4.1. Lecture 4 & 5. Riemann. f(t)dt. a = t 0 <t 1 < <t n 1 <b= t n (4.1) , n [t i 1,t i ] t i t i 1 (i =1,...,n) f(ξ i )(t i t i 1 ) (4.

Transkrypt

1 Lecture 4 & Riemnn t f(t) [, b] (Riemnn ) f(t)dt [, b] n 1 t 1,...,t n 1 t 0 <t 1 < <t n 1 <b t n (4.1), n [t i 1,t i ] t i t i 1 (i 1,...,n) δ, [t i 1,t i ] ξ i. n f(ξ i )(t i t i 1 ) (4.2) i1, δ 0 (4.1) Figure 4.1: Riemnn u(t), v(t) t w(t) u(t)+iv(t) w(t) [, b] w(t)dt u(t)dt + i v(t)dt (4.3) 43

2 (4.4): ( Re ( Im ) w(t)dt ) w(t)dt Re(w(t))dt (4.4) Im(w(t))dt (4.5) z 0 w(t)dt z 0 w(t)dt, z 0 (4.6) w(t)dt w(t) dt ( <b) (4.7) ( ) Re w(t)dt u(t)dt Rew(t)dt (4.8) (4.5). (4.6):z 0 x 0 + iy 0 z 0 w(t)dt {(x 0 u(t) y 0 v(t)) + i(x 0 v(t)+y 0 u(t))} dt (x 0 u(t) y 0 v(t))dt + i (x 0 v(t)+y 0 u(t))dt x 0 u(t)dt y 0 y 0 v(t)dt + ix 0 v(t)dt + iy 0 u(t)dt ( ) (x 0 + iy 0 ) u(t)dt + i v(t)dt z 0 w(t)dt (4.9) (4.7) w(t)dt r 0 e iθ 0 w(t)dt r 0 44

3 r 0 e iθ 0 Re w(t)dt e iθ 0 w(t)dt e iθ 0 w(t)dt Re(e iθ 0 w(t))dt e iθ 0 w(t) dt w(t) dt (4.10) ( ) 4.2 x(t), y(t) t( t b) z(t) x(t)+iy(t) ( t b) (rc) : z(t) x(t) +iy(t) ( t b), z(), z(b) z() z(b) z(b) z() z(t) Figure 4.2:, Jordn ( ) ( <t 1 t 2 <b t 1,t 2 z(t 1 ) z(t 2 ) ) z(), z(b),, Jordn ( ) 45

4 () (b) Figure 4.3: ()Jordn (b)jordn : z(t) x(t)+iy(t) ( t b), x(t), y(t) z(t) z (t) x (t)+iy (t) (4.11) t, b x (t), y (t) t b, ( 1 ) 4.1. z 0 z 1 z(t) z 0 (1 t)+z 1 t (0 t 1) (4.12) 4.2. z e iθ (0 θ 2π) (4.13) Jordn ( ) 4.3. z e iθ (0 θ 2π) (4.14), ( ) z z(t)( t b),z (t) 0,, t 0 <t 1 < <t n 1 <t n b, i : z(t) t i t t i+1 46

5 z 1 z 0 () (b) Figure 4.4: () (b) 1 c 2 c 3 Figure 4.5: z(), z(b) : z z(t) t b, z z( t) x( t)+iy( t), b t (4.15) z(b), z() 4.3 : z(t) x(t)+iy(t) ( t b) f(z) f(z)dz f(z(t))z (t)dt (4.16) 0,..., n 1 f(z)dz f(z)dz + + f(z)dz (4.17) 0 n 1 47

6 4.4. z(t) z 0 (1 t)+z 1 t (0 t 1) f(z) 1 1 dz (z 1 z 0 )dt z 1 z z(t) z 0 (1 t)+z 1 t (0 t 1) f(z) z n (n Z, n 1) 1 z n dz (z(t)) n z (t)dt d n +1dt z(t)n+1 dt zn+1 n +1 1 z0 n z e iθ (0 θ 2π) f(z) z n (n Z) 2π { 0 (n 1) z n dz ie i(n+1)θ dθ 2πi (n 1) 0 (4.18), L x (t) 2 + y (t) 2 dt z (t) dt dz (4.19) dz : z (t) dt (4.20) f(z)dz f(z( t))( z ( t))dt b b f(z(τ))( z (τ))( dτ) f(z(τ))z (τ)dτ f(z)dz (4.21) 48

