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1 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

2 ( ) ( ) ( ) ( ) ( ) 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

3 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 w = f (z)

4 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 n(n 0 1)zn02 (1z)

5 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 ab n02 + b n01 ) = lim a!b 1 a n01 + a n02 b + a n03 b ab n02 + b n01 = 1 n z(1=n)01 z 1=n z = 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)

6 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

7 ) 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 = z! z f = u +iv z = z 0 x y f (z) @y (2.10) Cauchy-Riemann z! z 0 z+1z z 1z =1x+i1y 4 z m=n

8 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 (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 @y 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

9 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 u v x, y v(x; 2 2 v 2 2 = = =0 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

10 =0 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 1u 2 2 u =202=0 u = =2y = =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)

11 ( x, y f = u +iv u = u(x; y) v = v(x; y) (2.16) ( D(u; v) D(x; = J (2.17) (J 6= 0) x = x(u; v) y = y(u; = u xv y 0 u y v x = u 2 x + v2 x = jf 0 f 0 (z) 6= f (z) g(w) w = f(g(w)) 1= dw dw = df ( ) dg dw (2.19) u v 2

12 )(r) r q F grad F (r) )(r) E E(r) )(r) (r) = (r) z 2 (r) 2 2 )(r) E(r) @y )(r) (x; y) 2 (x; (x; @y (x; (x; y) = =

13 v(r)!(r) = rotv! =0 rotv =0 v v = grad div(v) = const divv =0) div grad )8 = 0 8 v x, y z 2 v )8 = 0 v ; v 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 90

14 x, y 0y,x 0v y, v x v ; v y 8 9 f =8+i9 z = x +iy 8 9 f 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)

15 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( ) = lim( ) 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 ) f(z)

16 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

17 w = z 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 ( I w = u(x; y) + iv(x; y) u = u(x; y) ; v = v(x; y) (2.24)

18 38 2 (x; @u = jw 0 (2.25) (x; y) (u; f (x; y) 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

19 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

20 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 =( ) = 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)

21 =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

22 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) = @ = 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

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s ... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z

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1 : f(z = re iθ ) = u(r, θ) + iv(r, θ). (re iθ ) 2 = r 2 e 2iθ = r 2 cos 2θ + ir 2 sin 2θ r f(z = x + iy) = u(x, y) + iv(x, y). (x + iy) 2 = x 2 y 2 + 1.3 1.4. (pp.14-27) 1 1 : f(z = re iθ ) = u(r, θ) + iv(r, θ). (re iθ ) 2 = r 2 e 2iθ = r 2 cos 2θ + ir 2 sin 2θ r f(z = x + iy) = u(x, y) + iv(x, y). (x + iy) 2 = x 2 y 2 + i2xy x = 1 y (1 + iy) 2 = 1

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