 0; 1).

    , : 1 C bC N S2 n fNg S2. 1 bC N S2 .

    C ! S2 n fNg

    C [ f1g ! S2 n fNg [ fNg, , bC ! S2:

    , , bC S2 , , : bC S2 .

    S2 , Riemann.

    .

    1.2.1. [] R3 k+ l+m = n, () k; l;m 0. ( ) k; l;m; n S2; S2; S2 ; , S2 ( k; l;m; n) S2.[] l C bl = l[f1g bC. bC ,

    9

  • , C. C 1 . bC , , bC. bC bC - 1. bC; ( - : , , .)[] bC S2 N . bC S2 - N . , , bC S2.

    1.2.2. S2 bC:[] fz j jzj < 1g, fz j jzj = 1g, fz j jzj > 1g [ f1g.[] fz j 0g, fz j

  • 2

    C.

    2.1 .

    b

    b

    b

    b

    A

    Ac

    2

    D(z; r) = fw j jw zj < rg; C(z; r) = fw j jw zj = rg; D(z; r) = fw j jw zj rg

    , z r > 0.

    . D(z; r) r- z.

    A. Ac C nA A C.

    . z A r > 0 D(z; r) A. z A r > 0 D(z; r) Ac. z A r > 0 D(z; r) \ A 6= ; D(z; r) \Ac 6= ;.

    , [] - [] . , . 2 . , [] - [], .[] A A A Ac.

    11

  • [] A A , , : , , A. C : , A.[] A Ac. A Ac. A Ac.[] A A A Ac, A. : ( , ) A ( , ) Ac.

    2 . C . A Ac . , A Ac . , A ( Ac) A( Ac) . A ( Ac) .

    . z A r > 0 D(z; r) \A 6= ;. :

    [] A A. , A A.

    2 A .

    2.1.1. A = D(z; r) A = D(z; r). A z r. , A D(z; r) C(z; r), . : A D(z; r), A C(z; r) A D(z; r). A fw j jw zj > rg..

    Ao = fz j z [email protected] = fz j z AgA = fz j z Ag

    Ao A, A A @A A.

    2.1. [] @A = @(Ac).[] Ao = A n @A.[] A = A [ @A.[] (A)c = (Ac)o.

    . [] A Ac.[] z A, A

    12

  • A. Ao A n @A. , z A A, A. Ao A n @A.[] z A, A ( A) A. A A [ @A. , z A A, A A, A. A A [ @A.[] z A A Ac.

    . A . A .

    2 A A .

    2.2. [] A A \ @A = ;.[] A @A A.. [] A - . A .[] A , A, . A .

    2.1.2. .

    2.3. .

    .

    A .

    m A A.

    m Ac A.

    m Ac A.

    m Ac Ac.

    m Ac .

    , : .

    13

  • 2.4. [] A A.[] Ao A.

    . [] A A., A . z A r > 0 ( 3). z A, z0 2 A D(z; r). , D(z; r) , r0 > 0 D(z0; r0) D(z; r). , z0 A, w 2 A D(z0; r0) , , D(z; r).

    b

    b

    b

    z A

    z0 2 A

    w 2 A

    3

    r > 0 w 2 A D(z; r). z A, z 2 A. A A, A ., A A. B A B. z 2 A, A. r > 0 D(z; r) \ A 6= ;,, A B, D(z; r) \ B 6= ;. z B , B , z 2 B. z 2 A z 2 B, A B.[] , Ao A., Ao . z 2 Ao, A. r > 0 D(z; r) A. z0 2 D(z; r) r0 > 0 D(z0; r0) D(z; r) , , D(z0; r0) A ,, z0 A. z0 2 D(z; r) A, z0 2 Ao. D(z; r) Ao, z Ao. z 2 Ao Ao, Ao ., Ao A. B B A. z 2 B. B , z B, r > 0 D(z; r) B , , D(z; r) A. z A, z 2 Ao. z 2 B z 2 Ao, B Ao. 2.5. A , A .

    . A , A D(z; r). A A D(z; r) - A, A D(z; r), A .

    1.. r- 1 bC

    D(1; r) = fw j jwj > 1rg [ f1g 0 1r 1.

    14

  • r ( , , ) r-D(1; r) 1, r- D(z; r) z . -, r 0+ D(1; r) f1g.

    - A. r > 0 r-D(1; r) 1 A, 1 A bC. , , - A bC A C 1 A bC A C 1.

    @bCA = @A [ f1g; AbC = A [ f1g A -:

    , A, A - D(0; r), 1r - D(1; 1r ) 1 A. , 1 A - A bC A C A bC A C.

    @bCA = @A; AbC = A A :

    .

    2.1.1. [] ; C .[] , , .[] fz1; : : : ; zng, , ( ), ( ), , , - , .

    2.1.2. [] (0; 1) = fz 2 C j 0 < z < 1g C; , , (0; 1).[] [0; 1] = fz 2 C j 0 z 1g.[] fx+ iy jx; y 2 Qg.[] f 1n+iy jn 2 N; 0 y 1g f 1n+iy : n 2 N; 0 y 1g[fiy j 0 y 1g.2.1.3. [] z 2 A A r > 0 A \D(z; r) = fzg. A A.[] z A r > 0 (Anfzg)\D(z; r) 6=;. , z /2 A, z A A A. , z 2 A, z A A.

    2.1.4. [] . - .[] . .

    2.1.5. diamA = supfjz wj j z; w 2 Ag diamA - A.[] A B, diamA diamB.[] diamA = diamA.[] A diamA < +1.

    15

  • 2.1.6. d(z;A) = inffjz wj jw 2 Ag d(z;A) z - A. :[] d(z;A) = d(z;A).[] d(z;A) = 0, z 2 A.[] jd(z0; A) d(z00; A)j jz0 z00j.2.1.7. A , (@A)o = ;. A (@A)o = C.

    2.2 .

    (zn). (zn) z zn ! z > 0 n0

    n n0 jzn zj < , , zn 2 D(z; ). (zn) z D(z; ) z (zn) .

    , (zn) 1 zn ! 1 > 0 n0 n n0 jznj > 1 , , zn 2 D(1; ). (zn) 1 D(1; ) 1 (zn) .

    2.2.1. (2)n.

    R : (2)2k = 22k ! +1 (2)2k1 =22k1 ! 1., 1, (2)n = 2n ! +1. 2.2.2. (zn). jzj < 1, jzn 0j = jzjn ! 0, zn ! 0. jzj > 1, jznj = jzjn ! +1, zn !1. z = 1, zn = 1! 1., jzj = 1 z 6= 1 zn ! w. jznj = jzjn = 1 n, w 1 jwj = 1. zn ! w, zn+1 ! w, zn+1

    zn ! ww = 1. , zn+1

    zn = z n, z = 1 .:

    zn

    8>>>>>>>:! 0; jzj < 1! 1; z = 1!1; jzj > 1 ; jzj = 1; z 6= 1

    (zn). z = 0 z = 1, (zn) . 0 < r = jzj < 1 = Arg z, < z = r(cos + i sin ). zn = rn

    cos(n)+ i sin(n)

    zn+1 = rn+1

    cos(n+ )+ i sin(n+ )

    n.

    , zn 0 zn jj , > 0, , < 0. = 0, . , 6= 0, zn 0 0, , = 0, zn 0 [0; 1]. r = jzj > 1. zn +1. , 6= 0,

    16

  • zn 0 1, , = 0, zn 1 [1;+1]., jzj = 1 z 6= 1. zn C(0; 1) (, 0 0) jj , > 0, , < 0.

    . 2.6. zn = xn + iyn n z = x + iy. zn ! z xn ! x yn ! y..

    0 jxn xj; jyn yj jzn zj jxn xj+ jyn yj: (2.1) zn ! z, jznzj ! 0 (2.1) jxnxj ! 0 jyn yj ! 0 , , xn ! x yn ! y. , xn ! x yn ! y, jxnxj ! 0 jynyj ! 0, (2.1) jznzj ! 0, , zn ! z.

    : (zn) , . - 2.6 2.6. . zn ! z wn ! w, zn + wn ! z + w; zn ! z; zn wn ! z w; znwn ! zw; 1zn ! 1z ; znwn ! zw :,

    zn ! z; jznj ! jzj: z; w ( ) 1 . C R: zn ! 0, 1zn !1.

    , zn ! z, znk ! z (znk) (zn). C.

    2.7. Cauchy C .. (zn) Cauchy, jzn zmj ! 0 n;m ! +1. (2.1)

    0 jxn xmj; jyn ymj jzn zmj jxn xmj+ jyn ymj: (2.2) (2.2) jxn xmj ! 0 jyn ymj ! 0 n;m! +1, (xn) (yn) Cauchy. R (xn) (yn) 2.6 (zn).

    .

    2.2.1. zn ! z zn 6= 0 n z 6= 0.[] Arg z 6= , Arg zn ! Arg z.[] (1 + in), (1 in) (1 + (1)n in). ; ;[] Arg z = , (i) Arg zn ! , (ii) Arg zn ! , (iii) (zn) (zn0k) (zn00k ) Arg zn0k ! Arg zn00k ! .2.2.2. [] z A (zn) A zn ! z.[] 2.1.3.[] z A (zn) A zn ! z zn 6= z n.

    17

  • 2.3 .

    Bolzano - Weierstrass .

    2.1. .

    . jznj M n

    zn = xn + iyn n.

    jxnj M; jynj M n:

    Bolzano - Weierstrass , - (xnk) (xn) .

    xnk ! x:

    (ynk) (yn). (ynk) jynk j M k. Bolzano - Weierstrass , (ynkl ) (ynk) .

    ynkl ! y: xnk ! x,

    xnkl ! x: z = x+ iy

    znkl ! z:

    (znkl ) (zn).

    - .

    2.8. - 1.. (zn) (jznj) . , (jznk j) (jznj)

    jznk j ! +1, ,

    znk !1:

    2.1 2.8 bC.. K C K - K . , K (zn) K (znk) znk ! z z 2 K. 2.1. z A (zn) A zn ! z.

    18

  • . zn 2 A n zn ! z. r > 0 n0 n n0 zn 2 D(z; r). r > 0 r- D(z; r) z A , , z A., z A. n 2 N , zn, A 1n - D(z; 1n) z. , zn 2 A

    jzn zj < 1n n: (zn) A zn ! z.

    .

    2.2. .

    . K . K . z K. (zn) K

    zn ! z: K , (znk) (zn) , w, K:

    znk ! w w 2 K: zn ! z,

    znk ! z: z = w , , z 2 K. K K. K . K . n 2 N zn 2 K

    jznj > n:, (zn) K

    zn !1: (zn) , , 1, (zn) - K. K . K ., K . K -, K K. (zn) K. K , (zn) - , Bolzano - Weierstrass, (znk) (zn) :

    znk ! z: (znk) K, z K. K , -

    z 2 K: (znk) (zn) K.

    2.3.1. D(z; r) .

    R(z; r1; r2) = fw j r1 jw zj r2g . .

    19

  • 2.3.2. ( )

    R(z; r1;+1) = fw j r1 jw zjg

    , . D(z; r) , .

    2.3.3. C , , z C, D(z; r) D(z; r) C. C , , C., ;, C, . ; . !, C , , . ; ( ), ., C . C, (zn) zn = n n, 1 , , C ( 1). ; ; 2.3.4. C . ( ) bC, ., bC. - 1, bC (1). 1, C, C 2.1 (Bolzano -Weierstrass) 2.8 bC. bC bC , , bC . bC Riemann bC C (1). 2.3.5. f 1n jn 2 Ng 0, . ., f0g[f 1n jn 2 Ng (;) , .

    2.9 , , .

    2.9. - K - F . K F , > 0 jz wj z 2 K w 2 F ( 4).. . n zn 2 K;wn 2 F

    jzn wnj < 1n :

    K , (znk) (zn) , z, K:

    znk ! z z 2 K:

    20

  • 0 jwnk zj jwnk znk j+ jznk zj < 1nk + jznk zj

    k , ,wnk ! z:

    (wnk) F , z F . F ,

    z 2 F: z 2 K z 2 F K F .

