1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A M A M (1) 1.4 ( ). A M M 1. M A M 2. M M M < M > M A M sup A 1.5 ( ). A m ( ) A m (2) 1.6 ( ). A 1. A 1
2. A 3. 4. 5. { n } n N A = { n } n N 1.7 ( ). 1. { n } J, { n} 10 1, 2, 3,, 10 { n } 10 2. { k } k=1 { k} k=1 n 1.8 ( ). { n } 1. { n } 1 2 3 < { n } 2. { n } 1 2 3 > { n } 3.. 1.9 ( ). { n } 1. { n } sup n := sup{ n : n N} n N inf n := inf{ n : n N} n N k = 1, 2, sup n := sup{ n : n k} n k 2. { n } lim sup n := inf n lim inf n := sup n inf n := inf{ n : n k} n k k N k N ( sup n k ( inf n k n ) n = inf ) = sup { } sup n : k N n k { } inf n : k N n k 2
3. { n } () lim sup n = lim inf n R n n (b) lim inf n n = (c) lim sup n = n lim n = lim sup n = lim inf n (3) n n n ( ) lim sup n = n ( ) lim inf n = n lim n = (4) n lim n = (5) n lim sup n lim inf n lim n n n n 1.10 (ε-δ ( ) ). 1. { k } k=1 α ϵ N n n > N n α ϵ [ ] lim sup n = lim inf n n n 2. { k } k=1 ϵ N n, m n, m > N n m ϵ [ ] lim sup n = lim inf n n n 3. { k } k=1 K > 0 N n n > N n > K 3
[ ] lim inf n n = 4. { k } k=1 K > 0 N n n > N n < K [ ] lim sup n = n 1.11 ( ). { n } 1. 2. 3. n lim N N n n n n n 1.12 (2 ). 2 { m,n } m, m, n 2 1.13 ( ). A 1. A N N {1, 2,, N} A 2. A A 3. A N A 4. A A 5. A A 1.14 (R n ). R n n x R n x 1, x 2,, x n x = (x 1, x 2,, x n ) x = (x 1, x 2,, x n ) x 1, x 2,, x n R x R n 1.15 (R n ). x = (x 1, x 2,, x n ), y = (y 1, y 2,, y n ) R ε > 0 1. x + y = (x 1 + y 1, x 2 + y 2,, x n + y n ) 4
2. x = ( x 1, x 2,, x n ) 3. x = x 2 1 + x 2 2 + + x 2 n = n x 2 k 4. ε > 0 B(x; ε) = {y R n : x y < ε}, B(x; ε) = {y R n : x y < ε} (6) x = 0 x; B(ε), B(ε) 1.16 ( ). A R n k=1 1. A x R n ε > 0 B(x; ε) A 2. A A c = {x R n : x A } 3. A R > 0 A B(R) 4. A A R n 1.17 ( ). A R n B R m f : A B A f() B 1.18 ( ). A R n B R m f : A B 1. A 0 A B f(a 0 ) = {f() B : A 0 } 2. B 0 B A f 1 (B 0 ) = { A : f() B 0 } 1.19 ( ). I f lim f(x) = α (7) x inf δ>0 ( sup f(x) α x I, x <δ ) = 0 (8) 1.20 (R n 1 ). A R n f : A R m 5
