main.dvi
3 1 1 1 z 43 44 3 3.1 1 3.1.1 1 n 0 a0;a1;a2;111 (a0 6= 0) P(z) = a0zn + a1zn01 + 111+ an01z + an (3.1) n z P(z), Q(z) z Q(z) 6� 0 P(z) Q(z) = a0zn + a1zn01 + 111+ an01z + an b0zm + b1zm01 + 111+ bm01z + bm (3.2) a;b;c;d 1 w = az + b cz + d ; (ad 0bc 6= 0) (3.3) 1 1 1 1 (3.3) w = (bc 0ad)=c cz + d + a c (3.4) 1 1. w = z + 2. w = z 3. w = 1=z 3 3 (1) , (2) O P OP = 1=OQ Q P Q 3.1 1 45 3.1 1=z z 1=�
main.dvi
7 f(z) H f(z)dz = 0 f(z) , I f(z)dz = 0 19 123 124 7 7.1 . 7.1 7.1.1 32 f(z) D D 1 C C z 7.1 f(z) = 1 2�i I C f(�) � 0z d� (7.1) f(z) �- C f(�)=(�0 z) 1 � = z � = z Res(z) = lim �!z (� 0z) f(�) � 0 z = f(z): (7.2) 7.1 (7.1) C f(z) C 7.1 125 7.1.2 1 f(z) z = a jz 0aj = r C z = a r � Ref(a) > Ref(a + rei� ): (7.3) C z 0a = rei� ; dz = irei� d� = i(z 0 a)d� f(a) = 1 2� Z 2� 0 d�f(a + rei� ): (7.4) (7
main.dvi
5 z 1 1u = 0 u(x; y) 1827 A.L. Cauchy (1789-1857) 1814 1700 1850 150 75 76 5 5.1 (a) . (b) . 1 5.1 z t z = z(t) 1 a � t � b z(a) z(b) z(a) = z(b) t1 6= t2 z(t1) 6= z(t2) Jordan z(a) = z(b) z 2 D D 1 E.T. 5.2 77 5.2 C 1. ( 5.1) 5.2 5.2.1 D f(z) C C C z0 z C z0 z z1; z2, 111 , zN01 1 = fz0; z1; z2; 111; zN01; zN = zg (5.1) 1 zj01 zj �j S1 = N X j=1 f(�j)(zj 0zj01) (5.2) 5.2) 30 : 1 zjzj01 f(z) S1 78
main.dvi
6 2 2 z 9 105 106 6 6.1 : 40 I = Z 2� 0 d� 5 04 cos � : (6.1) z = ei� cos � = 1 2 (z + 1 z);dz = iei�d� = izd� � 0 2� z jzj = 1 1 I I = I jzj=1 dz iz 1 5 02(z + 1 z) = i I jzj=1 dz 1 2z2 05z + 2 = i 2 I jzj=1 dz 1 (z 0 1 2 )(z 02) : 1 z = 1=2 I = i 2 12�iRes � 1 2 � = 0� lim z!1=2 (z 0 1 2 ) 1 1 (z 0 1 2 )(z 02) = 2� 3 : (6.2) 41 I = 1 2 I jzj=2 jdzj jz 01j2 : (6.3) z = 2ei� dz = 2iei�d�;jdzj = 2d
main.dvi
8 1 1 f (x) x ! z f (x) 1 9 3 8.1 37 D f (z) D a D fzkg f (zk) = 0 (8.1) D f (z) = 0 141 142 8 8.1 jz 0aj < R a b . ) f (z) D a R jz 0aj < R f (z) = c0 + c1(z 0a) + c2(z 0a) 2 + 111 (8.2) f (z) z = a c0 = f (a) = lim k!1 f (zk) = 0: (8.3) f1(z) = f (z) z 0a = c1 + c2(z 0a) + c3(z 0a) 2 + 111 (8.4) f1(z) f1(zk) = f (zk)=(zk 0 a) = 0 c1 = f1(a) = lim k!1 f1(zk) = 0: (8.5) c0 = c1 = c2 = 111 = 0 (8.6
main.dvi
9 5.4.3 19 149 150 9 1 9.1 9.1.1 z 0(z) = Z 1 0 e0t tz01dt (9.1) 0 2 z z = x > 0 0(x + 1) = Z 1 0 e0t tx dt = 0e0t tx 1 t=0 +x Z 1 0 e0t tx01dt = x0(x) (9.2) x 0 n 0(1) = Z 1 0 e0t dt = 1 (9.3) 0(n + 1) = Z 1 0 e0t tn dt = n! (9.4) 0! = 1 z 0 < " � Rez � M < 1 0 < t � 1 je0t tz01j � e0t t"01 (9.5) 1 � t < 1 je0t tz01j � e0t tM01 (9.6) Z 1 0 e0t t"01dt; Z 1 1 e0t tM01dt Z 1 0 e0t tz01dt (9.7) 1 ( 9
1
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