    K ,

    F

    4

    2.3.6. K = D(2; 1) F = fx+ iy jx 0g . () z0 = 1 K w0 = 0 F . jz wj 1 z 2 K, w 2 F . 2.3.7. K = D(0; 2) F = R(0; 3;+1) . z0 2 C(0; 2) K w0 = 32z0 2C(0; 3) F , jz0 w0j = 1 jz wj z 2 K, w 2 F . , z0, w0 K;F .

    2.3.8. F1 = fx + iy : y 0g F2 = fx + iy : x > 0; y 1xg ( 5). , > 0 jz wj z 2 F1, w 2 F2. , j(x+ i0) (x+ i 1x)j , ,1x x > 0. 1x ! 0 x! +1. F1 F2 , .

    b

    b

    (x; 0)

    (x; 1/x)

    F1

    F2

    5

    21

  • .

    2.3.1. .

    2.3.2. , [z0; z1] [ [z1; z2] [ [ [zn1; zn], .2.3.3. K F F K. F .

    2.3.4. A. A @A .

    2.3.5. [] (zn) zn ! z zn 6= z n. fzn jn 2 Ng .[] (zn) zn ! z. fzn : n 2 Ng [ fzg .2.3.6. , 2.1.4, .

    2.3.7. (Kn) Kn+1 Kn n. -

    T+1n=1Kn 6= ;.

    2.3.8. [] 2.1.5 , - K , z0; w0 2 K jz0 w0j = diamK.[] 2.1.6 , - F , w0 2 F jz w0j = d(z; F ).[] - K - F , z0 2 K w0 2 F jz0 w0j jz wj z 2 K, w 2 F .

    2.4 .

    A

    B C

    6

    . A;B;C. B;C A (i)B[C = A,(ii) B \ C = ;, (iii) B 6= ;, C 6= ;, (iv) B;C ( 6).

    (i), (ii), (iii), B;C A. .

    2.4.1. B = D(0; 1), C = D(3; 1) A = B [ C ( 7). B;C A.

    2.4.2. B = D(0; 1), C = D(2; 1) A = B [ C ( 8). B;C , , A.

    22

  • 2.4.3. B = D(0; 1), C = D(2; 1) A = B [ C ( 9)., B;C A. B 1 C.

    b b

    0 3b b

    0 2b b

    0 2

    7 8 9

    . A , B;C (i) - (iv)

    2.4.4. A 2.4.1 A 2.4.2 - . , , A 2.4.3 . B;C A. A - , , A.

    2.4.5. ; fzg -. .

    2.2. B;C B \ C = ; B;C . A A B [ C, A B A C.. ( 10)

    B0 = A \B; C 0 = A \ C:, B0 [ C 0 = A B0 \ C 0 = ;., z 2 B0. z 2 B, z C. r > 0 C D(z; r) , C 0 C, C 0 D(z; r). z C 0. B0 C 0. C 0 B0.

    A

    B0 C 0B C

    b

    z

    10

    B0 6= ; C 0 6= ;, B0; C 0 A , A . B0 = ; C 0 = ; , , A C A B, . 2.10. A A - . A .

    23

  • . A0 A A0 .

    b

    z0

    A

    A

    A

    11

    , , z0 A ( 11). A0 , B;C B [ C = A0,B \ C = ;, B 6= ;, C 6= ; B;C ., z0 2 A0, z0 2 B z0 2 C. z0 2 B ( z0 2 C). A A A0 , , A B [ C. 2.2, A B C. : A C, z0 B. , , A B. A0 A B. A0 B C - A. A0 .

    2.11. A A D A, D .. D , B;C B [ C = D,B \ C = ;, B 6= ;, C 6= ; B;C . A D, A B [ C. A , 2.2 A B A C. A B. ( A C.) D A, D A , , B (A B). D C ( C B). C - D. D .

    . a; b r > 0. fz0; z1; : : : ; zn1; zng z0 =a, zn = b jzk zk1j < r k = 1; : : : ; n r- a; b ( 12). , , zk 2 A k = 0; 1; : : : ; n, r- A.

    b

    b

    b

    b

    b

    b

    b

    b

    b b

    b

    b

    b

    b

    b

    b

    b

    b

    b b

    b

    b

    b

    b

    b

    a

    b

    12

    2.3. K. K z; w 2 K r > 0 r- K z; w.

    24

  • . K . z; w 2 K r > 0 r- K z; w. r- K z; w - .

    B = fb 2 K j r- K z; bg;

    C = fc 2 K j r- K z; cg: B [ C = K, B \ C = ;, B 6= ; ( z 2 B) C 6= ; ( w 2 C). B , b, C. ( b 2 B) r- K z; b , , ( b C) c 2 C jcbj < r. r- K z; b ( b) c, r- K z; c. , , c 2 C. B C. , , C , c, B. ( c B) b 2 B jc bj < r ( b 2 B) r- K z; b. r- K z; b ( b) c, r- K z; c. , , c 2 C. C B., B;C K K . r- K z; w., z; w 2 K r > 0 r- K z; w. K . K , B;C B [C = K, B \C = ;, B 6= ;, C 6= ; B;C . z B. B K, z K. K , z 2 K. z /2 C ( C B) z 2 B. B , , . , B K K , B . B, , . C . B;C , , 2.9, r > 0 jb cj r b 2 B c 2 C. B 6= ;, C 6= ;, z 2 B w 2 C. , r- K z; w, ., r- fz0; z1; : : : ; zn1; zng K z0 = z,zn = w jzk zk1j < r k = 1; : : : ; n. z0 2 B, zn 2 C, k zk1 2 B, zk 2 C. jzk zk1j < r jb cj r b 2 B, c 2 C. 2.4.6. [a; b] , z; w [a; b] r > 0, [a; b], z w, < r. ( r .) [a; b] .

    25

  • 2.4.7. 2.4.6 2.10 [z0; z1] [ [z1; z2] [ [ [zn1; zn] ., [z0; z1], [z1; z2] , z1, [z0; z1][ [z1; z2] . , [z0; z1][ [z1; z2] [z2; z3] , z2, [z0; z1] [[z1; z2] [ [z2; z3] . - .

    2.4.8. , ( , ) ., I l. - [an; bn] I . . , 2.10 I .

    2.4.9. , . , , 2.4.6. , - 2.4.8 .

    . A A A .

    2.12. .

    . A z0 2 A.

    z0

    zz

    z

    Lz

    Lz Lz

    A

    13

    z 2 A Lz ( 13) A z0 z. , , Lz A. , z 2 A Lz ,, , A Lz . A Lz ., Lz z0, , A, .

    2.4.10. A , , . , ., A - A.

    26

  • 2.4.11. A z0 2 A z 2 A . [z0; z] A. z0 A ( 14). , A .

    b

    z0

    A

    14

    A , , ., A A : . z0 . z0 .

    2.4.12. .

    2.4.13. A = D(0; 1) [D(2; 1). 2.4.3 A -, 1.

    2.4. .

    . U . U , , 2.12, ., U . z; w 2 U U z; w. .

    B = fb 2 U j U z; bg;C = fc 2 U j U z; cg:

    B [ C = U , B \ C = ;, B 6= ; ( z 2 B) C 6= ; ( w 2 C). B , b, C. ( b 2 B) U z; b. U , r > 0 D(b; r) U ( b C) c 2 D(b; r)\C. U z; b ( ) . [b; c] ( D(b; r), U ), U z; c. , , c 2 C. B C., , C , c, B. U , r > 0 D(c; r) U . ( c B) b 2 D(c; r) \ B. , ( b 2 B) U z; b , ( ) . [b; c] ( D(c; r), U ), U z; c. ,, c 2 C. C B. B;C U U . U z; w.

    27

  • . .

    . C A A C : C C 0 A C 0 , C = C 0.

    , C A A A.

    . C A B A C \B 6=;. C [ B , , C C [B A. C A, C [B = C, , B C. : A A .

    C1,C2 A C1\C2 6=;. C1 A C2 A, C1 C2. C2 C1 , , C1 = C2. , C1 \ C2 = ;. : A .

    b

    b

    15

    2.13. ( ) ( 15).

    . A A - A. z 2 A C(z) B A z. ( fzg.)

    C(z) =[fB jB A z 2 Bg:

    C(z) A, B A. C(z) z , B z., C(z) C 0 A C 0 , C 0 B A z, C(z) C 0 C(z), , C(z) = C 0. C(z) A z.

    A A - A.

    2.4.14. A = D(0; 1) [D(3; 1). D(0; 1) D(3; 1) A. 2.2 B = D(0; 1) C = D(3; 1), - A D(0; 1) D(3; 1). A D(0; 1) D(3; 1). D(0; 1) D(3; 1) A.

    28

  • 2.4.15. Z fng n 2 Z. fng . fng C 0 Z C 0 6= fng. C 0 = fng[C 0 nfng. fng C 0 n fng C 0, . C 0 . , fng Z. Z , .

    2.14. .

    . C U z 2 C. z 2 U , U , r > 0 D(z; r) U . D(z; r) U C A, D(z; r) C. z C. C .

    , 2.13 2.14, ( ).

    2.15. .

    . C F . C F F C C, C C F . 2.11, C C F , C = C. C .

    .

    2.4.1. ; - .[] D(0; 1) [D(3; 1) [ [1; 2].[] D(0; 1) [D(3; 1) [ [1; 2].[] C n C(i; 2).[] fz j jz ij jzjg.[] fz jRe z < 1g [D(3 + i; 1).[] C n [1; 1 + i] [ [1 + i; 2i].[] C n [1; 1 + i] [ [1 + i; 2i] [ [2i; 1].[]S+1n=1D(ni;

    12).

    [] f 1n jn 2 Ng.[] f0g [ f 1n jn 2 Ng.[] [0; 1] [S+1n=1[ in ; 1 + in ].[] (0; 1] [S+1n=1[ 1n ; 1n + i].[] (0; 1] [ [0; i] [S+1n=1[ 1n ; 1n + i].[] (0; 1] [ [ i2 ; i] [

    S+1n=1[

    1n ;

    1n + i].

    []S+1n=1C(0; 1 +

    1n).

    [] D(0; 1) [S+1n=1C(0; 1 + 1n).[] Q.[] fx+ iy jx; y 2 Qg.2.4.2. [] U a 2 U . U n fag .

    29

  • [] []. U a1; : : : ; an 2 U . U n fa1; : : : ; ang .2.4.3. n An An \ An+1 6= ; n.

    S+1n=1An .

    2.4.4. .

    2.4.5. [] A @A .[] A Ao .

    2.4.6. U . U - U - .

    2.4.7. A ( ). r > 0 z; w 2 A r- A z; w.2.4.8. () A C. A B;C B [ C = A, B \ C = ;, B;C 6= ;.() A C. A B;C B [ C = A, B \ C = ;, B;C 6= ;.2.4.9. U . U ( ).

    2.4.10. A , , .

    30

  • 3

    .

    3.1 .

    A C f : A! C:

    f . f(A) R ,, f(z) 2 R z 2 A, f . f : A ! C ,

    Re f : A! R; Im f : A! R

    (Re f)(z) = Re (f(z)) = f(z) + f(z)2

    ; (Im f)(z) = Im (f(z)) = f(z) f(z)2i

    :

    Re f f Im f f ., f : A! C

    f : A! C

    f(z) = f(z):

    Re f = f + f2

    ; Im f = f f2i

    ; f = Re f + i Im f; f = Re f i Im f

    ., z = Re z + i Im z z = x+ iy,

    x; y 2 R , , x = Re z y = Im z, f(z)

    f(z) = u(z) + iv(z)

    u(z); v(z) 2 R , , u(z) = Re (f(z)) v(z) = Im (f(z)).,

    z = x+ iy = (x; y); f = u+ iv = (u; v)

    , ,

    f(z) = u(z)+iv(z) = (u(z); v(z)); f(x+iy) = u(x+iy)+iv(x+iy) = (u(x+iy); v(x+iy));

    f(x; y) = u(x; y) + iv(x; y) = (u(x; y); v(x; y)):

    31

  • . z A r > 0 D(z; r) \(A n fzg) 6= ;.

    . z A, r- z A z, r- z A , , z A. , z A, A.

    . z A A, r- z A z ( z A), r- z A z , , z A. , z /2 A A, A. z A A. , z A r- z A ( z A) z, A r- z. .

    . z A z 2 A r > 0 D(z; r) \A = fzg.

