.1 1,... ( )

Transcription

1 1 δ( ε )δ 2 f(b) f(a) slope f (c) = f(b) f(a) b a a c b [ ] nobuo@math.kyoto-u.ac.jp, [URL] nobuo 1

2 .1 1,... ( ) ( ) I R d C I

3 6 II I ( ) ( ) I ( ) II ( ) ( ) III II ( ) II ( ) ( ) ( ) ( ) ( ) Γ- B ( ) ( )Γ- B

4 13 ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ] [Rud] Rudin, W.: Principles of Mathmatical Analysis-3rd Edition McGraw-Hill Book Company. [ ] I,II [ ] [ 1] [ 2] 4

5 [ ] [ ] 1 exp, sin, cos... [ 1] [ 2] 3 5

6 1 ( ) ( ) P, Q P = Q P Q Q = P P Q P = Q P = Q x,... x... x, x 1 1x, x P def. Q P Q P def. = Q,, ( ) X, Y X Y, X Y, X\Y X Y = {z ; z X z Y }, X Y = {z ; z X z Y }, X\Y = {z ; z X z Y }. 6

7 1.1.3 ( ) X, Y φ x X x Y φ(x) φ X Y φ X Y φ : X Y x X φ(x) Y x φ(x) φ : X Y, A X φ(a) Y φ A φ(a) = {φ(x) ; x A}. φ(x) = Y φ φ x, y X, φ(x) = φ(y) = x = y. φ φ φ y φ(x) y = φ(x) x X x = φ 1 (y) φ 1 : φ(x) X (y φ 1 (y)) φ Z ψ : Y Z φ ψ ψ φ ψ φ : X Z (x ψ(φ(x))) 7

8 1.1.4 (a) X = {x 1, x 2, x 3 }, Y = {y 1, y 2 }, φ(x 1 ) = y 1, φ(x 2 ) = φ(x 3 ) = y 2 φ : X Y (b) X = {x 1, x 2 }, Y = {y 1, y 2, y 3 }, φ(x 1 ) = y 1, φ(x 2 ) = y 2 φ : X Y (c) X = {x 1, x 2, x 3 }, Y = {y 1, y 2, y 3 }, φ(x j ) = y j, j = 1, 2, 3 φ : X Y φ φ 1 (x j ) = x j, j = 1, 2, (16.1 )

9 1.2.1 R R N, Z, Q N = {, 1, 2,...}, ( ) Z = {a b ; a, b N}, ( ) Q = {a/b ; a Z, b N, b }. ( ) a a >, a <, a a, a +, a { { a, a, a = a + a, a, = a = a, a., a. {, a, a, a. a, a +, a,, R 2 +, R = R {, + } + ± a R { } a < +, a R {+ } < a. +, R = R {, + } ± a R {+ } a + = + a =, a R { } a = a + ( ) = + a =, < a + a = a =, ( a) = ( a) = a( ) = ( )a =. a R a/ = a/( ) =. < a + a a = + 1,, / 9

10 1.2.3 a, b R (a, b) R, [a, b] R (a, b) = {x R ; a < x < b}, [a, b] = {x R ; a x b}, (a, b) [a, b] [a, b) R (a, b] R [a, b) = {x R ; a x < b}, (a, b] = {x R ; a < x b}. (a, b), [a, b], [a, b), (a, b] a, b l, r R, l < r (l, r). x x (l, r), (l =, r = ), l + 1, ( < l, r = ), x = r 1, (l =, r < ), (l + r)/2, ( < l, r < ) A R, m R a A a m m A a A m a m A ( ) m A A m A max A \( )/ m A A m m A min A A A A A A R A R A R ± max A = { { A, A = { }, min A = A, A = { } A R max A, min A A n N φ : {x N ; 1 x n} A A n A 1.