7 f(z)dz dz + f(z)dz (4.22) z 0 f(z) z 0 f(z)dz (4.23) (f(z)+g(z))dz f(z)dz + g(z)dz (4.24) L, f(z) M M f(z)dz f(z) dz ML (4.25) (4.25) f(z)dz b f(z(t))z (t)dt f(z(t))z (t) dt f(z(t)) z (t) dt f(z) dz M dz ML (4.26) : z Re iθ (R>0, π θ π) z 1 exp(( 1)Logz) exp(( 1)Logz)dz (4.27) 4.2. z 1 dz (4.28) z 1 z 1 1 ( ) 49

8 4.4 uchy-gourst f(z) f(z) u(x, y)+iv(x, y). f(z) (4.16) f(z)dz f(z(t))z (t)dt (u + iv)(x + iy )dt (ux vy )dt + i (udx vdy)+i (uy + vx )dt dx x (t)dt, dy y (t)dt (udy + vdx) (4.29) Green xy D D D, D, D D (Fig. 4.6) Figure 4.6: D D 2 1 ϕ 1 (x) <ϕ 2 (x) (α<x<β) R (Fig. 4.7()) R

9 . P (x, y) x 9 ( ) P (x, y)dx P (x, y)dx + 1 β α β 2 (P (x, ϕ 1 (x)) P (x, ϕ 2 (x))) dx α R ϕ2 (x) P dx ϕ 1 (x) y dy P dxdy (4.30) y, 2 1 ψ 1 (y) <ψ 2 (y) (α<y<β) R (Fig. 4.7 (b)) R Q(x, y) y ( Q(x, y)dy β α β α R ) Q(x, y)dy ( Q(ψ 1 (y),y)+q(ψ 2 (y),y)) dy ψ2 (y) Q dy ψ 1 (y) x dx Q dxdy (4.31) x D, P x, D P (x, y) x P (x, y)dx D P dxdy (4.32) y 9 : z(t) x(t)+iy(t) ( t b) x, y P (x, y)dx : Q(x, y)dx : P (x(t),y(t))x (t)dt P (x(t),y(t))y (t)dt 51

10 () (b) Figure 4.7: D Q Q(x, y)dy dxdy (4.33) x D Figure 4.8: Jordn 4.1. (Green ), Jordn, R. R P (x, y), Q(x, y) Pdx+ Qdy R ( Q x P ) dxdy (4.34) y 52

11 Figure 4.9: Jordn R P y, Q x, A 1 ( ydx + xdy) 2 R dxdy (4.35) R zdz (x iy)(dx + idy) xdx + ydy + i( ydx + xdy) (4.36) Green (xdx + ydy) (0 0)dxdy 0 (4.37) R A A 1 2i zdz (4.38) 4.3., b : (x, y) ( cos θ, b sin θ) (0 θ 2π) zdz (4.39) (4.16) z x + yi (4.38) Green Stokes 3, 3 (x, y, z) A(x, y, z) S A A A S A ds ( A) ds (4.40) S 53

12 . Stokes, xy Jordn, S xy ds (dx, dy, 0), ds (0, 0,dxdy) A (P (x, y),q(x, y), 0) (4.40) (4.34) uchy R f(z) u(x, y)+iv(x, y) f(z)dz udx vdy + i vdx + udy (4.41) u, v R, Green f(z)dz udx vdy + i vdx + udy ( v x u y )dxdy + (u x v y )dxdy (4.42) R R uchy-riemnn (uchy ) Jordn f(z), f (z) f(z)dz 0 (4.43) f (z) uchy-gourst (4.43) ( ) Jordn f(z) D 1, D (simply connected) Jordn f(z) D D Jordn f(z)dz 0 (4.44) (uchy ) 54

13 1, 2 D, f(z) D f(z)dz f(z)dz (4.45) 1 2 () (b) Figure 4.10: uchy D, (multiply connected) Jordn Jordn Jordn, f(z) 1, f(z) f(z)dz f(z)dz (4.46) 1 1 n, n +1 Jordn n Jordn j (j 1,,n), R j j R n +1. f(z)dz f(z)dz + + f(z)dz (4.47) 1 n 55

14 Figure 4.11: Figure 4.12: 4.5 D f(z), D F (z) f(z) (4.48), D F (z) D f(z) (primitive function, ntiderivtive) 4.3. f(z) D, F (z) D f(z) D : z z(t) ( t b) f(z) f(z)dz F (z(b)) F (z()) (4.49) 56

