n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v
3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt + i v(t)dt 4 c = α + iβ 5 cf(t)dt = c f(t)dt b f(t) f(t)dt f(t) dt θ 4 c = e iθ [ ] b Re e iθ f(t)dt = Re[e iθ f(t)]dt f(t) dt θ = rg f(t)dt 6 z = z(t), t b f(z) f(z(t)) f(z)dz = f(z(t))z (t)dt z = z( t), b t f(z)dz = f(z( t))( z ( t))dt = f(z(t))(z (t))dt b b f(z)dz = f(z)dz z 7 z z fdz = fdz x, y 8 x, y
fdx = 2 ( ) fdz + fdz, fdy = ( ) fdz fdz 2i f = u + iv 6 (udx vdy) + i (udy + vdx) 9 f dz = f(z(t)) z (t) dt f dz = f dz fdz f dz 5 fdz = f(z(t))z (t)dt f(z(t))z (t) dt = f(z(t)) z (t) dt = f dz 3.. p(x, y)dx + q(x, y)dy p, q Ω Ω Ω pdx + qdy Ω U/ x = p, U/ y = q U(x, y) ( )Ω U/ x = p, U/ y = q U(x, y) ( U pdx + qdy = x x (t) + U ) d y y (t) dt = U(x(t), y(t))dt = U(x(b), y(b)) U(x(), y()) dt ( ) (x, y ) Ω (x, y) Ω U(x, y) U(x, y) = pdx + qdy y x
U(x, y) = x p(x, y)dx + C * 2 U/ x = p U/ y = q du = ( U/ x)dx + ( U/ y)dy pdx + qdy f(z)dz = f(z)dx + if(z)dy Ω F (z) F (z) x = f(z), F (z) y F (z) = if(z) F (z) F (z) = i x y f(z) ( fdz ) F (x) 2 F (z) f(z) 2 f fdz f(z) Ω f(z) 3 n Z, n (z ) n dz = (z ) n (z ) n+ /(n + ) n < n ( ) n = C z = + ρe it, t 2π C dz 2π z = iρe it dt = 2πi ρeit log(z ) ρ < z < ρ 2 log(z ) 3..2 x b, c y d R 4 R R R R *2 C
4 f(z) R f(z)dz = η(r) = f(z)dz R R R 4 R (), R (2), R (3).R (4) η(r) = η(r () ) + η(r (2) ) + η(r (3) ) + η(r (4) ) ( 2) R (), R (2), R (3).R (4) R (k) (k =, 2, 3, 4) η(r (k) ) 4 η(r) R n R R R n... η(r n ) 4 η(r n ), η(r n ) 4 n η(r) R n z R δ >, n, z R n, z z < δ δ z z < δ f(z) ϵ > < z z < δ f(z) f(z ) z z f (z ) < ϵ f(z) f(z ) (z z )f (z ) < ϵ z z n δ 3 dz =, zdz = R n R n η(r n ) = (f(z) f(z ) (z z )f (z ))dz R n η(r n ) f(z) f(z ) (z z )f (z ) dz < ϵ z z dz R n R n z z R n d n L n d, L d n = 2 n d, L n = 2 n L η(r n ) < 4 n dlϵ, η(r) < dlϵ ϵ η(r) =
5 j R ζ j R f(z) R lim (z ζ j )f(z) = z ζ j R f(z)dz = R ζ j ζ R 9 3 R 9 f(z)dz = f(z)dz R R z ζ (z ζ)f(z) ϵ > R R f(z) ϵ z ζ 2 fdz R = fdz f(z) dz ϵ R R R dz z ζ R ζ 2k z R z ζ k ϵ dz ϵ R z ζ < ϵ dz = 8ϵ k R 3..3 3 n = f(z) Ω Ω 6 f(z) f(z)dz = f(z)dz 5 7 ζ j f(z) j lim z ζ j (z ζ j )f(z) =