11 1.2.7 a, b Z, a < b A = b a. A = {z Z ; a < z b} z z + a {z N ; < z b a} A. \( )/ ( ) A R max A, min A A A = {a 1,..., a n } max A = a 1 a 2... a n, min A = a 1 a 2... a n (1.1) ( ) l < r, (l, r) I [l, r] x I r x, x I l x. (1.2) max I r I = max I = r, min I l I = min I = l. : (1.2) :. (1.3) : x < r x l < r x l < y < r y ( 1.2.4) y y I x < y. x I. \( )/ ( ) D f : D R c R f, g : D R cf, f + g,fg, f/g (cf)(x) = cf(x), (f + g)(x) = f(x) + g(x), (fg)(x) = f(x)g(x), (f/g)(x) = f(x)/g(x). f/g D(f/g) = {x D ; g(x) } 11

12 1.3.2 ( ) D R, f : D R f, x, y D, x < y = f(x) f(y) f x, y D, x < y = f(x) < f(y). f ( ) f f, f f f ( ) E D {f(x) ; x E} f E E ( ) f D f D ( ) E D max E max E f, min f E f = max f(e), min f = min f(e). (1.4) E (1.4) D R f : D R. f f 1. z, w f(i) z < w f 1 (z) < f 1 (w) f 1 (z) f 1 (z) z = f f 1 (z) f f f 1 (w) = w ( ) I R f : I R f \( )/ x, y I, α, β >, α + β = 1 f(αx + βy) αf(x) + βf(y). (1.5) (1.5) < f f f. f f. 12

13 (1.5) f f(x) = x n (n = 1, 2) R (n ). n = 1 (1.5) n = 2 x, y R α, β >, α + β = 1 2xy x 2 + y 2. (αx + βy) 2 = α 2 x 2 + 2αβxy + β 2 y 2 (α 2 + αβ)x }{{} 2 + (β 2 + αβ)y }{{} 2. =α =β \( )/ I R, f, g : I R (i) f + g (ii) f, g fg f(x) = x n (n ) [, ), n R I, J R f : I J f 1 : J I x, y R, n = 1, 2,.. n 1 x n y n = (x y) x k y n 1 k. (1.6) k= (a) x, y r x n y n nr n 1 x y. (1.7) (b) y x ny n 1 (x y) x n y n nx n 1 (x y). (1.8) (1.8) x = y, n = 1 = (c) x (, ) 1 + (x 1)n x n. (1.9) 13

14 (1.6):p(x, y) def = n 1 k= xk y n 1 k n 2 n 1 xp(x, y) = x k+1 y n 1 k + x n, yp(x, y) = y n + x k y n k. k= (1.6) (1.7): n 1 n 1 x k y n 1 k x k y n 1 k nr n 1. }{{} k= k= r n 1 (1.8) : (1.6) y x x = y, n = 1 (1.8) : (1.9):(1.8) y = 1 x 1 (1.9) (1.8) x, y y = 1 x 1 (1.9) k=1 \( )/ n N n!, n, k N (k n) ( n k )! = 1, n! = n(n 1) 1 (n 1), (1.1) ( n ) n(n 1) (n k + 1) n! = = k k! k!(n k)!. (1.11) ( 1.4.1) ( n ) ( ) n + k k + 1 ( ) n (k + 1) + k x, y R, n = 1, 2,.. (x + y) n = = ( n k ) k = n n k= ( n + 1 k + 1 ), (1.12) ( n k ). (1.13) ( n k ) x k y n k. (1.14) n n = (1.14) n (1.14) x(x + y) n = y(x + y) n = n k= n k= ( n k ) x k+1 y n k = x n+1 + ( n k ) x k+1 y n+1 k = n k=1 n k=1 ( ) n x k y n+1 k, k 1 ( n k ) x k y n+1 k + y n+1,(1.12) (1.14) n n + 1 \( )/ (1.12) (1.13). 14