15 F (z) U(x, y)+iv (x, y), z(t) x(t)+iy(t) d dt F (z(t)) d dt U(x(t),y(t)) + i d dt V (x(t),y(t)) U x x (t)+u y y (t)+iv x x (t)+iv y y (t) U x x (t) V x y (t)+iv x x (t)+iu x y (t) (U x + iv x )(x (t)+iy (t)) df dz z (t) (4.50) f(z)dz f(z(t))z (t)dt df (z(t)) dt dt F (z(b)) F (z()) (4.51) ( ) f(z) D. z 0 D z D z 0 z D f(z) f(z)dz. z z 0 f(z)dz 4.4. f(z) D, z 0 z D d dz z z 0 f(s)ds f(z) (4.52) f(z) F (z) z z 0 f(z)dz 57

16 F (z) z z 0 f(s)ds z z + z z D F (z + z) F (z) F (z + z) F (z) z z z 0 z+ z f(z) 1 z z z+ z f(s)ds z 0 f(s)ds f(s)ds (4.53) z+ z f(s), ϵ δ s z <δ f(s) f(z) <ϵ z z <δ F (z + z) F (z) f(z) z 1 z F (z) f(z) ( ) z z+ z z (f(s) f(z))ds (4.54) ϵds 1 ϵ z ϵ (4.55) z, 4.5. z z 0 f(z)dz 4.7. f(z) z, F (z) z f(z) Logz, D: z > 0, π <Argz <π 1 z 58

17 4.6 uchy 4.6. Jordn f(z), z 0. (uchy ) f(z 0 ) 1 2πi f(z) z z 0 dz (4.56) z 0, ρ z z 0 ρ dz 2π iρe iθ dθ z z 0 ρe iθ z z 0 ρ 0 2πi f(z) z z 0 ϵ δ z z 0 <δ f(z) f(z 0 ) <ϵ ρ <δ 0 : z z 0 ρ, z z 0 ρ f(z) f(z 0 ) <ϵ Figure 4.13: uchy z 0 f(z) z z 0 f(z) f(z) dz dz z z 0 z z 0 f(z) dz 2πif(z 0 ) z z f(z) f(z 0 ) dz dz z z 0 0 z z 0 0 f(z) f(z 0 ) z z 0 dz (4.57)

18 . ( ) f(z) dz 2πif(z 0 ) z z 0 < 0 0 f(z) z z 0 dz 2πif(z 0 ) f(z) f(z 0 ) z z 0 dz ϵ ρ dz ϵ 2πρ 2πϵ (4.58) ρ 4.9. z 2 dz z 2 +1 dz 1 ( 1 z 2 2i z i 1 ) z + i 1 (2πi 2πi) 0 (4.59) 2i 4.4. z 2 dz z 4 +1 (4.60) 4.7. (Gourst ) Jordn D f(z) D, n f (n) (z)(n 1, 2, ), f (n) (z) D f (n) (z) f (n) (z) n! 2πi f(s) ds (4.61) (s z) n+1 uchy f(z + z) f(z) z f(z) 1 f(s) 2πi s z ds 1 2πi z 1 2πi ( f(s) s z z f(s) ) ds s z f(s) ds (4.62) (s z)(s z z) 60

19 1 f(s) 2πi (s z)(s z z) ds 1 2πi 1 f(s) 2πi (s z) ds 2 f(s) (s z) ds 1 f(s) z 2 2πi (s z z)(s z) ds 2 z d. s z d. z 0 < z <d s z z s z z d z > 0 f(z) M f(s) z (s z z)(s z) ds 2 f(s) z (s z z)(s z) 2 ds M z ML z ds 0( z 0) (d z )d2 (d z )d2 (4.63). f (z) 1 f(s) 2πi (s z) ds 2 Figure 4.14: Gourst n f (n) (z + z) f (n) (z) z n! z2πi ( ) 1 (s z z) 1 f(s)ds n+1 (s z) n+1 61 (4.64)