    : z 2 A A, A - A., , -.

    : z A A, z A A ( A).

    3.1.1. A , , ( ).

    3.1.2. A = D(0; 1)[f2g ( , , ), 2 A A D(0; 1)[f2g ( , 2 ) A D(0; 1).

    z = 1. ( 1, ) A , 1 A. , , 1 - A. , A , 1 A bC , 1 /2 A ( A C), 1 A bC. , A , 1 ( - ) A bC. A , 1 ( ) A bC.

    . z f z0

    32

  • A f z0 z0. z0 A z0 , , z0 A.

    .

    . f : A ! C z0 2 bC A. w0 2 bC f z0 f w0 z0 > 0 > 0 f(z) 2 D(w0; ) z 2 D(z0; ) \ (A n fz0g).

    limz!z0

    f(z) = w0 f(z)! w0 z ! z0:

    f(z) 2 D(w0; ) z 2 D(z0; ) \ (A n fz0g) , , f A - z0, z0, - w0.

    : 1. z0; w0 2 C. limz!z0 f(z) = w0 > 0 > 0 jf(z)w0j < z 2 A 0 < jz z0j < . 2. z0 2 C, w0 =1. limz!z0 f(z) = 1 M > 0 > 0 jf(z)j > M z 2 A 0 < jz z0j < . 3. z0 =1, w0 2 C. limz!1 f(z) = w0 > 0 N > 0 jf(z)w0j < z 2 A jzj > N . 4. z0 = w0 =1. limz!1 f(z) =1 M > 0 N > 0 jf(z)j > M z 2 A jzj > N .

    limz!z0 f(z) = w0. w0 2 C, f w0 , w0 = 1, f 1. f w0 w0 f . f z z0, f .

    - . , , -.

    3.1. ( ), .

    3.2. f : A! C z0 2 bC A. f = u+ iv, u; v : A! R f .[] w0 = u0 + iv0 2 C, :

    limz!z0

    f(z) = w0 limz!z0

    u(z) = u0; limz!z0

    v(z) = v0:

    [] w0 =1, :

    limz!z0

    f(z) =1 limz!z0

    jf(z)j = limz!z0

    pu(z)2 + v(z)2 = +1:

    3.3. f; g : A ! C z0 2 bC A. z ! z0.

    33

  • [] f(z) ! p0 g(z) ! q0 p0 q0 ( p0, q0 1),

    f(z) g(z)! p0 q0:[] f(z)! p0 g(z)! q0 p0q0 ( p0, q0 0 1),

    f(z)g(z)! p0q0:[] f(z)! p0,

    1f(z) ! 1p0 :

    [] f(z) ! p0 g(z) ! q0 p0q0 ( p0, q0 1 0),

    f(z)g(z) ! p0q0 :

    [] f(z)! p0, jf(z)j ! jp0j:

    [] f(z)! p0, f(z)! p0:

    .

    3.1.3.

    p(z) = anzn + an1zn1 + + a1z + a0;

    a0; : : : ; an 2 C an 6= 0. p C. z0 2 C

    limz!z0

    p(z) = p(z0):

    - limz!z0 c = c limz!z0 z = z0. 1, n 1 an 6= 0,

    limz!1 p(z) =1:

    p(z) = znan + an1 1z + + a0 1zn

    an zn 1. - 1z ! 0 z ! 1. zn jznj = jzjn ! +1 z ! 1 (jzj ! +1) : zn = z z !1 1 =1. 3.1.4.

    r(z) =p(z)

    q(z)=

    anzn + + a1z + a0

    bmzm + + b1z + b0 ;

    a0; : : : ; an; b0; : : : ; bm 2 C an 6= 0 bm 6= 0. r C n fz1; : : : ; zkg, z1; : : : ; zk q. 0 k m. , :

    limz!1 r(z) =

    8>:1; n > manbm; n = m

    0; n < m

    34

  • r(z) = znman + an1 1z + + a0 1zn

    bm + bm1 1z + + b0 1zm

    :

    z0 2 C q(z0) 6= 0, :limz!z0

    r(z) = r(z0):

    , z0 2 C q(z0) = 0. z z0 q(z), k 1 q1(z) z

    q(z) = (z z0)kq1(z) q1(z0) 6= 0:, , z0 q(z) k. , p(z0) = 0 p(z0) 6= 0, l 0 p1(z) z

    p(z) = (z z0)lp1(z) p1(z0) 6= 0:, p(z0) = 0, l 1 z0 p(z) , p(z0) 6= 0, l = 0 ( z0 p(z) ) p1(z) = p(z). z q(z)

    r(z) = (z z0)lk p1(z)q1(z)

    p1(z0) 6= 0; q1(z0) 6= 0:

    p1(z0)q1(z0) 1 0,

    limz!z0

    r(z) =

    8>:1; k > lp1(z0)q1(z0)

    ; k = l0; k < l

    , , .

    3.4. f : A ! B, g : B ! C, z0 A, w0 - B, limz!z0 f(z) = w0 limw!w0 g(w). , , f(z) 6= w0 z 2 A z0.

    limz!z0

    gf(z)

    = lim

    w!w0g(w):

    , .

    3.5. f : A! C z0 A. limz!z0 f(z) = w0 (zn) A zn ! z0 zn 6= z0 n f(zn)! w0.

    .

    3.1.1. limz!0 Re z, limz!1 Re z, limz!0 zjzj , limz!1zjzj ;

    3.1.2. limz!0 f(z) = w0 f( in) =(1)n2n n 2 N.

    w0.

    3.1.3. f; g : D(0; 1) ! C g(z) = f(iz) z 2 D(0; 1). limz!0 f(z) = w0 limz!0 g(z) = w0.

    3.1.4. limz!1 f(z) = w0 limz!0 f(1z ) = w0.

    3.1.5. f : C! C f( 1n) f( 1n) = (1)n n 2 N. limz!0 f(z); f ; f ;

    35

  • 3.2 .

    .

    . f : A ! C z0 2 A ( z0 2 C). f z0 > 0 > 0 f(z) 2 D(f(z0); ) z 2 D(z0; ) \ A ,, jf(z) f(z0)j < z 2 A jz z0j < .

    f(z) 2 D(f(z0); ) z 2 D(z0; ) \ A , , f A - z0 - f(z0).

    . 1. z0 A, r0 > 0 D(z0; r0) \A = fz0g. > 0 = r0 > 0 z 2 A jz z0j < = r0, z = z0, jf(z) f(z0)j = jf(z0) f(z0)j = 0 < . f z0.:A z0 A, f () z0. 2. z0 A , , A. f z0. > 0 > 0 jf(z) f(z0)j < z 2 A jz z0j < . , > 0 > 0 jf(z)f(z0)j < z 2 A 0 < jzz0j < . limz!z0 f(z) =f(z0)., limz!z0 f(z) = f(z0). > 0 > 0 jf(z) f(z0)j < z 2 A 0 < jz z0j < . z = z0 jf(z) f(z0)j = jf(z0) f(z0)j = 0 < . > 0 > 0 jf(z) f(z0)j < z 2 A jz z0j < . f z0. : z0 A, f z0

    limz!z0

    f(z) = f(z0):

    , , .

    . f : A ! C , , A.

    3.2.1. p (, C)., limz!z0 p(z) = p(z0) z0. r , limz!z0 r(z) = r(z0) z0 r, z0 r.

    .

    3.6. f : A ! C z0 2 A. f = u + iv, u; v : A ! R f . f z0 u; v z0.

    3.7. f; g : A ! C z0 2 A. f; g z0, f + g; f g; fg; jf j; f : A ! C z0. , , g(z) 6= 0 z 2 A, fg : A! C z0.

    36

  • 3.8. f : A! B, g : B ! C, z0 2 A w0 = f(z0) 2 B. f z0 g w0, g f : A! C z0. 3.9. f : A ! C z0 2 A. f z0 (zn) A zn ! z0 f(zn)! f(z0).

    .

    3.2.1. 0;

    f(z) =

    (Re z1+jzj ; z 6= 00; z = 0

    f(z) =

    ((Re z)2jzj ; z 6= 0

    0; z = 0f(z) =

    8

  • . K . K R, supremum infimum () . u = supK. , n, u 1n K, un 2 K u 1n < un u. (un) K un ! u. u K , K , u 2 K. u supremum K K, K. , infimum K, K .

    3.1 3.1.

    3.11. f : A ! R A. A , f A.

    . A , f(A) , , . f(A) R, , , , f A.

    3.11 , f : [a; b]! R [a; b], [a; b]. , . [a; b] C.

    .

    3.3.1. l. z Pl(z) z l. Pl : C! l.[] jPl(z) Pl(w)j jz wj z; w.[] Pl : C! l .[] .

    3.3.2. z 2 C n f0g R(z) = zjzj z - C(0; 1). R : C n f0g ! C(0; 1).[] R : C n f0g ! C(0; 1) .[] - .

    3.3.3. K, L f : K ! C K f(z) /2 L z 2 K. > 0 jf(z) wj z 2 K w 2 L.3.3.4. f : C ! C C limz!1 f(z) = w0 w0 2 C. f C.

    3.3.5. f : A! C A > 0 > 0 jf(z0) f(z00)j < z0; z00 2 A jz0 z00j < .[] , f A, f A.[] f : D(0; 1)! C f(z) = 11z D(0; 1); D(0; 1);[] , A f A, f A.[] f : A! C -Lipschitz A M 0 jf(z0)f(z00)j M jz0z00j z0; z00 2 A. , f -Lipschitz A, f A.

    38

  • [] f : C! C C limz!1 f(z) = w0 w0 2 C. f C.[] f : A ! C A. z0 2 A n A. (zn) A zn ! z0 limn!+1 f(zn) (zn). , - f(z0) = limn!+1 f(zn). f A n A, f f : A ! C. f : A! C A.

    3.4 .

    3.2. f : A ! C A. A , f(A) .

    . f(A) . B0; C 0 B0 [ C 0 = f(A), B0 \ C 0 = ;, B0 6= ;, C 0 6= ; B0; C 0 . B0; C 0 A,

    B = f1(B0) = fz 2 A j f(z) 2 B0g; C = f1(C 0) = fz 2 A j f(z) 2 C 0g: ( , -) B [ C = A, B \ C = ;, B 6= ;, C 6= ;., B C, b 2 B b C. (cn) C cn ! b. f b, - f(cn) ! f(b). (f(cn)) C 0 , , f(b) C 0. , f(b) 2 B0 ( B0 C 0). B C. B C. C B. B;C A. A . f(A) .

    3.2.

    3.12. f : A ! R A. A , f A.

    . f(A) R - . f(A) R., , () u1; u2 f A. z1; z2 2 A

    f(z1) = u1; f(z2) = u2:

    u1; u2 f(A), (, ) u u1 0. a0; : : : ; an 2 A a0 = z, an = w jak1akj < jf(ak1) f(ak)j < k = 1; : : : ; n.3.4.6. A C. A f : A! Z A .

    40

  • 4

    4.1 .

    , - R , , R. - R , , C.

    . : A! C A R t0 2 A A. t0 limt!t0

    (t)(t0)tt0 (,

    1). t0

    0(t0) =d

    dt(t0) = lim

    t!t0

    (t) (t0)

    t t0 :

    , A R t0 . , , , .

    - , R.

    4.1. : A ! C A R t0 2 A A. , = x + iy, x y . , x; y : A! R

    (t) = x(t) + iy(t)

    t 2 A. t0 2 A x y t0 , ,

    0(t0) = x0(t0) + iy0(t0):

    .

    (t)(t0)tt0 =

    (x(t)+iy(t))(x(t0)+iy(t0))tt0 =

    x(t)x(t0)tt0 + i

    y(t)y(t0)tt0 (4.1)

    x(t)x(t0)tt0 y(t)y(t0)

    tt0 , , , (t)(t0)tt0 . 3.2[] limt!t0

    (t)(t0)tt0 -

    limt!t0x(t)x(t0)

    tt0 limt!t0y(t)y(t0)

    tt0 -. (4.1) t! t0.

    41

  • . . - - 4.1.

    4.2. : A! C A R t0 2 A A. t0, t0.

    4.3. 1; 2 : A ! C A R t0 2 A A.