15 2 ( ) ( ) (R ) R A R, m R A, U(A),L(A) U(A) = a A [a, ], L(A) = a A[, a]. (2.1) 1 (2.1) A, B R U(A B) = U(A) U(B), L(A B) = L(A) L(B). (2.2) 2.2 max A, min A A R, m R m = max A m A U(A) = [m, ], (2.3) m = min A m A L(A) = [, m]. (2.4) max A, min A, (2.3) : m A max U(A) = [m, ], x R (1) a A, a x m x. (1) : a = m m x. (1) : m U(A) a A a m. a A a x. (2.3) : (2.4): (2.3) \( )/ 15

16 2.1.3 ( ) A R, m R, A = { a ; a A} (2.5) m U(A) m L( A), (2.6) m L(A) m U( A), (2.7) m = max A m = min( A), (2.8) m = min A m = max( A). (2.9) (2.6), (2.7) A ( ) m A m A. (2.6) ( ) (2.8) (2.7) ( ) (2.9) \( )/ (2.6), (2.7) U(A) = L( A), L(A) = U( A). (2.1) A max A max A A A A R, l, r R, l < r. (a) (l, r] I [l, r], A I, r U(A) U(A I) = U(A), (2.11) m = max(a I) = m = max A. (2.12) (b) [l, r) I [l, r], A I, l L(A) L(A I) = L(A), (2.13) m = min(a I) = m = min A. (2.14) (a):(2.11) (A I) U(A I) A U(A). (2.12) (2.11) A + = A I, A = A\I. I [l, ] (1) = A + [l, ]. R\I [, l] (r, ), A (r, ) = 16

17 (2) A [, l]. (1) (2) (3) U(A + ) U(A ). A = A + A (2.2) (4) U(A) = U(A + ) U(A ). (3),(4) U(A) = U(A + ), (2.11). (b): (a) ( (a) ) \( )/ = A Z (a) A b Z max A (b) A b Z min A (a),(b) b Z b R 2.3.2) : (a): A a A A [a, b] max(a [a, b]) ( 1.2.8) max A (b): (a) ( (a) ) \( )/ ( ) d N\{}, x Z x = qd + r (q, r) Z {,.., d 1} ( ) d 1 = x, d = d d n 1 d n d n+1 n =, 1, 2,.. d n 1 d n+1 d n 1. d N 1 d N+1 = N N x d d N ( ) x, d, q q x/d [q, q + 1) (2.1)

18 (a) x x < n n (b) x 2 = 2 (a),(b) 2.3.1, A R, m R U(A), L(A) (2.1), m A, sup A U(A) = [m, ]. (2.15), m A, inf A L(A) = [, m]. (2.16) sup A = min U(A), inf A = max L(A). (2.17) ( ) l < r, (l, r) I [l, r] sup I = r, inf I = l. (2.18) ( ) A R, m R (a) m = max A m = sup A A (b) m = min A m = inf A A \( )/ (2.1) 18

19 2.2.4 ( ) A R, m R, A = { a ; a A} m = sup A m = inf( A), (2.19) m = inf A m = sup( A). (2.2) (2.9) A U(A) m = min U(A) m = max( U(A)) (2.1) max( U(A)) = max L( A). (2.17) (2.19) (2.2) \( )/ (AC) : (AC) A R sup A (AC) A U(A) sup A 4 (AC) (AC) (AC ) (AC ) A R inf A : (AC) (AC ): A A (AC) sup( A) inf A sup( A) (AC ) (AC): \( )/ 4 ( 2.2.4) 19

20 2.2.7 (a) A R sup A R, inf A R { sup A b b A, (b) b R b inf A b A. (c) b R (d) { { b < sup A a A, b < a, inf A < b a A, a < b. sup A < + A, inf A > A. (e) = A [l, u] l u. inf A sup A. : (a): sup A (2.15) { +, A sup A =, A =, { }, A R A, { } A R (AC) m = sup(a R). A A (, ] = A R (a) I = (, ], U(A) = U(A (, ]) = U(A R) = [m, ] m = sup A. (b): (2.15),(2.16) (c): b R (b) (d): (b) (e): \( )/ A R, γ R\{}, β R γa + β = {γa + β ; a A} γ (, ) sup(γa + β) = γ sup A + β, inf(γa + β) = γ inf A + β, sup( γa + β) = γ inf A + β, inf( γa + β) = γ sup A + β = A (, ] sup{1/a ; a A} = 1/ inf A, inf{1/a ; a A} = 1/ sup A. 1/ = A 1, A 2 R sup(a 1 A 2 ) = sup A 1 sup A 2, inf(a 1 A 2 ) = inf A 1 inf A ( ) B, B R, R = B B, b B, b B b < b (B, B ) R R (B, B ) 2