f(z)dz = 3.2 3.2. 8 dz z 2πi z = z(t), α t β h(t) = t α z (t) z(t) dt [α, β] z (t) h (t) = z (t) z(t) g(t) = e h(t) (z(t) ) g(t) g(t) g (t) = h (t)e h(t) (z(t) ) + e h(t) z (t) = g(t) = g(α) e h(t) (z(t) ) = e h(α) (z(α) ) = z(α) z(β) = z(α) e h(β) = h(β) 2πi 9 2. n(, ) = n(, ) n(, ) = dz 2πi z 2. n(, ) = ( 6 ) 3. n(, )
2 2 z, z 2 z z 2,z 2 z 2 z z 2 2 n(, ) = 3.2.2 f(z) F (z) = f(z) f() z z z = lim z F (z)(z ) = lim (f(z) f()) = z 7 f(z) f() f(z) dz = dz = f() z z dz = 2πi n(, )f() z 22 f(z) n(, )f() = f(z) 2πi z dz ( ) 7 n(, ) = f() = f(z) 2πi z dz n(, ) = f(z) n(, ) = f(z) 23 f(z) n(, z) = f(z) = 2πi ζ z dζ f(z) 2 3.2.3 23
24 φ(ζ) φ(ζ) F n (z) = (ζ z) n dζ F n(z) = nf n+ (z) φ(ζ) 7 f(z) F (z) z z z < δ z z z < δ/2 ζ ζ z > δ/2 F (z) F (z ) = (z z ) φ(ζ)dζ (ζ z)(ζ z ) z z φ(ζ) ζ z ζ z dζ < z z 2 δ 2 φ dζ z z F (z) z φ(ζ) φ(ζ)/(ζ z ) F (z) F (z ) φ(ζ) = z z (ζ z)(ζ z ) dζ z z F 2 (z ) F (z) = F 2 (z) ( ) Ω f(z) Ω δ Ω C f(z) C z n(c, z) = f(z) = 2πi ζ z dζ 24 f (z) = 2πi C C (ζ z) 2 dζ, f (n) (z) = n! dζ 2πi C (ζ z) n+ Ω Ω 25 Ω Ω 3.3 3.3. 7 24 f(z)
26 Ω Ω f(z) Ω Ω Ω f(z) lim z (z )f(z) = ( ) ( ) C C Ω z C f(z) = 2πi C ζ z dζ 24 C z f(z) z = Ω 2πi C ζ dζ f(z) f() F (z) = f(z) f() z z = lim (z )F (z) = z z F (z) f () z F (z) z = f () f (z) f (z) z f (z) f () z = z f () f 2 (z) n f n (z) f n (z) = f n () + (z )f n (z) z = f(z) = f() + (z )f () + (z ) 2 f 2 () + + (z ) n f n () + (z ) n f n (z) n z = f (n) () = n!f n () 27 f(z) Ω Ω Ω f n (z) Ω f(z) = f() + f () (z ) + + f (n ) ()! (n )! (z )n + f n (z)(z ) n f n (z) C C z f n (z) = 2πi (ζ ) n (ζ z) dζ C
3.3.2 28 Ω f(z) Ω f() f (n) () f(z) f n (z) f(z) f() f (h) () 29 Ω f(z) f() = Ω f (h) () h f(z) = (z ) h f h (z) f h (z) f h () (f h () = (z ) ) f h (z) f h (z) z = 3 f(z) f(z) < z < δ f(z) z < δ f() lim z f(z) = f(z) f() = < z < δ f(z) δ δ g(z) = /f(z) g(z) g(z) g() = g(z) g(z) = (z ) h g h (z), g h () h f h (z) = /g h (z) f h () f(z) = (z ) h f h (z) α. lim z α f(z) = z 2. lim z α f(z) = z z 3 (i) α f(z) (ii) h Z α > h α < h 2 (iii) α 2 (i) (ii) h f(z) f(z) f() h
(z ) h f(z) z = φ(z) (z ) h f(z) = B h + B h (z ) + + B (z ) h + φ(z)(z ) h z (z ) h f(z) = B h (z ) h + B h (z ) h+ + + B (z ) + φ(z) φ(z) f(z) z = 2 (iii) f(z) *3 f(z) = e z z = g(z) = f(/z) = e /z z = (z g(z) + z g(z) z g(z) ) 3 3.4 2 3.4., 2,..., n fdz = fdz + fdz + fdz + 2+ + n 2 n + 2 + + n f 2 2 *3 f(z) z = f(/z) z =
