2 f(z) 0 f(z 0 ) lim z!z 0 z 0 z 0 z z 1z = z 0 z 0 1z z z 0 1z 21
22 2 2 (1642-1727) (1646-1716) (1777-1855) (1789-1857) (1826-1866) 18 19 2.1 2.1.1 12 z w w = f (z) (2.1) f(z) z w = f(z) z z 0 w w 0 " 0 < jz0z 0 j < z jf (z) 0 w 0 j <" (2.2) lim f(z) =w 0 (2.3) z!z 0 w 0 w z! z 0 13 lim z!z0 f (z) =f(z 0 ) " (") jz 0 z 0 j <(") z jf(z) 0 f (z 0 )j <" (2.4) f(z) z = z 0 1
2.1 23 2.1. z w = f(z) ". 2 (x; y) 2 (u; v) f(z) =u(x; y) iv(x; y) x =(z +z)=2, y =(z0z)=(2i) f = u(z; z)+iv(z; z) z jzj z w = f (z) =u(z) +iv(z) w 1 = z 2 w 2 = x 2 + y 2 w 2 = z z = jzj 2 z z jzj 2 1 f(z), g(z) lim z!z0 f(z), lim z!z0 g(z) 1. lim z!z0 (f(z) +g(z)) = lim z!z0 f (z) + lim z!z0 g(z) 2. lim z!z0 cf(z) =c lim z!z0 f(z) 3. lim z!z0 (f(z)g(z)) = lim z!z0 f(z)lim z!z0 g(z) f(z) 4. lim z!z0 g(z) = lim f(z) z!z 0 lim g(z) lim g(z) 6= 0 z!z0 z!z 0 1 2 w = f (z)
24 2 2.2 2.2.1 14 z z = z 0 jz0z 0 j < w = f (z) f (z) 0 f(z 0 ) lim (2.5) z!z 0 z 0 z 0 f(z) z = z 0 f 0 (z 0 ) df z=z0 f 0 (z 0 )= df f (z) 0 f(z 0 ) = lim (2.6) z=z0 z!z0 z 0 z 0 z z 0 3 1f f(z +1z) 0 f(z) =f 0 (z)1z + 1z ;! 0(1z! 0) : (2.7) 15 D D regular, holomorphic) f(z) z 0 z 0 z 0 f(z) f(z) z = z 0 z 0 10 z n 2 (z +1z) n = z n + nz n01 1z + 1 2 n(n 0 1)zn02 (1z) 2 111 3
2.2 25 n = lim (z +1z) n 0 z n 1z!0 1z = nz n01 11 z 1=n (z +1z) 1=n = a, z 1=n = b 1=n (z +1z) 1=n 0 z 1=n = 1z = a 0 b a n 0 b n a 0 b (a 0 b)(a n01 + a n02 b + a n03 b 2 + 111ab n02 + b n01 ) = lim a!b 1 a n01 + a n02 b + a n03 b 2 + 111ab n02 + b n01 = 1 n z(1=n)01 z 1=n z =0 2 1. 2. 3. 4. d d d fcf (z)g = cdf(z) ff(z) +g(z)g = df (z) ff(z)g(z)g = df (z) d f (z) g(z) = df(z) + dg(z) dg(z) g(z) +f(z) g(z) 0 f(z) dg(z) g 2 (z) 5. g(z) z = z 0 f(w) w = w 0 = g(z 0 ) f (g(z)) z = z 0 df(g(z 0 )) = df (w(z 0)) dg(z 0 ) dw (2.8)
26 2 2.2 w = z 1=3. f (g(z)) g 1g f 1f 1f = f 0 (g)1g + 1g 1g = g 0 (z)1z + 0 1z 1g! 0! 0 1z! 0 0! 0 1f 1z =(f 0 (g) +)(g 0 (z) + 0 )=f 0 (g)g 0 (z) + f 0 (g) 0 + g 0 (z) + 0 1z! 0 0 (2.8) 3 z m=n n, m n 6= m m=n = m n z(m=n)01 (2.9) (2.9), f(w) =w m, g(z) =z 1=n z = e i 0 2 z m=n =0 =2
2.2 27 2.2) z m=n z 1 z =0 z = z 1 z m=n 1 z 1 z 2 z m=n 2 z 1 z 2 z =0 z z m=n z m=n z =0 4 2.2.2 z! z 0 1 13 f = u +iv z = z 0 x y f (z) =u +iv @u @x = @v @y @u @y = 0 @v @x (2.10) Cauchy-Riemann z! z 0 z+1z z 1z =1x+i1y 4 z m=n