21 (i)b B (ii): (i) sup B, inf B (iii) m = sup B = inf B B = (, m] ( B B = (, m) ( B = [m, )) = (m, )), ( ) R (B, B ) max B, min B = A R A (i) max(r\u(a)), max(r\u(a)) (ii) sup A B def = R\U(A), B def = U(A) R R ( ) Z Z (AC) m def = sup Z R. m 1 < m m 1 U(Z). z Z, m 1 < z. z m < z + 1 Z. m U(Z) (AC ) \( )/ Z Z x x < z z Z (a) = A Z A max A A min A (b) ( ) a R q (a 1, a], a [q, q + 1) q Z q a a [a] a Q (a): max A A = m Z, A [, m] = max A (b):(a 1, a] 2 b, c (a 1, a] Z, b < c 1 = a (a 1) > c b 1 (a 1, a] Z 2 21

22 (1) A def. = Z (, a] max A (2) m = max A a 1 < m. m (a 1, a] Z (1) (a) A : A (, a] A : z Z, z a., z A. (2) m a 1, m+1 a m < m+1 A. m A \( )/ D R a < b (a, b) D D R R R ( ) Q R : a < b (b a) 1 < n, na + 1 < nb n N\{} m (na, na + 1] m Z m (na, nb), m/n (a, b) Q. \( )/ ( ) x R N = 1, 2, x p q < 1 Nq p Z q {1,, N} N + 1 x j = jx jx [, 1) (j =, 1,.., N) x p/q Diophantus 22

23 D, f : D R E D sup f, inf f E E sup f = sup f(e), E inf E f = inf f(e). max f(e), min f(e) max E inf f, max f, min E E E f, min f sup E E f f sup f(x) x E f E sup f <, E f E sup f <, E f E inf E sup, inf sup inf A f : A R, β R (a) inf A f sup A f. (b) sup A f β a A f(a) β. (c) inf A f β a A f(a) β. : \( )/ A f : A R, γ R\{}, β R sup A (γf+β) = γ sup A f+β, inf A (γf+β) = γ inf A f+β, sup A ( γf+β) = γ inf A f+β, inf A ( γf + β) = γ sup A f + β A, B f : A R,g : B R (i) a A f(a) g(b) b B sup A f sup B g. (ii) a A f(a) g(b) b B inf B (iii) A B inf B g inf A g sup A g sup B g. g inf A f. (iv) A = B a f(a) g(a) inf A f inf A g, sup A f sup A g A f : A R, g : A R inf A f + inf g inf A A (f + g) sup A (f + g) sup A f + sup g.. A 23

24 2.4.4 ( ) B b B A b R sup ( b B A b ) = sup sup A b, b B inf ( b B A b ) = inf b B inf A b ( ) A,B f : A B R (i)(sup sup ) (ii)(inf inf ) sup (a,b) A B inf (a,b) A B (iii) ( sup inf inf sup ) sup a A f(a, b) = sup b B f(a, b) = inf sup a A inf b B a A inf f(a, b) inf b B f(a, b) = sup a A f(a, b) = inf sup b B a A f(a, b). inf a A b B sup f(a, b). b B f(a, b) ( ) {, 1} A B A B [, 1], f(a, b) = a b sup inf f(a, b) = < 1/2 inf a A b B sup b B a A f(a, b) ( ) (a, b ) f, f(a, b ) f(a, b ) f(a, b), (a, b) A B sup inf f(a, b) = inf a A b B sup b B a A f(a, b) = f(a, b ). 24