    1; 2 t0, 1 + 2, 1 2, 12 t0., 2(t) 6= 0 t 2 A, 12 t0. :

    (1 + 2)0(t0) = 10(t0) + 20(t0); (1 2)0(t0) = 10(t0) 20(t0);

    (12)0(t0) = 10(t0)2(t0) + 1(t0)20(t0);

    1

    2

    0(t0) =

    10(t0)2(t0) 1(t0)20(t0)

    (2(t0))2:

    .

    4.4. : B ! A : A ! C B R A R s0 2 B B t0 = (s0) 2 A A. s0 t0, : B ! C s0

    ( )0(s0) = 0((s0))0(s0) = 0(t0)0(s0):

    4.2 .

    : [a; b] R [a; b].

    . : [a; b] ! C, [a; b] R [a; b]. C.

    : [a; b] ! C (: [a; b] R [a; b]), ,

    = f(t) j t 2 [a; b]g C C , [a; b] . (a) (b) . t 2 [a; b] . t [a; b], (t) .,

    z = (t); t 2 [a; b] .

    , , . -,

    . 1 2 , 1 = 2.

    42

  • 4.2.1. z0; z1 2 C, z0 6= z1. : [a; b]! C

    z = (t) =t ab az1 +

    b tb az0; t 2 [a; b]

    [z0; z1] z0; z1. z0 z1. , : [c; d]! C

    z = (t) = tcdcz1 +dtdcz0; t 2 [c:d];

    [z0; z1] z0; z1. [0; 1],

    : [0; 1]! C

    z = (t) = tz1 + (1 t)z0; t 2 [0; 1]: z0 z1.

    4.2.2. z0 r > 0. : [0; 2]! C

    z = (t) = z0 + r(cos t+ i sin t); t 2 [0; 2]

    C(z0; r). z0+r. , . : [0; 2]! C [0; 2]

    z = (t) = z0 + r(cos(2t) + i sin(2t)); t 2 [0; 2] ( - ). , C(z0; r) z0 + r, . - , . - (). , , () .

    , , : [a; b]! C , (a) = (b), .

    , : [a; b] ! A , A C, (t) 2 A t 2 [a; b] , , A. A A.

    : [a; b] ! C t0 2 [a; b]. t0 t0,

    0(t0) = limt!t0

    (t)(t0)

    tt0 2 C:

    t0 = a, , , , t0 = b, .

    : [a; b] ! C t0 2 [a; b] I . 0+(t0) 6= 0. l+

    z = "+(t) = (t0) + (t t0)0+(t0); t t0;

    43

  • (t0) 0+(t0). -

    limt!t0+j(t)"+(t)jj"+(t)(t0)j = limt!t0+

    (t)(t0)(tt0)0+(t0)(tt0)0+(t0)

    = 1j0+(t0)j limt!t0+

    (t)(t0)tt0 0+(t0)

    = 0:, t t0 , (t) ( ) "+(t) ( l+) "+(t) (t0) l+. l+ (t0). l+

    (t0) 0+(t0) ( ) (t0) .

    : [a; b] ! C t0 2 [a; b] [a; b]. 0(t0) 6= 0. l

    z = "(t) = (t0) + (t t0)0(t0); t t0;

    (t0) 0(t0) ( ).

    limt!t0j(t)"(t)jj"(t)(t0)j = limt!t0

    (t)(t0)(tt0)0(t0)(tt0)0(t0)

    = 1j0(t0)j limt!t0

    (t)(t0)tt0 0(t0)

    = 0:, t t0 , (t) ( ) "(t) ( l) "(t) (t0) l. l (t0) l (t0) 0(t0) ( ) (t0) .

    t0 I , l+ l (t0), - 0+(t0) 0(t0). 0+(t0) = 0(t0) =

    0(t0) 6= 0, l - (t0). 0(t0) (t0) 0(t0) . l

    z = "(t) = (t0) + (t t0)0(t0); t 2 R:

    0(t0) 6= 0. 0(t0), . , z = (t0) +(t t0)0(t0) = (t0) .

    , 0+(t0) 6= 0(t0), l+ l .

    44

  • (t0), 180 , 0 < .

    , , . t , 0(t0) () (t) (t0). , - j0(t0)j arg(0(t0)) () (t) (t0).

    4.2.3. z0; z1 2 C, z0 6= z1. : [a; b]! C z = (t) = tabaz1 +

    btbaz0; t 2 [a; b]

    [z0; z1]. - (t)

    0(t) = z1z0ba

    .

    4.2.4. z0 r > 0. : [0; 2]! C

    (t) = z0 + r(cos t+ i sin t); t 2 [0; 2]

    C(z0; r).

    (t)

    0(t) = r( sin t+ i cos t): (t) z0 0(t) ( ) ,

    ((t) z0) 0(t) = (r cos t; r sin t) (r sin t; r cos t) = r2 cos t sin t+ r2 sin t cos t = 0:

    j0(t)j = r:, (t) r . : [0; 2]! C

    (t) = z0 + r(cos(2t) + i sin(2t)); t 2 [0; 2]; C(z0; r). (t)

    0(t) = 2r( sin(2t) + i cos(2t)):

    j0(t)j = 2r:, (t) (t) . [0; 2] (t) (t) .

    : [a; b] ! C. ,

    0(t) t 2 [a; b] 0 : [a; b] ! C [a; b].

    () =Z baj0(t)j dt: (4.2)

    45

  • 4.2.5. z0 6= z1 : [a; b]! C

    z = (t) = btbaz0 +tabaz1; t 2 [a; b]:

    () =R ba j0(t)j dt =

    R ba

    z1z0ba

    dt = z1z0ba R ba dt = jz1z0jba (b a) = jz1 z0j; [z0; z1].

    4.2.6. z0 r > 0 : [0; 2]! C

    z = (t) = z0 + r(cos t+ i sin t); t 2 [0; 2]:

    () =R ba j0(t)j dt =

    R 20 jr( sin t+ i cos t)j dt =

    R 20 r dt = 2r;

    C(z0; r)., : [0; 2]! C

    z = (t) = z0 + r(cos(2t) + i sin(2t)); t 2 [0; 2];

    () =R ba j0(t)j dt =

    R 20 j2r( sin(2t) + i cos(2t))j dt =

    R 20 2r dt = 4r;

    C(z0; r).

    ( ) (4.2) -. = ft0; t1; : : : ; tn1; tng [a; b], a = t0 < t1 < : : : < tn1 < tn = b.

    f(t0); (t1); : : : ; (tn1); (tn)g : [a; b] ! C (t0) = (a) (tn) = (b). ,,

    j(t1) (t0)j+ + j(tn) (tn1)j =Pn

    k=1 j(tk) (tk1)j: (4.3)

    k 2 [xk1; xk], , 0 -, (tk)(tk1)tktk1

    0(k) (tk)(tk1)tktk1

    0(k). Pn

    k=1 j0(k)j(tk tk1): (4.4)

    , (4.3) (4.4) 0 0. (4.4) Riemann j0j [a; b] = ft0; t1; : : : ; tn1; tng = f1; : : : ; ng . , 0 (4.3) Riemann (4.4)

    R ba j0(t)j dt. , , 0,

    (4.2).

    46

  • . : [a; b]! C ( 0 [a; b]) 0(t) 6= 0 t 2 [a; b].

    , (t) - 0(t)( ) t [a; b] ( (t) ).

    . : [a; b] ! C a = t0 0 s 2 [c; d]. , [c; d] (c) = a (d) = b. .

    2 = 1 : [c; d]! C [c; d], . 2 1 : 1 t 2 [a; b]

    2 s 2 [c; d].

    2 = f2(s) : s 2 [c; d]g = f1((s)) : s 2 [c; d]g = f1(t) : t 2 [a; b]g = 1:

    , . , : 2(c) = 1((c)) = 1(a) 2(d) =

    1((d)) = 1(b). , , . s [c; d], t = (s) [a; b]

    2(s) = 1((s)) = 1(t) . . 1 , 2 . ,

    20(s) = (1 )0(s) = 10((s))0(s) = 10(t)0(s)

    s 2 [c; d] t = (s) 2 [a; b]. 0(s) > 0 s 2 [c; d], 20(s) 2(s) = 1((s)) =

    1(t) 10(t) 1(t). -, (

    47

  • ) . , : 2(s) 1(t) -. , , . .

    , , , -: . , t = (s)

    (2) =R dc j20(s)j ds =

    R dc j10((s))jj0(s)j ds

    =R dc j10((s))j0(s) ds =

    R ba j10(t)j dt = (1):

    4.2.7. 1 : [a; b]! C 2 : [c; d]! C

    z = 1(t) =btbaz0 +

    tabaz1; t 2 [a; b]; z = 2(s) = dsdcz0 + scdcz1; s 2 [c; d]:

    2 1. , : [c; d] ! [a; b]

    t = (s) = dsdca+scdcb; s 2 [c; d]

    2(s) = 1((s)) s 2 [c; d]:

    , . [z0; z1]. jz1 z0j . .

    .

    , : [a; b] ! C [c; d] [a; b], [c; d] [a; b]. - : [c; d] ! [a; b]. .

    t = (s) = dsdca+scdcb; s 2 [c; d]:

    .

    1 : [a; b]! C 2 : [b; c]! C 1(b) = 2(b). , 1 2. 1 2 ( ) 1

    + 2 : [a; c]! C

    (1+ 2)(t) =

    (

    1(t); a t b

    2(t); b t c

    1+ 2 ( [a; c]) 1 2.

    , 1 2 , 1+ 2 .

    t [a; c], a b (1+ 2)(t)

    1(t) 1 1(a) 1(b) ,

    48

  • , b c (1+ 2)(t) 2(t)

    2 2(b) 2(c). , , (1+ 2)

    1 2.

    , - .

    4.2.8. , , - .

    - () ().

    :

    (1+ 2) =

    R ca j(1

    + 2)

    0(t)jdt = R ba j(1 + 2)0(t)jdt+ R cb j(1 + 2)0(t)jdt=R ba j10(t)jdt+

    R cb j20(t)jdt = (1) + (2):

    .

    , : [a; b]! C. : [a; b]! C ()(t) = (a+ b t); t 2 [a; b]:

    ( [a; b]) . , , .

    ()(a) = (b) ()(b) = (a). . , t [a; b] (t) (a) (b) ()(t) . , ., , s = a+b t:

    () = R ba j()0(t)j dt = R ba j0(a+ b t)j dt= R ab j0(s)j ds = R ba j0(s)j ds = ():

    .

    4.2.1. C : [a; b] ! . > 0 j(t) zj t 2 [a; b] z /2 .4.2.2. - :[] .[] 2 1, 1 2.[] 2 1 3 2, 3 1.

    4.3 -.

    - R R.

    49

  • . f : [a; b]! C u = Re f : [a; b]! R v = Im f : [a; b]! R f , ,

    f = u+ iv:

    f (Riemann) [a; b] u; v (Riemann) [a; b] , , (Riemann) f [a; b] Z b

    af(t) dt =

    Z bau(t) dt+ i

    Z bav(t) dt: (4.5)

    R ba u(t) dt

    R ba v(t) dt (

    ),

    ReZ baf(t) dt =

    Z baRe f(t) dt; Im

    Z baf(t) dt =

    Z baIm f(t) dt:

    , = ft0; : : : ; tng [a; b] = f1; : : : ; ng k 2 [tk1; tk] Riemann

    Pnk=1 f(k)(tk tk1) 0

    Riemann R ba f(t) dt. :

    lim ()!0

    nXk=1

    f(k)(tk tk1) =Z baf(t) dt:

    Pnk=1 f(k)(tk tk1) =

    Pnk=1 u(k)(tk tk1) + i

    Pnk=1 v(k)(tk tk1);

    lim ()!0Pn

    k=1 u(k)(tk tk1) =R ba u(t) dt;

    lim ()!0Pn

    k=1 v(k)(tk tk1) =R ba v(t) dt

    , (4.5). 4.3.1. f [a; b], u = Re f v = Im f [a; b], [a; b] , , f [a; b].