1. A inf δ>0 ( lim f(x) = α (9) x, x A sup f(x) α x <δ, x A 2. A f ( inf δ>0 ) sup f(x) f() x A, x <δ ) = 0 = 0 (10) lim f(x) = f() (11) x, x A 3. f A f 1.21 ( ). A R n 1. A A A := {x R n : ε > 0 B(x, ε) A } 2. A Int(A) = (A c ) c 1.22 ( ). I {f n } f ( ) lim sup f(x) f n (x) = 0 (12) n x I 1.23 (). U {f n } B U B 1.24 ( ). 1. {U λ } λ Λ R n () Λ (b) λ Λ R n U λ {U λ } λ Λ R n 6
2. R n {U λ } λ Λ R n U λ = {x R n : λ Λ x U λ } (13) λ Λ 3. R n {U λ } λ Λ R n U λ = {x R n : λ Λ x U λ } (14) λ Λ 4. R n {U λ } λ Λ R n A A U λ (15) λ Λ () {U λ } λ Λ R n U λ (b) {U λ } λ Λ R n Λ (c) Λ 0 Λ {U λ } λ Λ0 {U λ } λ Λ 1.25 (). A A 1.26 ( ). A R n f : A R m f A ( ) lim δ 0 sup f(x ) f(x) x,x A, x x <δ = 0 (16) 1.27 ( ). A R n 1. A U, V A = (A U) (A V ), (A U) (A V ) =, (A U), (A V ). (17) A A 2. A A 3. A p, q A γ : [0, 1] A γ(0) = p, γ(1) = q γ 7
4. 1.28 ( ). I = (, b) f c (, b) f (c) = lim x c f(x) f(c) x c (18) lim f,c(x) = 0 (19) x c f(x) = f(c) + (x c)f (c) + (x c) f,c (x) (20) 1.29 ( ). n 2 f (n) f (n 1) f 1.30 (C 1 - C k - C - )., b R < b f : (, b) R C 1 - f (, b) f f k f C k -f C k ((, b)) C - k C k - 1.31 ( ). n (x x 0 ) n, x 0 R n=0 1.32 (e x, sin x, cos x ). 1. e x = exp(x) = 1 + x + x2 2 + x3 6 + + xn n! + 2. sin x = x 1 6 x3 + 1 120 x5 + ( 1)k (2k + 1)! x2k+1 + = 3. cos x = 1 x2 2 + x4 24 x6 720 + + ( 1)n (2n)! x2n + k=0 ( 1) k x 2k+1 (2k + 1)! e e = e 1 = 1 + 1 + 1 2 + 1 6 + + 1 n! + (21) 1.33 ( ). 1. log x e x 2. x = e x log, > 1 8
3. log b = log b/ log, (0, 1) (1, ), b > 0 1.34 ( ). cos x = 0, 0 < x < 2 2 π 1.35 (tn). tn x = sin x cos x 1.36 ( ). 1. sin : [ π 2, π ] [ [ 1, 1] sin 1 : [ 1, 1] π 2 2, π ] 2 2. cos : [0, π] [ 1, 1] cos 1 : [ 1, 1] [0, π] ( 3. tn : π 2, π ) ( R tn 1 : R π 2 2, π ) 2 1.37 ( ). < < b < 1. {x i } N i=0 [, b] x 0 =, x N = b D[, b] 2. [, b] {y i } M i=0 [, b] {x i} N i=0 {y i} M i=0 {x i} N i=0 3. {ξ i } N i=1 [, b] {x i} N i=0 x i 1 ξ i x i, i = 1, 2,, N 4. {x i } N i=0 D[, b] = sup x i x i 1 (22) i=1,2,,n 1.38 ( ). f : [, b] R 1. [, b] = {x i } N i=0 f S (f) = s (f) = N i=1 N i=1 2. (x i x i 1 ) sup x i 1 x x i f(x) (23) (x i x i 1 ) inf x i 1 x x i f(x) (24) f(x) dx = f(x) dx = 9 inf S (f) (25) D[,b] sup s (f) (26) D[,b]