28 2 (1) 1x! 0; 1y =0 (2) 1x =0; 1y! 0 (1) f 0 +1x; y) 0 u(x; y) v(x +1x; y) 0 v(x; y) (z) = lim fu(x +i g 1x!0 1x 1x = @u @x +i@v @x (2) f 0 y +1y) 0 u(x; y) v(x; y +1y) 0 v(x; y) (z) = lim fu(x; +i g 1y!0 i1y i1y = 0i @u @y + @v (2.11) @y @u @x = @v @y @u @y = 0 @v @x u x ; u y ; v x ; v y (2.10) q u = u 0 + a1x + b1y + " (1x) 2 +(1y) 2 v = v 0 0 b1x + a1y + " 0q (1x) 2 +(1y) 2 ", " 0 j1zj!0 f = u +iv f = f 0 +(a 0 ib)1z + " 00q (1x) 2 +(1y) 2 j1zj!0 " 00! 0 f 0 = a 0 ib
2.2 29 f = u +iv u, v ( ) 5 12 f(z) =z 2 u(x; y) = x 2 0 y 2, v(x; y) =2xy u; v 7 7.2 u v x, y v(x; y) @ 2 u @x 2 = @2 v @x@y @ 2 u @y 2 = 0 @ 2 v @x@y 2 u @ 2 u @x 2 + @ =0 (2.12) @y2 2 v @ 2 v @x 2 + @ =0 (2.13) @y2 2 u v 2 5 u(x; y) (x; y) 1u u(x +1x; y +1y) 0 u(x; ) =a1x + b1y + " p (1x) 2 +(1y) 2 a,b 1x, 1y (1x) 2 +(1y) 2! 0 "! 0
30 2 16 2 2 u @ 2 u @x 2 + @ =0 (2.14) @y2 1 u(x; y) f(z) =u + iv v(x; y) u, v 13 f(z) =u +iv u(x; y) =x 2 0 y 2 @ 2 1u = @x 2 + @ 2 @y 2 u =202=0 u u v @v @x = 0@u @y =2y @v @y = @u @x =2x v(x; y) = = Z x dx(2y) =2xy + (y) Z y dy(2x) =2xy + (x) (y) = (x) = f(z) =(x 2 0 y 2 )+i2xy = z 2 14 f(z) z = z 0 f 0 (z 0 ) 6= 0 z = z 0 w = f(z) z = g(w) df =1= dg dw (2.15)
2.2 31 ( 2 1 2 2 x, y f = u +iv u = u(x; y) v = v(x; y) (2.16) ( D(u; v) D(x; y) = @u @x @v @x @u @y @v @y = J (2.17) (J 6= 0) x = x(u; v) y = y(u; v) (2.18) @u @u J = @x @y = u xv y 0 u y v x = u 2 x + v2 x = jf 0 (z)j 2 @v @x @v @y f 0 (z) 6= 0 1 2 f (z) g(w) w = f(g(w)) 1= dw dw = df ( ) dg dw (2.19) 2.2.3 u v 2
32 2 14 (r) ( @2 @x 2 + @ 2 @y 2 + @ 2 )(r) =0 @z2 r q F grad F (r) =0qgrad(r) =0q( @ @x ; @ @y ; @ @z )(r) E E(r) =0grad(r) =0( @ @x ; @ @y ; @ @z )(r) (r) = (r) z 2 (r) E(r) ( @2 @x 2 + @ 2 )(r) =0 @y2 E(r) =0grad(r) =0( @ @x ; @ @y )(r) (x; y) 2 (x; y) @ (x; y) = @ @x @y (x; y); @ @ (x; y) =0 @y @x (x; y) = =
2.2 33 15 v(r)!(r) = rotv! =0 rotv =0 v v = grad8 8 8 t @ + div(v) =0 @t = const divv =0) div grad8 = ( @2 @x 2 + @ 2 @y 2 + @ 2 @z 2 )8 = 0 8 v x, y z 2 v 8 2 ( @2 @x 2 + @ 2 @y 2 )8 = 0 v x = @8 @x ; v y = @8 @y 2.3 1 A P C A P 90 v v n A P C Z P 9(P) = v n ds A C P A P 2 C C 0 C C 0 9(x; y) 9 = 0 9 v n = @9=@s 90