    , , -. -. 4.5. f1; f2 : [a; b] ! C [a; b]. f1 + f2 : [a; b] ! C [a; b] Z b

    a(f1(t) + f2(t)) dt =

    Z baf1(t) dt+

    Z baf2(t) dt:

    . u1 = Re f1, v1 = Im f1, u2 = Re f2, v2 = Im f2. - [a; b], Re (f1 + f2) = u1 + u2 Im (f1 + f2) = v1 + v2 [a; b]. f1 + f2 [a; b] R b

    a (f1(t) + f2(t)) dt =R ba (u1(t) + u2(t)) dt+ i

    R ba (v1(t) + v2(t)) dt

    =R ba u1(t) dt+

    R ba u2(t) dt+ i

    R ba v1(t) dt+ i

    R ba v2(t) dt

    =R ba f1(t) dt+

    R ba f2(t) dt:

    50

  • 4.6. f : [a; b] ! C [a; b] 2 C. f :[a; b]! C [a; b] Z b

    af(t) dt =

    Z baf(t) dt:

    . u = Re f , v = Im f = Re, = Im. u; v [a; b], Re (f) = u v Im (f) = u + v [a; b]. f [a; b] R b

    a f(t) dt =R ba (u(t) v(t)) dt+ i

    R ba (u(t) + v(t)) dt

    = R ba u(t) dt

    R ba v(t) dt+ i

    R ba u(t) dt+ i

    R ba v(t) dt

    = (+ i) R b

    a u(t) dt+ iR ba v(t) dt

    =

    R ba f(t) dt:

    4.7. f : [a; c] ! C a < b < c. f [a; b] [b; c], f [a; c] Z c

    af(t) dt =

    Z baf(t) dt+

    Z cbf(t) dt:

    . u = Re f , v = Im f . u; v [a; b] [b; c], [a; c]. f [a; c] R c

    a f(t) dt =R ca u(t) dt+ i

    R ca v(t) dt

    =R ba u(t) dt+

    R cb u(t) dt+ i

    R ba v(t) dt+ i

    R cb v(t) dt

    =R ba f(t) dt+

    R cb f(t) dt:

    4.8. f : [a; b] ! C [a; b]. jf j : [a; b] ! R [a; b] Z b

    af(t) dt

    Z bajf(t)j dt:

    . u = Re f , v = Im f . u; v [a; b], jf j = pu2 + v2 [a; b]. .(i)

    R ba f(t) dt = 0, j

    R ba f(t) dtj

    R ba jf(t)j dt 0

    R ba jf(t)j dt

    jf(t)j 0 t 2 [a; b].(ii)

    R ba f(t) dt 6= 0.

    R ba f(t) dt,

    R ba f(t) dt =

    R ba f(t) dt

    (cos + i sin ) = R ba f(t) dt z;

    R ba f(t) dt (

    ) z = cos + i sin . z

    jzj = j cos + i sin j = 1:, R b

    a f(t) dt = z1 R ba f(t) dt = R ba (z1f(t)) dt: (4.6)

    (4.6) . (4.6) , , ! R b

    a f(t) dt = Re R ba (z1f(t)) dt = R ba Re (z1f(t)) dt R ba jz1f(t)j dt = R ba jf(t)j dt:

    51

  • 4.4 .

    R C.

    . : [a; b] ! C. [a; b] , , [a; b]. , , f : ! C = f(t) j t 2 [a; b]g. f : [a; b] ! C [a; b]. (f )0 [a; b] , , [a; b]. , Z

    f(z) dz =

    Z ba(f )(t)0(t) dt =

    Z baf((t))0(t) dt

    f .

    4.4.1. z0 6= z1 : [a; b]! C z = (t) = btbaz0 +

    tabaz1; t 2 [a; b]

    . = [z0; z1] , f : [z0; z1]! C [z0; z1],

    R

    f(z) dz R

    [z0;z1]f(z) dz. ,Z

    [z0;z1]f(z) dz =

    Z

    f(z) dz =

    z1 z0b a

    Z baf b tb az0 +

    t ab az1

    dt:

    f . [z0; z1] z0 z1.

    4.4.2. z0 r > 0. : [0; 2]! C

    z = (t) = z0 + r(cos t+ i sin t); t 2 [0; 2]; = C(z0; r). f : C(z0; r)! C C(z0; r),

    R

    f(z) dz

    RC(z0;r)

    f(z) dz. ,ZC(z0;r)

    f(z) dz =

    Z

    f(z) dz = ir

    Z 20

    fz0 + r(cos t+ i sin t)

    (cos t+ i sin t) dt:

    f C(z0; r) .

    4.4.3. z0 r > 0 A;B C(z0; r). A = z0 + r(cos t1 + i sin t1) t1 2 R B = z0 + r(cos t2 + i sin t2) t2 t1 < t2 < t1 + 2. : [t1; t2]! C

    z = (t) = z0 + r(cos t+ i sin t); t 2 [t1; t2]: t [t1; t2] (t) z0 ( ) C(z0; r) A B. f , R

    f(z) dz Z

    arc (A;B)f(z) dz =

    Z

    f(z) dz = ir

    Z t2t1

    fz0 + r(cos t+ i sin t)

    (cos t+ i sin t) dt:

    f arc (A;B) .

    52

  • : , .

    4.9. 1 : [a; b] ! C 2 : [c; d] ! C 2 1. f : 1 = 2 ! C . Z

    2

    f(z) dz =

    Z

    1

    f(z) dz:

    . : [c; d] ! [a; b] 2(s) = 1((s)) s 2[c; d]. , t = (s) , R

    2f(z) dz =

    R dc f(2(s))2

    0(s) ds =R dc f(1((s)))1

    0((s))0(s) ds

    =R ba f(1(t))1

    0(t) dt =R

    1f(z) dz:

    4.10. 1 : [a; b] ! C 2 : [b; c] ! C 1(b) = 2(b) f : 1

    [ 2 ! C . Z

    1

    +2

    f(z) dz =

    Z

    1

    f(z) dz +

    Z

    2

    f(z) dz:

    . f (1+ 2)

    = 1 [ 2 1+ 2.

    R

    1

    +2

    f(z) dz

    R

    1

    +2

    f(z) dz =R ca f(1

    + 2)(t)

    (1

    + 2)

    0(t) dt

    =R ba f(1(t))1

    0(t) dt+R cb f(2(t))2

    0(t) dt=R

    1f(z) dz +

    R

    2f(z) dz:

    4.7.

    4.4.4. z0, z1, z2 z1 - [z0; z2]. : [a; c]! C

    z = (t) = ctcaz0 +tacaz2; t 2 [a; c]:

    b [a; c] (b) = z1. , b =z2z1z2z0a +

    z1z0z2z0 c. , 1 : [a; b] ! C 2 : [b; c] ! C

    z = 1(t) =btbaz0 +

    tabaz1; t 2 [a; b] z = 2(t) = ctcbz1 + tbcbz2; t 2 [b; c]:

    (t) = 1(t) t 2 [a; b] (t) = 2(t) t 2 [b; c].,

    = 1+ 2:

    , f . [z0; z2], R

    f(z) dz =

    R

    1f(z) dz+

    R

    2f(z) dz

    4.4.1 Z[z0;z2]

    f(z) dz =

    Z[z0;z1]

    f(z) dz +

    Z[z1;z2]

    f(z) dz:

    53

  • 4.4.5. z0, z1, z2 .

    (z0; z1; z2)

    z0, z1, z2,

    @(z0; z1; z2) = [z0; z1] [ [z1; z2] [ [z2; z0]:

    1 . [z0; z1] z0 z1, 2 . [z1; z2] z1 z2

    3 . [z2; z0] z2 z0 , f @(z0; z1; z2), , 4.4.1,R[z0;z1]

    f(z) dz =R

    1f(z) dz;

    R[z1;z2]

    f(z) dz =R

    2f(z) dz;

    R[z2;z0]

    f(z) dz =R

    3f(z) dz:

    (4.7), = 1

    + 2

    + 3 @(z0; z1; z2)

    z0 z1 z2 z0. R

    f(z) dz

    [email protected](z0;z1;z2)

    f(z) dz =

    Z

    f(z) dz: (4.8)

    , R

    f(z) dz =

    R

    1f(z) dz +

    R

    2f(z) dz +

    R

    3f(z) dz, (4.7) (4.8)

    [email protected](z0;z1;z2)

    f(z) dz =

    Z[z0;z1]

    f(z) dz +

    Z[z1;z2]

    f(z) dz +

    Z[z2;z0]

    f(z) dz:

    4.4.6. z0 r > 0 A;B;C C(z0; r). B arc (A;C) - z0 A C. 1 arc (A;B) A B

    2 arc (B;C) B C,

    = 1

    + 2 arc (A;C) A C. , f

    arc (A;C), R

    f(z) dz =

    R

    1f(z) dz +

    R

    2f(z) dz , ,

    4.4.3, Zarc (A;C)

    f(z) dz =

    Zarc (A;B)

    f(z) dz +

    Zarc (B;C)

    f(z) dz:

    4.11. : [a; b]! C f : ! C . Z

    f(z) dz = Z

    f(z) dz:

    . f () = . R f(z) dz , , R

    f(z) dz =R ba f(()(t))()0(t) dt

    = R ba f((a+ b t))0(a+ b t) dt = R ab f((s))0(s) ds= R ba f((s))0(s) ds = R f(z) dz:

    54

  • 4.4.7. z0 6= z1 . [z0; z1] z0 z1. . z1 z0. 4.4.1, R

    [z0;z1]f(z) dz =

    R

    f(z) dz;

    R[z1;z0]

    f(z) dz =R f(z) dz:

    Z[z1;z0]

    f(z) dz = Z[z0;z1]

    f(z) dz:

    4.12. : [a; b]! C f1; f2 : ! C . Z

    (f1(z) + f2(z)) dz =

    Z

    f1(z) dz +

    Z

    f2(z) dz:

    . .R

    (f1(z) + f2(z)) dz =

    R ba (f1((t)) + f2((t)))

    0(t) dt

    =R ba f1((t))

    0(t) dt+R ba f2((t))

    0(t) dt=R

    f1(z) dz +

    R

    f2(z) dz:

    4.5.

    4.13. , : [a; b]! C f : ! C . Z

    f(z) dz =

    Z

    f(z) dz:

    . R

    f(z) dz =

    R ba f((t))

    0(t) dt = R ba f((t))

    0(t) dt = R

    f(z) dz:

    4.6.

    4.14. : [a; b] ! C f : ! C jf(z)j M z 2 . Z

    f(z) dz

    M ():. R

    f(z) dz = R ba f((t))0(t) dt R ba jf((t))jj0(t)j dt M R ba j0(t)j dt = M ():

    4.8.

    .

    4.4.1. R

    jzj dz,

    i i.[] 1(t) = it t 2 [1; 1].[] 2(t) = cos t+ i sin t t 2 [2 ; 2 ].[] 3(t) = cos t+ i sin t t 2 [2 ; 2 ]. .

    55

  • 4.4.2. A;B C(z0; r) f : C(z0; r) ! C .

    Rarc (A;B) f(z) dz,

    Rarc (B;A) f(z) dz R

    arc (A;B) f(z) dz ;

    4.4.3. z0; z1; z2 f : @(z0; z1; z2) ! C .

    [email protected](z0;z1;z2)

    f(z) dz z0; z1; z2. - ;

    4.4.4. f R(0; 0; r0) = fz j 0 < jzj < r0g R(0; r0;+1) =fz j r0 < jzj < +1g. M(r) = maxfjf(z)j j jzj = rg rM(r) ! 0 r ! 0+ r ! +1, . r(t) = r(cos t + i sin t) t1 t t2,

    R

    rf(z) dz ! 0 r ! 0+ r ! +1, .

    56

  • 5

    .

    5.1 .

    . f : A ! C z0 A, r > 0 D(z0; r) A. f z0 limz!z0 f(z)f(z0)zz0 . , , f z0

    f 0(z0) =df

    dz(z0) = lim

    z!z0f(z) f(z0)

    z z0 :

    5.1.1. c C. , z0

    dcdz (z0) = limz!z0

    cczz0 = limz!z0 0 = 0:

    C 0 C.

    5.1.2. z C. , z0

    dzdz (z0) = limz!z0

    zz0zz0 = limz!z0 1 = 1:

    C 1 C.

    5.1.3. z2 C. , z0

    dz2

    dz (z0) = limz!z0z2z02zz0 = limz!z0(z + z0) = 2z0:

    z2 C 2z C.