3. 1.39 ( ). f(x) dx = f(x) dx = f(x) dx = f(x) dx (27) f(x) dx (28) 1. f f 2. f(x) dx f(x) f 1 f(x) dx = x f(u)du 1.40 (f x ). f : [, b] R x ( ) ( ) f(x) = lim δ 0 f(x) = lim δ 0 ( sup y [,b] (x δ,x+δ) f(y) inf f(y) y [,b] (x δ,x+δ) ) = lim δ 0 ( = lim δ 0 sup y (x δ,x+δ) F (y) inf F (y) y (x δ,x+δ) ) (29) (30) f(b) F (x) = f(x) f() (x b ) ( x b ) (x ) 1.41 (0 ). E R 0 ε > 0 I 1, I 2,, I j, E I j, j=1 l(i j ) < ε (31) j=1 1.42 ( ). 10
1. f : [, ) R b > f [, b] f(x) dx = lim R R f(x) dx (32) f(x) dx 2. f : [, c) R b < c f [, b] c f(x) dx = lim R c R f(x) dx (33) c f(x) dx 3. f : [, c) (c, b] A < c, c < B b A, B f [, A] [B, b] f(x) dx = lim A c A f(x) dx + lim B c 4. B f(x) dx (34) f(x) dx 1.43 ( ). γ(t) = (x 1 (t), x 2 (t),, x n (t)) [, b] R n γ L(γ) N L(γ) = sup n (x k (t j ) x k (t j 1 )) 2 : {t j } N j=0 [, b] j=1 k=1 1.44 ( ). 1, 2,, n, (35) ( n ) (36) { n } (37) 11
1.45 ( ). { n } Q ε > 0 N N m, n N m n < ε (38) 1.46 ( ). { n } {b n} [ ] ε > 0 N N n N n b n < ε 1.47 ( ). { n } M n M, n N 1.48 ( ). { n }, {b n} 1. { n } + {b n} { n + b n } 2. { n } {b n} { n b n } 3. { n } {b n} { nb n } 4. {b n } 0 { n} {b n} { n/b n } 1.49 ( ). { n } {b n} { n} {b n } ε > 0 N n N b n n > ε 1.50 ( ). {{ m,n } } m=1 m N { m,n } 1.51 ( ). m = 1, 2, α m = { m,n } {α m} m=1 = {{ m,n } } m=1 ε = {ε n} 0 N n n, m > N ε < α m α n < ε (39) 1.52 ( ). α = {α n } {α n } m=1 = {{ m,n} } m=1 α ε = {ε n } 0 N n n, m > N ε < α m α < ε (40) α {α n } m=1 = {{ m,n} } m=1 {α n } m=1 = {{ m,n} } m=1 12
2 2.1 ( ). θ R, n Z 2.2 (). 1. { n } (cos θ + i sin θ) n = cos nθ + i sin nθ lim n n lim n = sup n n n N M R { n } n M, n = 1, 2, { n } 2. 2.3 ( ). { n } n N 1. lim n n 2. ε > 0 N N n, m > N n m < ε 2.4 ( ). { n } 1. 2. n sup N N N n < (41) n n n = (42) 2.5 ( ). { n } {b n} n b n n = 1, 2, b n n 13
2.6 (( ) ). { n } n n+1, n = 1, 2,, lim n n = 0 (43) ( 1) n 1 n 2.7 ( ). { n } α σ : N N σ(k) = α 2.8 ( 2 ). { m,n } m, 2 m, n N m,n ( ) ( ) ( n ) m,n = m,n = m,n+1 m. (44) m=1 m=1 2.9 ( 2 ). { m,n } m, 2 ( ) ( ) ( n ) m,n, m,n, m,n+1 m (45) m=1 m=1 ( ) ( ) ( n ) m,n = m,n = m,n+1 m. (46) m=1 m=1 2.10 (R ). R 2.11 ( ). k=1 m=1 m=1 m=1 1. Z 2. A 3. A B 2.12 (). f : X Y X Y A, B X, C, D Y 1. f(a B) = f(a) f(b) 2. f(a B) f(a) f(b) 3. f 1 (C D) = f 1 (C) f 1 (D) 4. f 1 (C D) = f 1 (C) f 1 (D) 14