= + 2 2 + + n n, j N, j ( ) = 2 j *4 Ω 3.5 32 Ω / Ω, n(, ) = (mod Ω) (mod Ω) 2 2 Ω Ω, (mod Ω) (mod Ω ) 33 f(z) Ω Ω f(z)dz = Ω δ > δ Ω Q j, j J Ω J δ Q j Q j Q j Γ δ = j J Q j Q j Q j Q j Ω δ 4 Ω Ω δ δ ζ Ω Ω δ Q j, j J Q ζ Q, ζ / Ω ζ ζ ζ Q Ω δ Ω δ n(, ζ) = n(, ζ ) = Γ δ ζ n(, ζ) = f Ω z Q j 4 5 n *4 = i i p i, q i p = q j, p 2 = q j2,..., p n = q jn, {j, j 2,..., j n } = {, 2,..., n} i=
2πi Q j ζ z dζ = f(z) = 2πi Γ δ f(z) (j = j ) (j j ) ζ z dζ z z Ω δ f(z)dz = ( 2πi Γ δ ) ζ z dζ dz Γ δ n(, ζ) = f(z)dz = Γ δ ( ) 2πi ζ z dz dζ = Ω Ω z < R Ω Ω Ω Ω n(, ) = (mod Ω ) Ω Ω Ω 4 n= n 2 = π2 6 *5 34 n 2 = π2 6 x2 n= log x π2 dx = x 6 log x x x dx x = e 2y dx dy = 2e 2y = 2(x ) x = e 2δ, x 2 = e 2Y x2 log x Y dx = 2 log ( e 2y )dy x δ x x, x 2 + δ +, Y + *5
log ( e 2y )dy = π2 2 π log (sin x)dx e 2iz = 2ie iz sin z e 2iz = e 2y (cos 2x + i sin 2x) (z = x + iy) x = nπ, y log ( e 2iz ) log ( e 2iz ) π π, π, π + iy, iy π δ 5 *6 π π log ( e 2Y (cos 2x + i sin 2x))dx log ( e 2Y (cos 2x + i sin 2x)) dx (Y + ) log ( e 2iz )dz e2iz log z dz + log z dz e 2iz lim = 2 δ log z log δ + π z z 2 log z dz 4 δ log δ + π δ (δ ) 8 π π log ( 2ie ix sin x)dx = π (log 2 + log ( i) + log (sin x) + log (e ix ))dx = π π log ( i) = iπ 2, log (eix ) = ix π log 2 π2 2 i + π π log (sin x)dx + π2 2 i = log (sin x)dx = π log 2, π 2, π 2 + iy, iy π 2 δ *7 6 5 log (sin x) x = π 2 π 2 log (sin x)dx = π log 2 2 *6 5 *7 π 2
π 2 log ( 2ie ix sin x)dx = π 2 (log 2 + log ( i) + log (sin x) + log (e ix ))dx = π2 8 i δ +, Y + Y 2 3 2 δ Y δ Y δ Y δ Y log ( e 2i iy ) idy + Y log ( e 2y )dy + log ( e 2y )dy + log ( e 2y )dy + δ Y δ δ 2Y log ( e 2i( π 2 +iy) ) idy π2 8 i log ( + e 2y )dy π2 8 (log ( e 4y ) log ( e 2y ))dy π2 8 2δ log ( e 2y )dy = π2 8 log ( e 2t ) π2 dt 2 8 log ( e 2y )dy = π2 2 (t = 2y ) n Z, n lim x + xn+ log x = lim x + lim [ ] x n log xdx = n + xn+ log x (n + ) 2 xn+ = (n + ) 2 N x n log xdx = n 2 (N ) N n= n= x N x log xdx = N n= n 2 (N ) * 8 x N log x x dx *9 f(x) = x log x x < x < f (x) = log x x + ( x) 2 < f(x) =, lim f(x) = < x < < f(x) < lim N x,x 2 + x x2 x x2 x x N dx < x2 x x N log x x dx < x N dx = lim N N = x N log x lim N x dx = *8 *9 2
n= n 2 = lim x N π2 log xdx = N x 6 5 n= n 2 = π2 6 n 2 = log x x dx L. V. / n=