    - - . : dcdx = 0,

    dxdx = 1,

    dx2

    dx = 2x. .

    5.1. f : A! C z0 A. f z0.

    .

    f(z) = f(z)f(z0)zz0 (z z0) + f(z0) z 2 A; z 6= z0:

    57

  • f 0(z0) 2 C, limz!z0 f(z) = f 0(z0) 0 + f(z0) = f(z0):

    f z0.

    5.2 .

    5.2. f; g : A ! C z0 A. f +g; f g; fg : A! C z0. , g(z) 6= 0 z 2 A, fg : A! C z0. ,

    (f + g)0(z0) = f 0(z0) + g0(z0); (f g)0(z0) = f 0(z0) g0(z0);

    (fg)0(z0) = f 0(z0)g(z0) + f(z0)g0(z0);fg

    0(z0) =

    f 0(z0)g(z0) f(z0)g0(z0)(g(z0))2

    :

    . - .

    5.3 .

    5.3. f : A! B z0 A g : B ! C w0 = f(z0) B. g f : A! C z0. ,

    (g f)0(z0) = g0(w0)f 0(z0):. - .

    5.1.4. 5.2 , - C : p(z) = a0 + a1z + a2z2 + + anzn, p0(z) = a1 + 2a2z + + nanzn1. 5.1.5. .

    5.1.6. h(z) = (z23z+2)153(z23z+2)2, , ,h0(z) =

    15(z2 3z + 2)14 6(z2 3z + 2)(2z 3).

    .

    5.4. f : A ! B -- A B, f1 : B ! A. z0 A w0 = f(z0) B. f z0 f 0(z0) 6= 0 f1 w0, f1 w0

    (f1)0(w0) =1

    f 0(z0):

    . w = f(z) z = f1(w) , f1 w0, z = f1(w)! f1(w0) = z0 w ! w0. :

    limw!w0f1(w)f1(w0)

    ww0 = limz!z0zz0

    f(z)f(z0) = limz!z01

    f(z)f(z0)zz0

    = 1f 0(z0) :

    f1 w0 (f1)0(w0) = 1f 0(z0) .

    58

  • . f : A ! C z0 A. f z0 r > 0 f D(z0; r) (, ,D(z0; r) A).

    -: .

    5.1.7. C.

    5.1.8. . , r(z) = p(z)q(z) z1; : : : ; zn q(z), r(z) A = Cnfz1; : : : ; zng. r(z) A A . z0 2 A r > 0 D(z0; r) A, r(z) D(z0; r) , , r(z) z0.

    5.1.9. f : C ! C f(z) = z. z0

    limz!z0f(z)f(z0)

    zz0 = limz!z0zz0zz0

    , f z0. zz0zz0 z z0, zz0zz0 z z0 z0 , , zz0zz0 z z0 z0. z0 = x0 + iy0,

    limx!x0(x+iy0)(x0+iy0)(x+iy0)(x0+iy0) = limx!x0

    xx0xx0 = limx!x0 1 = 1

    limy!y0(x0+iy)(x0+iy0)(x0+iy)(x0+iy0) = limy!y0

    iy+iy0iyiy0 = limy!y0(1) = 1:

    , limz!z0 zz0zz0 , , f z0. f , .

    5.1.10. f : C! C f(z) = jzj2.

    limz!0 f(z)f(0)z0 = limz!0jzj2z = limz!0 z = 0:

    f 0 f 0(0) = 0. z0 6= 0

    limz!z0f(z)f(z0)

    zz0 = limz!z0jzj2jz0j2zz0

    , f z0. jzj

    2jz0j2zz0 z z0,

    jzj2jz0j2zz0 z z0

    z0 , , jzj2jz0j2zz0 z

    z0 z0. z0 = x0 + iy0,

    limx!x0jx+iy0j2jx0+iy0j2(x+iy0)(x0+iy0) = limx!x0

    x2x02xx0 = limx!x0(x+ x0) = 2x0

    59

  • limy!y0jx0+iyj2jx0+iy0j2(x0+iy)(x0+iy0) = limy!y0

    y2y02iyiy0 = i limy!y0(y + y0) = 2iy0:

    z0 6= 0, . limz!z0 jzj2jz0j2zz0 , -

    , f z0. f 0, .

    . f - f .

    5.5. f : A ! C B A f . f B. , f .

    . U f . z 2 U , r > 0 f D(z; r), D(z; r) B. z B, z 2 Bo., z 2 Bo, r > 0 D(z; r) B, f D(z; r). f z, z 2 U . U = Bo.

    5.1.11. C.

    5.1.12. .

    5.1.13. 5.1.9 5.1.10, f : C ! C f(z) = z f : C ! C f(z) = jzj2 .

    . f : A! C A. f , , f .

    , f f . , , f , f . , z 2 r > 0 D(z; r) . f D(z; r), z. , f , .

    .

    5.1.1. Re z, Im z jzj C.5.1.2. C f : ! C. = fz j z 2 g f : ! C f(z) = f(z) z 2 . , f z0 2 , f z0 2 .5.1.3. U; V C f : V ! U , g : U ! C, h : V ! C h 1-1 V h = g f . h w0 2 V , g z0 = f(w0), g0(z0) 6= 0 f w0, f w0 f 0(w0) = h

    0(w0)g0(z0) .

    60

  • 5.2 Cauchy - Riemann.

    f : A ! C ( z) z0 A u = Re f : A! R v = Im f : A! R ( (x; y)) (x0; y0). 5.1. f : A ! C z0 = (x0; y0) A u; v f A. f z0, u; v x y (x0; y0)

    @u

    @x(x0; y0) =

    @v

    @y(x0; y0);

    @u

    @y(x0; y0) = @v

    @x(x0; y0): (5.1)

    .

    limz!z0f(z)f(z0)

    zz0 = f0(z0) (5.2)

    f 0(z0) = + i; ; 2 R: (5.3)

    f(z)f(z0)zz0 z z0, f(z)f(z0)zz0 z z0 z0 , , f(z)f(z0)zz0 z z0 z0. (5.2) , ,

    limx!x0f(x;y0)f(x0;y0)

    xx0 = f0(z0); limy!y0

    f(x0;y)f(x0;y0)iyiy0 = f

    0(z0): (5.4)

    (5.4) (5.3)

    limx!x0u(x;y0)+iv(x;y0)u(x0;y0)iv(x0;y0)

    xx0 = + i;

    , ,

    limx!x0u(x;y0)u(x0;y0)

    xx0 = ; limx!x0v(x;y0)v(x0;y0)

    xx0 = ;

    @[email protected](x0; y0) = ;

    @[email protected](x0; y0) = : (5.5)

    , (5.4) (5.3)

    limy!y0u(x0;y)+iv(x0;y)u(x0;y0)iv(x0;y0)

    iyiy0 = + i;

    , ,

    limy!y0v(x0;y)v(x0;y0)

    yy0 = ; limx!x0u(x;y0)u(x0;y0)

    yy0 = ;

    @[email protected] (x0; y0) = ;

    @[email protected] (x0; y0) = : (5.6)

    (5.5) (5.6) (5.1).

    (5.1) Cauchy - Riemann (x0; y0). : (C-R).

    , f z0 (5.3), (5.5) (5.6) 5.1

    f 0(z0) [email protected]

    @x(x0; y0) + i

    @v

    @x(x0; y0) =

    @v

    @y(x0; y0) [email protected]

    @y(x0; y0):

    5.1 (C-R) 5.2.

    61

  • 5.2. f : A ! C z0 = (x0; y0) A u; v f A. u; v x y (x; y) (x0; y0) (x0; y0) (C-R) (x0; y0), f z0.

    . (C-R) (5.1), :

    = @[email protected](x0; y0) [email protected]@y (x0; y0); = @[email protected] (x0; y0) = @[email protected](x0; y0): (5.7)

    > 0. @[email protected] ,@[email protected] (x0; y0) A, r > 0

    @[email protected](x; y)

    < 4 ; @[email protected] (x; y) + < 4 (x; y) 2 D((x0; y0); r): (5.8) (x; y) 2 D((x0; y0); r)

    u(x; y) u(x0; y0) = u(x; y) u(x0; y) + u(x0; y) u(x0; y0): (5.9) , x0 x; x0

    u(x; y) u(x0; y) = @[email protected](x0; y)(x x0) (5.10) y0 y; y0

    u(x0; y) u(x0; y0) = @[email protected] (x0; y0)(y y0): (5.11)

    x0; y0 x; y. , , (x0; y),(x0; y

    0) D((x0; y0); r). , (5.8), @[email protected](x

    0; y) < 2 ; @[email protected] (x0; y0) + < 2 : (5.12) (5.9), (5.10) (5.11),

    u(x; y) u(x0; y0)(x x0) (y y0)

    =u(x; y) u(x0; y) (x x0)

    +u(x0; y) u(x0; y0) + (y y0)

    [email protected]@x(x

    0; y) (x x0) + @[email protected] (x0; y0) + (y y0), (5.12),u(x; y) u(x0; y0) (x x0) (y y0)

    @[email protected](x0; y) jx x0j+ @[email protected] (x0; y0) + jy y0j< 4 jx x0j+ 4 jy y0j< 2

    p(x x0)2 + (y y0)2:

    (5.13)

    , v, (5.13) v(x; y) v(x0; y0) (x x0) + (y y0) < 2p(x x0)2 + (y y0)2: (5.14) (5.13) (5.14) (x; y) 2 D((x0; y0); r). (5.13) (5.14) , ,

    f(z) f(z0) (+ i)(z z0) = f(x; y) f(x0; y0) (+ i)(x x0) + i(y y0)

    :

    62

  • , (5.13) (5.14)

    jf(z) f(z0) (+ i)(z z0)j < p(x x0)2 + (y y0)2 = jz z0j

    z 2 D(z0; r) , ,f(z)f(z0)zz0 (+ i)

    < z 2 D(z0; r):

    limz!z0f(z)f(z0)

    zz0 = + i;

    f z0 f 0(z0) = + i.

    5.2.1. 5.1.3 (C-R). f(z) = z2 f(z) = (x+ iy)2 = x2 y2 + i2xy,

    u(x; y) = Re f(x; y) = x2 y2; v(x; y) = Im f(x; y) = 2xy:

    @[email protected](x; y) = 2x;

    @[email protected] (x; y) = 2y; @[email protected](x; y) = 2y; @[email protected] (x; y) = 2x

    (C-R) . 5.2, f(z) = z2

    f 0(z) = @[email protected](x; y) + [email protected]@x(x; y) = 2x+ i2y = 2z:

    5.2.2. 5.1.9 . f(z) = z. f(z) = x+ iy = x iy,

    u(x; y) = Re f(x; y) = x; v(x; y) = Im f(x; y) = y:

    @[email protected](x; y) = 1;

    @[email protected] (x; y) = 0;

    @[email protected](x; y) = 0;

    @[email protected] (x; y) = 1

    (C-R) (x; y). 5.1, f , , .

    5.2.3. 5.1.10. f(z) = jzj2 f(z) = jx+ iyj2 = x2 + y2,

    u(x; y) = Re f(x; y) = x2 + y2; v(x; y) = Im f(x; y) = 0:

    @[email protected](x; y) = 2x;

    @[email protected] (x; y) = 2y;

    @[email protected](x; y) = 0;

    @[email protected] (x; y) = 0

    (C-R) (0; 0). 5.1, f , , 0. , (C-R) (0; 0), 5.2 f 0

    f 0(0) = @[email protected](0; 0) + [email protected]@x(0; 0) = 0 + i0 = 0:

    63

  • 5.2.4. f(z) = (Re z)2. f(z) = (Re (x+iy))2 = x2,

    u(x; y) = Re f(x; y) = x2; v(x; y) = Im f(x; y) = 0:

    @[email protected](x; y) = 2x;

    @[email protected] (x; y) = 0;

    @[email protected](x; y) = 0;

    @[email protected] (x; y) = 0:

    (C-R) (0; y) y 2 R. , 5.1, f , , (0; y) . , (C-R) (0; y) y 2 R, 5.2 f . f . ( ) , f .