2.13 ( ). I f 1. lim x f(x) = α. 2. I { n } lim n n = lim n f( n) = α. 3. ε > 0 δ > 0 y (x δ, x + δ) I f(y) α < ε 2.14 (ε-δ ). A R n A f : A R m (1) lim f(x) = α x, x A (2) ε > 0 δ > 0 x A x < δ f(x) α ε (3) A { n } n N {f( n )} n N α 2.15 ( ). A R n f : A R m 1. f A 2. U R m f 1 (U) A 2.16 ( ). f : A R m 1. f A 2. E R n f(e A A) f(e A) 2.17 ( ). I {f n } f f 2.18 ( ). R > 0 [ R, R] n = {x = (x 1, x 2,, x n ) : i = 1, 2,, n R x i R} (47) 2.19 ( ). A R n 1. A 2. A 15
2.20 ( ). A R n f : A R m f(a) 2.21 ( ). 2.22 ( ε-δ ). A R n A f : A R m 1. f A 2. ε > 0 δ > 0 x, x A x x < δ f(x) f(x ) ε 2.23 ( ). K R n {U λ } λ Λ δ > 0 x, y K x y δ x, y U λ λ Λ 2.24. δ {U λ } λ Λ 2.25 (). 2.26 (R ). R 2.27 ( ). R n 2.28 (R n R ). R n R 2.29 (R ). R 2.30 (R ). R R, (, ), (, ) 2.31 (R n ). G R n 1. G 2. G 3. G 2.32 (). A R n f : A R m f(a) 2.33 (). A R n f : A R f(a) 2.34 ( ). I = [, b] f : I R 16
(1) f(), f(b) f I γ R (f() γ)(f(b) γ) 0 c I f(c) = γ (2) f I 2.35 (m ). m 2 > 0 x m = x m 2.36 ( ). I = (, b) f f 2.37 ( ). f(x), g(x) 1. (f(x) + g(x)) = f (x) + g (x) 2. ( f(x)) = f (x) 3. (f(x)g(x)) = f (x)g(x) + f(x)g (x) ( ) f(x) 4. g (x) 0 = f (x) g(x) g(x) f(x)g (x) g(x) 2 = f (x)g(x) f(x)g (x) g(x) 2 k x k = k x k 1 0 2.38 (). I, J R f : I J, g : J R g f : I R f(g(x)) = f (g(x))g (x) (48) 2.39. I = (, b) f x = c I f (c) = 0 2.40 ()., b R < b f : [, b] R f (, b) < c < b, c f(b) f() b = f (c) (49) 2.41 ()., b R < b f, g : [, b] R f, g (, b) g (, b) 0 < c < b, f(b) f() g(b) g() = f (c) g (c) (50) c 17
2.42 ( )., b R < b f, g : [, b] R f, g (, b) g (, b) 0 p (, b) f(x) f(p) lim x p g(x) g(p) = lim f (x) x p g (x) (51) 2.43 ( ). f I C 1 - f (x) 0, x I f g : f(i) I C 1-2.44 ( ). A < x, < B n = 1, 2, f : (A, B) R n, x c f(x) = f() + (x )f () + + (x )n 1 f (n 1) () + 1 (n 1)! n! (x )n f (n) (c) (52) 2.45 (). f : (A, B) R M, R f (n) (x) MR n (53) A < x < B A < x < b f(x) = f() + (x )f () + + (x )n 1 f (n 1) () + (54) (n 1)! 2.46. f : (, b) R C 2 - f (c) = 0, f (c) > 0 f δ > 0 (c δ, c + δ) (, b) f(c) = min{f(x) : x (c δ, c + δ)} 2.47 ( ). n (x x 0 ) n R [0, ] 1 n = lim sup n (55) R n R n (x x 0 ) n 1. x x 0 > R 2. x x 0 < R n=0 n=0 n (x x 0 ) n n=0 n (x x 0 ) n n=0 18