34 2 2.3. x, y 0y,x 0v y, v x v x = @9 @y ; v y = 0 @9 @x 8 9 f =8+i9 z = x +iy 8 9 f 2.3 2.3.1 D w = f(z) D 3 z i, (i = 0; 1; 2) z 0 z i (i =1; 2) f w 0 w i (i =1; 2) z w z 0 w 0 z 1 0 z 0 = r 1 e i 1; z 2 0 z 0 = r 2 e i 2 w 1 0 w 0 = 1 e i 1; w 2 0 w 0 = 2 e i 2 (2.20)
2.3 35 2.4 3 zi, (i =0; 1; 2) w = f (z) wi (i =0; 1; 2) w = f (z) f 0 (z 0 )= w 1 0 w 0 w 2 0 w 0 lim = lim z 1!z 0 z 1 0 z 0 z 2!z 0 z 2 0 z 0 f 0 (z 0 )=lim 1 r 1 e i( 10 1 ) = lim 2 r 2 e i( 20 2 ) (2.21) 1 r 1 e i(101) = f 0 (z 0 )+ 1 2 r 2 e i( 20 2 ) = f 0 (z0 )+ 2 (2.22) 1, 2 z! z 0 r 1, r 2, 1, 2 0 z! z 0 lim 2 1 = lim r 2 r 1 ; lim( 2 0 1 ) = lim( 2 0 1 ) z 0 3 1z 0 z 1 z 2 w 0 3 1w 0 w 1 w 2 1z 0 z 1 z 2 1w 0 w 1 w 2 (2.23) 2.21 jf 0 (z 0 )j argf 0 (z 0 ) 1 0 1 f(z)
36 2 2.5 w = 1 z. f (z) (conformal) f(z) 17 D w = f (z) z 0 2 w 0 2 w = f (z) z 0 15 f(z) z 0 f 0 (z 0 ) 6= 0 w = f(z) z 0 f 0 (z 0 )=0 w = f(z) z = z 0 16 w = 1 z z = re i w =(1=r)e 0i z =0 w =0 z =0 w =0 z w z =0 2.5
2.3 37 2.6 w = z 2. 17 w = z 2 z y =0;x> 0 v =0;u > 0 z x =0;y >0 u =0;v < 0 z 1 w f 0 (0) = 0 z =0 u = x 2 0 y 2 ; v =2xy z x = a ) u = a 2 0 ( v 2a )2 y = b ) u =( v 2b )2 0 b 2 2 ( 2.6 2.3.2 2 I w = u(x; y) + iv(x; y) u = u(x; y) ; v = v(x; y) (2.24)
38 2 (x; y) @ 2 @x 2 + @ 2 n @u 2+ @u 2 o @ 2 @y 2 = @x @y @u 2 + @ 2 @v 2 (2.25) @u 2+ @u 2= jw 0 (z)j 2 @x @y (2.25) (x; y) (u; v) @ 2 @x 2 + @ 2 @y 2 f (x; y) =0, @ 2 @u 2 + @ 2 @v 2 f (x(u; v);y(u; v))=0: (2.26) 2 f =8+i9 z = z(w) w = w(z) z w f(z(w)) = 8 + i9 w w 8,9 u; v (w = u +iv w = u +iv 8(u; v), 9(u; v) z w 9 = 2 a ( C I I I(C) = v 1 ds = v s ds (2.27) C C
2.3 39 2.7. 2.8. 6 w = z + a2 z (2.28) z = ae i w =2acos z a w 4a 2.7 w 6 f = U(z + a 2 =z) + i(0=2)logz 3 3.4
40 2 f = Uw (2.29) w = u +iv 8(u; v) =Uu,9(u; v) =Uv u U v 0 v = f z f = U(z + a2 z ) (2.30) z!1 f! Uz z = ae i f = 2Uacos (2.31) 8=2U cos ; 9=0 (2.32) ( v =( 1 @8 r @ ) = 02U sin (2.33) r=a 2.8 w f = U(e 0i w + a2 e i w ) (2.34) z 2a z = w + a2 w z 2 f(z) =Az n (2.35)
2.3 41 2.9 =n. z = re i f 8=Ar n cos n; 9=Ar n sin n (2.36) 9 =0 = m=n (n =0; 1; 2; 111) m=n 2.9
42 2 2.4 2 1. 21. 4. 2. u(x; y) u (1)u = x 3 0 3xy 2 (2)u = 1 2 ln(x2 + y 2 ) (3)u =e x cos y 3. w(z) z = re i ;w(z) =Re i2 @R @r = R r @2 @ ; @R @ = 0rR@2 @r 4. w =1=z z x = a y = b w 5. w = z + a 2 =z = (z = re i ) 6. = z + b 2 =z r = (z = re i ) 7. w = z +e z z 0 Imz w Imz = 6 Imw = 6; Rew 01