20 ( ) n! 1 z2πi (s z z) 1 (n +1)! f(s) f(s)ds ds n+1 (s z) n+1 2πi (s z) n+2 ( ) n! (s z) n+2 (s z z) n+1 (s z) z(n +1)(s z z) n+1 f(s)ds z2π (s z z) n+1 (s z) n+2 (4.65) (s z) n+2 (s z z) n+1 (s z) z(n +1)(s z z) n+1 { (s z) n+2 (s z) (s z) n+1 (n +1) z(s z) n (n +1)n } + ( z) 2 (s z) n 1 + { } 2 z(n +1) (s z) n+1 (n +1) z(s z) n + (n +1)(n +2) ( z) 2 (s z) n + (4.66) 2 (4.65) n! (n +1)(n +2) f(s) z 2π 2 (s z z) n+1 (s z) ds + 2 O( z2 ) (n +2)! z ML 4π (d z ) n+2 d + 2 O( z2 ) 0( z 0) (4.67) ( ) uchy (4.56) n z 0 dn (z z 0 ) 1 dz n 0 n! 1 (z z 0 ) n+1 f (n) (z 0 ) n! f(z) dz (4.68) 2πi (z z 0 ) n z 1 e z dz (4.69) zn n 4.8. f(z) u(x, y)+iv(x, y) u(x, y), v(x, y) 62

21 f(z) f (z) f (z), u, v f (z) f (z) u, v 2 ( ) (Morer) f(z) D D Jordn f(z)dz 0 f(z) D D 2 z 1, z 2 z 2 z 1 f(z)dz, f(z) F (z) z z 0 f(s)ds, F (z) f(z) F (z) f(z) ( ) Morer uchy uchy : z z 0 R f(z) f (n) (z 0 ) n! f(z) dz (4.70) 2πi (z z 0 ) n+1 f(z) M R uchy n 1 f (n) (z 0 ) n!m R R n (4.71) f (z 0 ) M R R (4.72) Liouville z f(z) M M uchy f (z 0 ) M R R R f (z 0 )0 z 0 f(z) ( ) n P n (z) z + n z n ( n 0) 63

22 R 0, z >R 0 n 1,..., 0 < n z z n 2n n n z z n 2 n + n n n z z n 2 n 2 P n (z) n 2 z n P n (z), 1/P n (z) z >R 0 1/P n (z) 2 n z < 2 n n R 0 n z R 0 1/P n (z) P n (z) ( ) 4.7 : uchy-groust, 1 f(z) 3 f(z)dz 0 (4.73) ( ) 3 1, 2, 3, 4 4 f(z)dz f(z)dz (4.74) i i1 f(z)dz 4 f(z)dz (4.75) i i f(z)dz 1 i 4 f(z)dz i1 64

23 i (1) (1) f(z)dz 1 4 f(z)dz (2) (1) (2) 3 (1) (2) (k) f(z)dz 1 4 f(z)dz 1 (k) 4 (k 1) k f(z)dz k1 (k) z 0. z z 0 f(z) f(z) f(z 0 )+f (z 0 )(z z 0 )+η(z z 0 )(z z 0 ) η(z z 0 ) 0 (z z 0 ). ϵ δ z z 0 <δ η(z z 0 ) < ϵ K k K (k) U δ (z 0 ) f(z)dz f(z 0 ) dz + f (z 0 ) (z z 0 )dz + η(z z 0 )(z z 0 )dz (k) (k) (k) (k) η(z z 0 )(z z 0 )dz (4.76) (k) L, (k) L k L 2 k f(z)dz 4k f(z)dz (k) 4 k η(z z 0 )(z z 0 )dz 4k η(z z 0 ) z z 0 dz (k) (k) < 4 k ϵl 2 k ϵl 2 (4.77) ϵ f(z)dz 0 () 3 ( ) uchy 65

24 2 f(z) D, D ϵ, δ,, δ D Γ f(z)dz Γ δ f(z)dz <ϵ (4.78) U δ () {z, z w <δ w } w U δ (w) (4.79) 2 : L, δ D 1 2 δ U δ () 10 T, T D f(z) T 11 ϵ z z <δ f(z) f(z ) < ϵ 2L δ 0 <δ < min(δ, δ ) δ,, z k 1 z k <δ n f(z)dz f(z k )(z k z k 1 ) < ϵ 2 k1 z k 1 z k Γ k, ΓΓ 0 + +Γ n T n f(z)dz f(z k )(z k z k 1 ) Γ n (f(z) f(z k ))dz Γ k ϵ 2L L ϵ 2 (4.80) k1 k1 f(z)dz Γ f(z)dz f(z)dz + f(z)dz Γ n f(z k )(z k z k 1 ) n f(z k )(z k z k 1 ) < ϵ 2 + ϵ 2 ϵ(4.81) k1 k1 ( 2 )