3. x x 0 < R f(x) = n (x x 0 ) n n=0 f (x) = (n + 1) n+1 (x x 0 ) n (56) n=0 f n+1 2.48. { n } n N lim n n 2.49 ( ). lim n n n 1. (sin x) = cos x 2. (cos x) = sin x 3. (e x ) = e x 2.50 ( ). x, y 1. e x+y = e x e y 2. e x > 0 2.51. 1 x f : R R f (x) = f(x) f(0) = f(x) = exp x 2.52 (sin, cos ). α, β R sin(α + β) = sin α cos β + cos α sin β cos(α + β) = cos α cos β sin α cos β 2.53 (2 ). 2 x f : R R f (x) = f(x) f(0) =, f (0) = b f(x) = cos x + b sin x 2.54 ( ). 1. (x ) = x 1 2. (tn x) = tn 2 x + 1 = 1 cos 2 x 3. ( x ) = x log, > 0 4. (log x) = 1 x 19
5. (log x) = 1, (0, ) \ {1} x log 2.55 ( ). d 1 dx sin 1 x =, x ( 1, 1) 1 x 2 d dx tn 1 x = 1 1 + x 2, x R 2.56 ( ). f : [, b] R ε > 0 δ > 0 < δ f(x) dx ε < S (f) (57) 2.57 ( ). f : [, b] R ε > 0 δ > 0 < δ f(x) dx S (f) + f(x) dx s (f) < ε (58) 2.58 ( ). f, g : [, b] R f + g, f g l (f(x) + g(x)) dx = f(x) dx + g(x) dx, lf(x) dx = l f(x) dx (59) 2.59 ( ). < b < c f : [, c] R f [, b] [b, c] c f(x) dx = f(x) dx + c b f(x) dx (60) 2.60 ( I). f : [, b] R F (x) = x f(t) dt, x (, b) F (x) = f(x) (61) 20
2.61 ( ). I, b I f I f f (x) dx = f(b) f(),, b I (62) 2.62 ( ). n = 1, 2, f n f(x) = f()+(x )f ()+ + (x )n 1 (n 1)! f (n 1) 1 ()+ (n 1)! 2.63 (). 1 x 2 dx = 1 ( x ) 1 x 2 2 + sin 1 x + C dx = 1 x 2 sin 1 x + C dx = log(x + x 2 + 1) + C 1 + x 2 dx x 2 1 = log(x + x 2 1) + C 1 + x 2 dx = 1 2 x 2 1 dx = 1 2 x ( x 1 + x 2 + log(x + ) x 2 + 1) + C ( x x 2 1 log(x + ) x 2 1) + C (x s) n 1 f (n) (s) ds 2.64 (). f, g : [, b] R f(x) g(x), x [, b] (63) f(x) dx g(x) dx (64) 2.65 ( ). f : [, b] R f(x) dx f(x) dx (65) 2.66 ( ( ) ). f : [, b] R c [, b] f(x) dx = (b )f(c) (66) 21
2.67 ( ). 1 < p, q < 1 p + 1 q = 1 (67), [, b] f, g ( f(x)g(x) dx ) 1 ( p b ) 1 q f(x) p dx g(x) q dx 2.68 (). [, b] f, g (68) ( ) 1 f(x) + g(x) p p dx ( ) 1 ( f(x) p p b ) 1 dx + g(x) p p dx (69) 2.69 ( (Cuchy-Schwrz), ). [, b] f, g 1. 2. ( 2 f(x)g(x) dx) (f(x)) 2 dx ( 1/2 ( (f(x) + g(x)) dx) 2 (g(x)) 2 dx. 1/2 ( 1/2 f(x) dx) 2 + g(x) dx) 2. 2.70 (lim ). I = [, b] {f n } f lim t f n (t) dt = f(t) dt (70) 2.71 (lim ). I = (, b) C 1 - {f n } 1. {f n } f 2. {f n} g g = f lim n f n(x) = f (x), x (, b) 2.72 ( ). f : [, b] R 22