    5.2.5. (x0; y0) 5.2.

    f(z) = f(x; y) =

    8

  • , , ! f 0. f 0

    limz!0 f(z)f(0)z0 = lim(x;y)!(0;0)(xy)/px2+y2

    x+iy :

    (xy)/px2+y2

    x+iy (x; y) (0; 0) 0. y = ax , a 6= 0,

    limx!0(xax)/

    px2+(ax)2

    x+iax =a

    (1+ia)p1+a2

    limx!0 xjxj

    .

    , 5.2 5.2 .

    5.6. f : A ! C, u; v f A A. u; v (C-R) , f .

    . z - z ( , -). 5.2, f z. , f - , , f .

    .

    5.2.1. 5.1.1 (C-R).

    5.2.2. [] F (x; y) =

    pjxyj

    (C-R) 0 0.

    []

    G(x; y) =

    (x2y

    x4+y2; (x; y) 6= (0; 0)

    0; (x; y) = (0; 0)

    (C-R) 0, G(z)G(0)z0 z ! 0 - 0, G 0.

    5.2.3. f : A! C z0 = (x0; y0) A, u; v f A u; v x y (x; y) (x0; y0) (x0; y0).[] R limz!z0 Re

    f(z)f(z0)zz0 , f z0.

    [] R limz!z0f(z)f(z0)

    zz0, f z0

    f z0.

    5.2.4. C f . 2.4.6.[] Re f = 0 . f .[] , l f(z) 2 l z 2 . f .[] [], l, C f(z) 2 C z 2 ;

    65

  • 5.3 .

    . f : A ! C A : [a; b] ! A. f .

    f() = f : [a; b]! C;

    [a; b]. f() f .

    , f : A! C

    z0 A. f z0,

    f(z0) = w0; f0(z0) 6= 0:

    : [t0; b)! A; (t0) = z0:

    , z0 A.

    0(t0) 6= 0; z0.,

    f() : [t0; b)! C; f ,

    f()(t0) = (f )(t0) = f((t0)) = f(z0) = w0 w0

    f()0(t0) = (f )0(t0) = f 0((t0))0(t0) = f 0(z0)0(t0) 6= 0: (5.15)

    (5.15) .

    jf()0(t0)j = jf 0(z0)jj0(t0)j:

    , f() w0 - z0 jf 0(z0)j > 0. , : f z0 jf 0(z0)j > 0 , , f z0 jf 0(z0)j > 0.

    arg f()0(t0) = arg f 0(z0) + arg 0(t0): (5.16)

    , f() w0 - z0 arg(f 0(z0))., : f z0 arg(f 0(z0)) ,, f z0 arg(f 0(z0)).

    66

  • z0 : . , , , -, , ! .

    , 1 2. 1 2 z0

    arg 20(t0)) arg 10(t0))

    f(1) f(2) w0

    arg f(2)0(t0) arg f(1)0(t0):

    (5.16) 1 2

    arg f(2)0(t0) arg f(1)0(t0) = arg 20(t0) arg 10(t0);

    f(1) f(2) w0 1 2 z0. , : f z0.

    f f z0 , , f z0 f 0(z0) 6= 0.

    .

    5.3.1. w = az + b a 6= 0.[] -- z- w-.[] z- , -, w-.[] z- kx + ly = m k0x + l0y = m0. ; . w- . w = az + b .[] bC w- - 1 z-; -- z- w-. - [] : z- w-.

    5.3.2. w = z2.[] u0; v0 2 R, x2y2 = u0 2xy = v0 z- ( z = x+ iy). ; ;[] x0; y0 2 R x0; y0 6= 0, u = 14y02 v2y02 u = 1

    4x02v2 + x0

    2 w- ( w = u + iv). ; ;

    5.3.3. f : ! A C : A ! C A C. f() A f . Rf() (w) dw =

    R

    (f(z))f

    0(z) dz.

    67

  • 5.4 .

    . z = x+ iy x; y 2 R exp z = ex(cos y + i sin y):

    z 2 R, z = x + i0 x 2 R, exp z = ex(cos 0 + i sin 0) = ex = ez . ez exp z ez ez z . ,

    ez = exp z = ex(cos y + i sin y):

    . z = x+ iy x; y 2 R, jezj = jexjj cos y+ i sin yj = ex,

    jezj = eRe z arg(ez) = y + k2 k 2 Z,

    arg(ez) = Im z + k2 k 2 Z: ez = ex(cos y i sin y) = ex(cos(y) + i sin(y)) = ez ,

    ez = ez:

    ez . z1 = x1+iy1 z2 = x2+iy2 x1; x2; y1; y2 2R, , R - R,

    ez1ez2 = ex1(cos y1 + i sin y1)ex2(cos y2 + i sin y2)= ex1+x2(cos(y1 + y2) + i sin(y1 + y2)) = ez1+z2 ;

    z1 + z2 = (x1 + x2) + i(y1 + y2).

    ez1ez2 = ez1+z2 :

    .

    ez2 = ez1 () z2 z1 = k2i k 2 Z:, z2 z1 = k2i k 2 Z,

    ez2 = ez1ek2i = ez1(cos(k2) + i sin(k2)) = ez1 :

    , ez2 = ez1 z2 z1 = x+ iy x; y 2 R. ex(cos y + i sin y) = ez2z1 = ez2ez1 = 1

    , ,ex = 1; cos y = 1; sin y = 0:

    x = 0 y = k2 k 2 Z. z2 z1 = k2i k 2 Z., , 2i , ,

    --. 0 , z = x + iy

    jezj = ex > 0. , w 6= 0 . ez = w z,

    w = r(cos + i sin )

    w , z = x+ iy,

    68

  • ez = w

    mex(cos y + i sin y) = r(cos + i sin )

    mex = r; cos y = cos ; sin y = sin

    mx = ln r; y = + k2 k 2 Z;

    ln : (0;+1)! R (0;+1). , w 6= 0, ez = w

    z = ln r + i( + 2k) k 2 Z

    , ,z = ln jwj+ i argw:

    :ez = w () z = ln jwj+ i argw:

    exp : C! C n f0g:

    exp z

    u(x; y) = Re exp(x; y) = ex cos y; v(x; y) = Im exp(x; y) = ex sin y:

    u; v

    @[email protected](x; y) = e

    x cos y; @[email protected] = ex sin y; @[email protected] = ex sin y; @[email protected] = ex cos y;

    (C-R) C. , exp - C.

    , , w = ez z- w-. : z = x+iy w = u+iv x; y; u; v 2 R.

    y 2 R x R, z = x + iy hy z- C y- (0; y) w = ez = ex(cos y + i sin y) ry w- C, 0 ( 0) y u-. , z hy , x 1 +1, w = ez ry 0 1. y y, hy , ry 0 y. 0 < y < 2, z- hy hy+y w- ry ry+y ( ). y = 2, ry ry+y () z- hy hy+y w- ry = ry+y ( 0, ). , w-

    69

  • ( 0, ). y > 2, z- hy hy+y w- ( 0, ) .

    x 2 R y R, z = x+ iy vx z- C x- (x; 0), w = ez = ex(cos y + i sin y) C(0; ex), cx, w-C. , z vx , y 1 +1, w = ez cx . y 2, w = ez cx . y y < 2, w = ez y. , y > 2, w = ez cx . x x, vx , cx ex cx+x ex+x = exex. ex > 1. z- vx vx+x w- cx cx+x.

    . ,

    = (x1; y1;x2; y2) = fx+ iy jx1 < x < x2; y1 < y < y2g

    z- . - hy1 hy2 vx1 vy2 . (0 2, z- w- R = R(0; ex1 ; ex2) .

    .

    5.4.1. jez 1j ejzj 1 jzjejzj z.5.4.2. z ! 1 . , lim ez C. ;

    5.4.3. : fx+iy j a < x < b; < y < +g,fx+ iy j a < x < b; < y < +2g, fx+ iy jx < b; < y < + g, fx+ iy jx < b; < y < + 2g, fx+ iy j a < x; < y < + g, fx+ iy j a < x; < y < + 2g.

    70

  • 5.4.4.

    cos z = eiz+eiz2 ; sin z =eizeiz

    2i ; tan z =sin zcos z ; cot z =

    cos zsin z :

    [] C - R. ; ;

    sin2 z + cos2 z = 1;sin(z + w) = sin z cosw + cos z sinw; cos(z + w) = cos z cosw sin z sinw;j cos(x+ iy)j2 = cos2 x+ sinh2 y; j sin(x+ iy)j2 = sin2 x+ sinh2 y:

    [] .[] w = sin z fx + iy j 2 < x < 2 g w = cos z fx + iy j 0 < x < g. ( ) ;

    5.5 .

    , w 6= 0, ez = w () z = ln jwj+ i argw:

    .

    logw = ln jwj+ i argw w 6= 0 logw w.

    argw 2, logw i2. z = logw x = ln jwj vx x = ln jwj. logw vx : 2., . vx, 2 , - z = logw. , , 2 ( ), , w 6= 0, - z = logw. , 0

    Z0 = fx+ iy j 0 < y 0 + 2g Z0 = fx+ iy j 0 y < 0 + 2g; Z0 z = logw = ln jwj + i argw : y () argw ,,

    0 < 0 + 2 0 < 0 + 2: 0 =

    , Z0 = fx+ iy j < y g;

    () z = logw, , - .

    71

  • .

    Logw = ln jwj+ iArgw w 6= 0

    Logw w.

    , Logw logw Z < Argw .

    ,

    ez = w () z = logw:

    , logw ez = w

    5.5.1. 1 log 1 = ln j1j + i(0 + k2) = i2k k 2 Z. Log 1 = ln j1j+ i0 = 0. 1 log(1) = ln j 1j+ i( + k2) = i(2k + 1) k 2 Z. Log (1) = ln j 1j+ i = i. i log i = ln jij + i(2 + k2) = i(2k + 12) k 2 Z. Log i = ln jij+ i2 = i2 . i log(i) = ln j ij+ i(2 + k2) = i(2k 12) k 2 Z. Log (i) = ln j ij+ i(2 ) = i2 .

    , w = exp z = ez C Cnf0g --. , -- w 6= 0 z. , ( ).

    .

    log : C n f0g ! C

    exp : C! C n f0g. ,

    , w z = logw. - . , - ( ) -- . . , y = x2 (1;+1) [0;+1) -- (1;+1), [0;+1) ( ), -- x = py [0;+1) [0;+1) ( py - - y). , y = x2 (1; 0] ( ), -- x = py [0;+1) (1; 0]. - y = x2 x = py x = py. ( ) - x = py x = py, py - . - ,

    72

  • . - y = cosx, y = sinx, y = tanx, y = cotx x = arccos y, x = arcsin y, x = arctan y,x = arccot y.

    ( ) . , , y = x2 - . y [0;+1)( y = x2) x x2 = y. x, x = py x = py. : x = py y [0;+1) x = py y [0;+1). , , !! : x = py y [0;+1) x = py y [0;+1),, ,

    x =

    (py; 0 y 1py; 1 < y < +1

    y = x2, y = x2 (1;1) [ [0; 1]. x, x = py x = py, y. , , : ! : , [0;+1). , x = py y 2 [0;+1) x = py y 2 [0;+1) [0;+1). y = x2 ; . , , x = f(y) y = x2 [0;+1) ( y = x2) [0;+1). , f : [0;+1)! R [0;+1) f(y)2 = y y 2 [0;+1). f(0) = 0 f(0) =

    p0 f(0) = p0. y1; y2 > 0 y1 6= y2

    f(y1) =py1 f(y2) = py2. f [y1; y2] [y2; y1]

    , y : f(y) = 0. , y > 0 f(y) = py f(y) = py. y1; y2 > 0. , : f(y) = py y > 0 f(y) = py y > 0. , [0;+1), x = py x = py.

    .

    . A C n f0g f : A ! C. f A (i) f A (ii) w 2 A z = f(w) logw , , z = f(w) ez = exp z = w , , ef(w) = exp f(w) = w.

    A w = 0 log 0. 5.7 :

    .

    5.7. 0,

    A0 = fw = rei j 0 < r < +1; 0 < < 0 + 2g;

    73

  • w- ( C 0 w 0)

    Z0 = fx+ iy j 0 < y < 0 + 2g z-. , w A0 z Z0 ez = exp z = w.

    f : A0 ! Z0 w 2 A0 f(w) = z, z z Z0 ez = exp z = w. f (ii) A0 . f A0 , (i) . f A0 .