1. f 2. f B = {x [, b] : f(x) > f(x)} 0 2.73 ( ). 1. f, g : [, ) R b > f, g [, b] f(x) g(x) f(x) dx g(x) dx 2. f, g : [, c) R b < c f, g [, b] f(x) g(x) c f(x) dx c 2.74 ( ). n = 0, 1, 2, g(x) dx 1. Γ 2. Γ(x + 1) = xγ(x), x > 0. 3. 0! = 1 0 t n e t dt = lim R R n Γ(n) = (n 1)! Γ(x) = Γ(n + 1) = n! 2.75 ( ). 0 t x 1 e t dt, x > 0 0 t n e t dt = n!. (71) 0 t n e t dt 1. k N Γ (k) (α) = 0 0 (log t) k t α 1 e t dt (72) log t k t α 1 e t dt < (73) 2. Γ 2 () log φ (b) φ(1) = 1. 23
(c) φ(x) = (x 1)φ(x 1), x > 1. 2.76 ( ). α, β > 0 1. B(α, β) = 1 0 2. B(α, β) = Γ(α)Γ(β) Γ(α + β) B(α, β) 2.77 (Γ ( 1 2) ). 1. Γ 2. ( ) 1 = π. 2 e x2 dx = π. x α 1 (1 x) β 1 dx 2.78 (1/2 ). Γ(2x) = πγ(x)γ 2.79 ( ). sin πx = ( x + 1 ), x > 0 2 π Γ(x)Γ(1 x) x (0, 1) 2.80. γ : (A, B) R n C 1 - γ x 1, x 2,, x n L(γ) = n x k (t)2 dt 2.81 ( ). { n } Q k=1 1. ε > 0 N N m, n N m n < ε (74) 2. ε > 0 N N m, n > N m n < ε (75) 24
3. ε > 0 N N m, n N m n ε (76) 4. ε > 0 N N m, n > N m n ε (77) 4 2.82 ( N, ε ). { n } Q 1. ε > 0 N N m, n N m n < ε (78) 2. ε > 0 N N m, n N m n < 2ε (79) 3. ε > 0 N N m, n N + 1 m n < ε (80) 4. ε > 0 N N m, n N m n < 1 2 ε (81) 4 2.83 ( ). { n }, {b n}, {c n} 1. { n } { n} 2. { n } {b n} {b n} { n} 25
3. { n } {b n} {b n} {c n} { n } {c n} 2.84 ( ). 2.85 ( ). α = { n } α m = { m } α 2.86 ( ). 2.87 ( ). K = {k n } {α m} = {{ m,n } } α 1 α 2 α m α m+1 K (82) α = {α m } m=1 2.88 ( ). A M A M M M 0 M 0 1. A M 0 2. m < M 0 A > m 26
3 3.1 ( ). z = + bi,, b R 0 3.2 (, ). 3.3 ( ). A M 3.4 ( ). A M M 1. 2. M sup A 3.5 ( ). A m ( ) 3.6 ( ). A 3.7 ( ). 3.8 ( ). { n } 1. { n } 2. { n } 3.. 3.9 ( ). { n } 3.10 (ε-δ ( ) ). 1. { k } k=1 α 2. { k } k=1 3. { k } k=1 4. { k } k=1 3.11 ( ). { n } 1. n 27
2. n 3. n 3.12 (2 ). 3.13 ( ). A 1. A 2. 3. A 4. 5. 3.14 (R n ). 3.15 (R n ). x = (x 1, x 2,, x n ), y = (y 1, y 2,, y n ) R ε > 0 1. 2. 3. 4. ε > 0 B(x; ε) (83) x = 0 x; B(ε), B(ε) 3.16 ( ). A R n 1. A 2. A 3. A 4. A 3.17 ( ). A R n B R m f : A B 3.18 ( ). A R n B R m f : A B 1. A 0 A B f(a 0 ) = 28
2. B 0 B A f 1 (B 0 ) = 3.19 (R n 1 ). A R n f : A R m 1. A lim f(x) = α (84) x, x A 2. A f (85) lim f(x) = f() (86) x, x A 3. f A 3.20 ( ). A R n 1. A A 2. A 3.21 ( ). I {f n } f (87) 3.22 (). U {f n } 3.23 ( ). 1. {U λ } λ Λ R n () Λ (b) λ Λ R n U λ {U λ } λ Λ R n 2. R n {U λ } λ Λ R n U λ = {x R n : λ Λ x U λ } (88) λ Λ 29
3. R n {U λ } λ Λ R n U λ = (89) λ Λ 4. R n {U λ } λ Λ R n A A U λ (90) λ Λ () {U λ } λ Λ R n U λ (b) {U λ } λ Λ R n Λ (c) Λ 0 Λ {U λ } λ Λ0 3.24 (). A {U λ } λ Λ 3.25 ( ). A R n f : A R m f A (91) 3.26 ( ). A R n 1. A U, V (92) A A 2. A 3. A 4. 3.27 ( ). I = (, b) f c (, b) (93) lim f,c(x) = 0 (94) x c (95) 30
3.28 (C 1 - C k - C - )., b R < b f : (, b) R C 1-3.29 ( ). 3.30 (e x, sin x, cos x ). 1. e x = 2. sin x = 3. cos x = e e = e 1 = 1 + 1 + 1 2 + 1 6 + + 1 n! + (96) 3.31 ( ). 1. log x 2. x 3. log b = 3.32 ( ). 3.33 (tn). 3.34 ( ). 1. sin : [ π 2, π ] [ 1, 1] 2 2. cos : [0, π] [ 1, 1] ( 3. tn : π 2, π ) R 2 3.35 ( ). < < b < 1. {x i } N i=0 [, b] 2. [, b] {y i } M i=0 [, b] {x i} N i=0 3. {ξ i } N i=1 [, b] {x i} N i=0 4. {x i } N i=0 D[, b] = (97) 31
3.36 ( ). f : [, b] R 1. [, b] = {x i } N i=0 f 2. 3. 3.37 ( ). 1. f 2. f(x) dx S (f) = (98) s (f) = (99) f(x) dx = (100) f(x) dx = (101) 3.38 (f x ). f : [, b] R x f(b) F (x) = f(x) f() f(x) = (102) f(x) = (103) (x b ) ( x b ) (x ) 3.39 (0 ). E R 0 3.40 ( ). 1. f : [, ) R b > f [, b] f(x) dx (104) f(x) dx 32
2. f : [, c) R b < c f [, b] c f(x) dx (105) c f(x) dx 3. f : [, c) (c, b] A < c, c < B b A, B f [, A] [B, b] f(x) dx = (106) 4. f(x) dx 3.41 ( ). γ(t) = (x 1 (t), x 2 (t),, x n (t)) [, b] R n γ L(γ) L(γ) = 3.42 ( ). 3.43 ( ). { n } Q 3.44 ( ). { n } {b n} [ ] 3.45 ( ). { n } 3.46 ( ). { n }, {b n} 1. { n } + {b n} 2. { n } {b n} 3. { n } {b n} 4. {b n } 0 { n} {b n} 3.47 ( ). { n } {b n} { n} {b n } 33
3.48 ( ). {{ m,n } } m=1 3.49 ( ). m = 1, 2, α m = { m,n } {α m} m=1 = {{ m,n } } m=1 3.50 ( ). α = {α n } {α n } m=1 = {{ m,n} } m=1 α 34