    . f w A0 . (wn) A0

    wn ! w f(wn) 6! f(w): f(wn) 6! f(w) > 0 jf(wn) f(w)j > 0 n. n wn , , , ( ) (wn) (wn) A0

    wn ! w jf(wn) f(w)j > 0 n: (5.17), ,

    z = f(w) zn = f(wn) n;

    ez = w ezn = wn n (5.18)

    (5.17)

    wn ! w jzn zj > 0 n: (5.19) wn; w 2 A0 , wn; w

    w = jwjei wn = jwnjein n;

    0 < < 0 + 2 0 < n < 0 + 2 n:

    , f Z0 ,

    z = ln jwj+ i zn = ln jwnj+ in n: wn ! w, jwnj ! jwj , - , ln jwnj ! ln jwj. (ln jwnj) . , (n) , (0; 0+2). (zn) . , - Bolzano - Weierstrass (znk) z0 :

    znk ! z0: (5.20)

    74

  • znk Z0 , z0 , :

    z0 2 Z0 = fx+ iy j 0 y 0 + 2g: (5.21)

    (, ) (wn) , (5.19),

    wnk ! w: (5.18) (5.20), -

    wnk = eznk ! ez0

    , ,ez

    0= w:

    (5.18) ez

    0= ez;

    z0 z i2. - , z Z0 , z0 Z0 ( (5.21)) (, ) 2. f w A0 .

    , , 0, - z = logw A0 w- Z0 z-. , 0, 0+k2 k 2 Z, A = A0+k2 (!) , Z0+k2 k2. Z0+k2 k 2 Z z- ( y = k2). : w- 0, - A z = logw . A z- - 2. , ( A), , z- ( - ). , A, .

    5.5.2. 0 = .

    A

    w- u- ( w = u+ iv)

    Z = fx+ iy j < y < g:

    w 2 A z = Logw logw. ,

    Log : A ! Z: A w-, , . - A Z+k2, Z

    75

  • k2 k 2 Z. z = Logw ik2 ( k2)

    z = Logw + i2k:

    5.8. A C n f0g f : A ! C A. w0 A, f w0

    f 0(w0) =1

    w0:

    . z0 = f(w0) z = f(w) w 2 A. , ez0 = w0 ez = w. f , w ! w0 z ! z0. , z0, w ! w0,

    f(w)f(w0)ww0 =

    zz0ezez0 = 1

    ezez0zz0

    ! 1/(ez0) = 1/w0: f w0 f 0(w0) = 1w0 .

    , , f : A! C A, f A, f A. , A , f A. , : z = logw A A

    dz

    dw=

    d logwdw

    =1

    w A:

    A A.

    5.5.3. - A, w- 0. A. ,

    Log : A ! Z A.

    5.9. A C n f0g f1; f2 : A! C.[] f1 A f2(w) f1(w) = ik2 A, k (, w A) , f2 A.[] , , A f1 f2 - A, f2(w)f1(w) = ik2 A, k (, w A) ., f1, f2 w0 2 A, A.. [] f1 A A. f2 = f1 + ik2 A., ef1(w) = w w 2 A.

    ef2(w) = ef1(w)+ik2 = ef1(w)eik2 = w 1 = w

    w 2 A. f2 A.

    76

  • [] k : A! C k(w) = 1i2 (f2(w) f1(w)) w 2 A:

    w 2 A f2(w) f1(w) logw, k(w) .

    k : A! Z:, f1; f2 A, k A., k A, . w1; w2 2 A k(w1) 6= k(w2) k k(w1) k(w2), k . w1; w2 2 A k(w1) 6= k(w2) , , k A. , (, wA) k 1i2 (f2(w) f1(w)) = k , , f2(w) f1(w) = ik2 w 2 A. : f2(w0) = f1(w0), , f2 f1 A, f2(w) f1(w) = f2(w0) f1(w0) = 0 w 2 A f2(w) = f1(w) w 2 A.

    5.9 : A , A ik2 k 2 Z. A.

    ( ). A z0 w0 2 A, z0 . z0 logw0 , , ez0 = w0.

    5.5.4. A = A w- u- ( w =u+ iv). A z = 0 w = 1. A Log - z = Log 1 = 0 w = 1. 5.9[], A , A. () . z- A,

    Z+k2 = fx+ iy j + k2 < y < + k2g k 2 Z z = 0. k = 0,

    Z = fx+ iy j < y < g: f A Z

    f(w) = ln r + i w = rei r = jwj > 0; < < ; argw w (; ). , 5.9[], A , A.

    5.5.5. A = A w- u- ( w =u+ iv). A z = i4 w = 1. z- A,

    Z+k2 = fx+ iy j + k2 < y < + k2g k 2 Z

    77

  • z = i4. - k = 2,

    Z3 = fx+ iy j 3 < y < 5g: f A Z3

    f(w) = ln r + i w = rei r = jwj > 0; 3 < < 5; argw w (3; 5). 5.9[], A , A.

    5.5.6. A = A0 w- u- (w = u+iv). A z = i(2 + 4) w = i. z- A,

    Z0+k2 = fx+ iy j k2 < y < 2 + k2g k 2 Z z = i(2 + 4). k = 2,

    Z4 = fx+ iy j 4 < y < 6g: f A Z4

    f(w) = ln r + i w = rei r = jwj > 0; 4 < < 6; argw w (4; 6). 5.9[], A , A.

    5.10. C(0; r) r > 0. A 0.

    . C(0; r),

    f : C(0; r)! C C(0; r) w 2 C(0; r) ef(w) = w. Log , w- u-. Log C(0; r)

    B = C(0; r) n frg: f B Log B B. B , () k

    Logw = f(w) + ik2 w 2 B: (5.22) B. (w0n) (w00n)

    w0n = rei(/n); w00n = re

    i(+/n) n:

    78

  • C(0; r) r. r r. f C(0; r),

    f(w0n)! f(r); f(w00n)! f(r), ,

    f(w0n) f(w00n)! 0: (5.22)

    Logw0n Logw00n ! 0:,

    Logw0n Logw00n =ln r + i( /n) ln r + i( + /n) = i(2 2/n)! i2

    . C(0; r). f A 0, f , , .

    5.5.7. 0 ,, C n f0g

    .

    5.5.1. Log fw j r1 jwj r2g n [r2;r1], fw j 0 2n , z- r r+ w- ( 0, ) .

    r 2 (0;+1) R, z = rei C(0; r), w = zn = rnein C(0; rn) w- C. , z C(0; r) , 2, w = zn C(0; rn) n . 2n , w = zn C(0; rn) . z C(0; r) < 2n , w = z

    n C(0; rn) n. , > 2n , w = z

    n C(0; rn) . r , C(0; r) , C(0; rn) . z- C(0; r1) C(0; r2), 0 < r1 < r2, w- C(0; r1n) C(0; r2n).

    , w 6= 0, zn = w () z = n

    pjwj ei argwn :

    .

    w1n = n

    pjwj ei argwn w 6= 0

    w 1n n- w.

    z = w1n n

    n- C(0; npjwj) z-. ,

    , 2n , - z = w

    1n . , z-,

    2n , , w 6= 0, - z = w 1n . , 0

    A0 = frei j 0 < 0 + 2n g A0 = frei j 0 < 0 + 2n g;

    82

  • A0 z = w1n .

    ,

    zn = w () z = w 1n :, w 1n ez = w

    , w = zn C n f0g C n f0g n--. , .

    . n-

    C n f0g 3 w 7! z = w 1n 2 C n f0g n- C n f0g 3 z 7! w = zn 2 C n f0g.

    , , w z = w

    1n .

    . n- , ( ) -- .

    . A C n f0g f : A ! C n f0g. f n- A (i) f A (ii) w 2 A z = f(w) w 1n , , z = f(w) zn = w , , f(w)n = w.

    5.11 n- .

    5.11. 0,

    A0 = fw = sei j0 < < 0 + 2g; w- ( C 0 w 0)

    B0/n = fz = rei j 0n < < 0n + 2n g z-. , w A0 z B0/n zn = w.

    f : A0 ! B0/n w 2 A0 f(w) = z, z z B0/n zn = w. f (ii) A0 . f A0 , (i) . f n- A0 .

    . f w A0 . (wk) A0

    wk ! w f(wk) 6! f(w): f(wk) 6! f(w) > 0 jf(wk) f(w)j > 0 k. k (wk) , ,

    83

  • , ( ) (wk) (wk) A0

    wk ! w jf(wk) f(w)j > 0 k: (5.23)

    , ,z = f(w) zk = f(wk) k;

    zn = w zkn = wk k (5.24)

    (5.23)

    wk ! w jzk zj > 0 k: (5.25)

    wk; w 2 A0 , wk; w

    w = jwjei wk = jwkjeik k;

    0 < < 0 + 2 0 < k < 0 + 2 k:

    , f B0/n,

    z = npjwj ein zk = n

    pjwkj ei

    kn k:

    wk ! w, npjwkj ! npjwj. ( npjwkj) .

    (zk) . , Bolzano -Weierstrass (zkm) z0 :

    zkm ! z0: (5.26)

    zkm B0/n, z0 , :

    z0 2 B0/n = frei j 0n 0n + 2n g: (5.27)

    (, ) (wk) , (5.25),

    wkm ! w: n- (5.24) (5.26),

    wkm = zkmn ! z0n

    , ,z0n = w:

    (5.24) z0n = zn;

    z0 z , 0 , 2n . , z B0/n, z

    0 B0/n ( (5.27)) (, ) 2n . f w A0 .

    84

  • , , 0, n- z = w 1n A0 w- B0/n z-. , 0, 0 + k2 k = 0; 1; : : : ; n 1, A = A0+k2 , B(0+k2)/n - k 2n . n B(0+k2)/n k = 0; 1; : : : ; n 1 z- ( 0). : w- 0, - A n z = w 1n n- . A z- 2n . , n- ( A), , z- ( 0). , A, n- .

    5.6.2. n- 0 = .

    A w- u- ( w = u+ iv) n-

    B/n = frei j n < < ng: n-

    z = nps ei

    n w = sei < < :

    z = npjwj eiArgwn :

    A w-, n- , n n- . A B(+k2)/n k = 0; 1; : : : ; n 1, - B/n k 2n . z = f(w) eik

    2n (

    k 2n )

    z = f(w)eik2n = f(w)!n

    k;

    !n n- .

    5.12. A Cnf0g f : A! Cnf0g n- A. w0 A, f w0

    f 0(w0) =f(w0)

    nw0:

    . z0 = f(w0) z = f(w) w 2 A. , z0n = w0 zn = w. f , w ! w0 z ! z0. , n- z0, w ! w0,

    f(w)f(w0)ww0 =

    zz0znz0n = 1

    znz0nzz0

    ! 1/(nz0n1) = z0/(nw0) = f(w0)/(nw0): f w0 f 0(w0) = f(w0)nw0 .

    85

  • , , f : A ! C n- A, f A, f A. , A , f A. , : z = w 1n A A

    dz

    dw=

    dw1n

    dw=

    w1n

    nw A:

    n- A n- A.

    5.6.3. n n- A, w- 0. n- A.

    5.13. A C n f0g f1; f2 : A ! C n f0g. , , !n n- .[] f1 n- A f2(w)f1(w) = !n

    k A, k = 0; 1; : : : ; n1 (, w A), f2 n- A.[] , , A f1 f2 n- A, f2(w)f1(w) = !n

    k A, k = 0; 1; : : : ; n 1 (, w A)., f1, f2 w0 2 A, A.. [] f1 A A. f2 = f1!nk A., f1(w)n = w w 2 A.

    f2(w)n = f1(w)

    n(!nk)n = w(!n

    n)k = w 1 = w

    w 2 A. f2 n- A.[] w 2 A f2(w) f1(w) w 1n , w 2 A f2(w)

    f1(w)

    n= f2(w)

    n

    f1(w)n= ww = 1:

    w 2 A f2(w)f1(w) n- . f2f1

    : A! f1; !n; : : : ; !nn1g

    n , n- C(0; 1) 1. f2f1 A ,

    f2f1(A)

    . . f2f1 A , , f2(w)f1(w) = !n