4 4.1 ( ). 4.2 (). 1. { n } 4.3 ( ). { n } n N 1. lim n n 2. 4.4 ( ). { n } 1. n (107) 2. n n (108) 4.5 ( ). { n } {b n} 4.6 (( ) ). 4.7 ( ). { n } 4.8 ( 2 ). { m,n } m, 2 m, n N m,n 4.9 ( 2 ). { m,n } m, 2 4.10 (R ). 4.11 ( ). 4.12 (). f : X Y X Y A, B X, C, D Y 1. f(a B) f(a) f(b) 35
2. f(a B) f(a) f(b) 3. f 1 (C D) f 1 (C) f 1 (D) 4. f 1 (C D) f 1 (C) f 1 (D) 4.13. 1 x f : R R f (x) = f(x) f(0) = 4.14 (sin, cos ). α, β R sin(α + β) = cos(α + β) = 4.15 (2 ). 2 x f : R R f (x) = f(x) f(0) =, f (0) = b f(x) = 4.16 ( ). 1. (x ) = 2. (tn x) = 3. ( x ) = 4. (log x) = 5. (log x) = 4.17 ( ). d dx sin 1 x = d dx tn 1 x = 4.18 ( ). f : [, b] R ε > 0 δ > 0 < δ (109) 4.19 ( ). f : [, b] R ε > 0 δ > 0 < δ (110) 36
4.20 ( ). f, g : [, b] R f + g, f g l (111) 4.21 ( ). < b < c f : [, c] R f [, b] [b, c] (112) 4.22 ( I). f : [, b] R F (x) = x f(t) dt, x (, b) (113) 4.23 ( ). I, b I f I f (114) 4.24 ( ). n = 1, 2, f n 4.25 (). 1 x 2 dx = (115) dx 1 x 2 = dx 1 + x 2 = dx x 2 1 = 1 + x 2 dx = x 2 1 dx = 37
4.26 (). f, g : [, b] R f(x) g(x), x [, b] (116) 4.27 ( ). f : [, b] R (117) 4.28 ( ( ) ). f : [, b] R c [, b] 4.29 ( ). 1 < p, q < (118) 1 p + 1 q = 1 (119), [, b] f, g (120) 4.30 (). [, b] f, g (121) 4.31 ( (Cuchy-Schwrz), ). [, b] f, g 1. 2. ( ) 2 f(x)g(x) dx ( 1/2 (f(x) + g(x)) dx) 2 4.32 (lim ). I = [, b] {f n } f (122) 4.33 (lim ). I = (, b) C 1 - {f n } 38
1. 2. g = f 4.34 ( ). f : [, b] R 1. 2. 4.35 ( ). 1. f, g : [, ) R b > f, g [, b] f(x) dx 2. f, g : [, c) R b < c f, g [, b] c f(x) dx 4.36 ( ). n = 0, 1, 2, 1. Γ 2. Γ(x + 1) = 3. 0! = 1 0 t n e t dt = lim R R n Γ(n) = (n 1)! Γ(x) = Γ(n + 1) = n! 4.37 ( ). 1. k N 0 t x 1 e t dt, x > 0 0 t n e t dt = (123) 0 t n e t dt Γ (k) (α) = (124) (125) 39
2. Γ 2 () (b) (c) 4.38 ( ). α, β > 0 1. B(α, β) = 2. B(α, β) = B(α, β) 4.39 (Γ ( 1 2) ). 1. Γ 2. ( ) 1 =. 2 e x2 dx =. 4.40 (1/2 ). 4.41 ( ). 4.42. γ : (A, B) R n C 1 - γ x 1, x 2,, x n L(γ) = 4.43 ( ). { n } Q 4.44 ( N, ε ). 4.45 ( ). { n }, {b n}, {c n} 1. 2. { n } {b n} 3. { n } {b n} {b n} {c n} 4.46 ( ). 40
4.47 ( ). α = { n } α m = { m } α 4.48 ( ). 4.49 ( ). K = {k n } {α m} = {{ m,n } } α = {α m } m=1 4.50 ( ). A (126) 41