25 3 I ( ) 3.1 n = 1, 2,... n 1/n 1/n (ε- ) a R, ε (, ) (a ε, a + ε) (a R ), B(a, ε) def = (1/ε, ) (a = ), (, 1/ε) (a = ). (3.1) a ε a a+ε B(a,ε) 1/ε B(,ε) 1/ε B(,ε) B(a, ε) a a ε ( ) N n a n (a n ) n=, (a n ) n, a n. a n a a n : ε (, ) n 1 N n n 1 a n B(a, ε). (3.2) ((3.1) ) (3.2) lim a n = a, a n a n (3.2) a R (a n ) n a R (3.2) a = ± (a n ) n ± 25

26 (3.2) ε (, ), n 1 N, n n 1, a n B(a, ε). (3.3) 1 a n N, n = 1, 2,.. N [n, ). 2 (3.2) ε (, ) ε (, ) ε 3 a R, m N lim a n = a lim a m+n = a n n (a n ) (a) a n 1/a n 1/( a n ) = 1/a n. (b) c > n a n cn a n, 1/a n. (a): a n B(, ε) 1/a n B(, ε) 1/a n B(, ε) (b): (a) a n ε (, ) n 1 N, n n 1, n > 1 cε ). n n 1 a n cn > 1. ε \( )/ (a) p = 1, 2,.. lim n n p = r n (b) r > 1, p =, 1,.. lim n n =, p (a): n p n (b): s = r 1 >, n 2p + 1, c = sp+1 (p+1)! def a n = rn (1 + s)n = = 1 np n p n p n k= lim n p =. n lim n p n r =. n n(n 1) (n k + 1) s k k! > 1 n(n 1) (n p) s p+1 = cn n p (p + 1)! ( 1 1 ) ( 1 p ), n n n k= k = p +1 j = 1,..., p 1 j n 1 p 2p n 2p + 1 a n c 2 p n. 26

27 3.1.3 \( )/ ( I) K 1, K 2 N, K 1 K 2 = N, K i (i = 1, 2) k i () < k i (1) <... ( K 1 =, K 2 =, k 1 (n) = 2n, k 2 (n) = 2n + 1. K 1, K 2 ) a R (a),(b) (a) a n a, (b) K i i a ki (n) a. (a) (b): K i ε > n 1 N, n n 1, a n B(a, ε). n n k i (n) n n 1, a ki (n) B(a, ε). (a) (b): K 1 K 2 : K 1,m 1 = max K 1 m = k 2 (n 2 ) N [m 1 + 1, ) = {k 2 (n) ; n n 2 }. lim n a n = a lim n a k2 (n) = a K 1,K 2 : ε > n 1 N, n n 1, a k1 (n) B(a, ε), n 2 N, n n 2, a k2 (n) B(a, ε). n k 1 (n 1 ) k 2 (n 2 ) n n = k 1 (n ) (n n 1 ), n = k 2 (n ) (n n 2 ) a n B(a, ε) \( )/ a n = ( 1) n a R, a n a a n a (a n ) lim n a n = a R n a n = a ( ) D R R a R D (a n ) a n a 27

28 a, b, c R, (a n ), (b n ), (c n ) ( ) a n a (a) a a n (b) a + a n (c) a R a n : (a): b R, m, n m, b < a n (, a = b = 1 a b = a 1 ) n =, 1,.., m 1 n a n (b):(a) min{a, a 1,..., a m 1 } a n. m def. = min{a, a 1,..., a m 1, b} a n. (c): (a),(b) \( )/ a n a, b n b n a n b n a b. : b < a c R, b < c < a ( 1.2.4). a n a n 1 N, n n 1, c < a n, b n b n 2 N, n n 2, b n < c. n n 1 n 2, b n < c < a n. \( )/ ( ) (a n ) n a R : a, b R, a n a, a n b a b a b. \( )/ 28

29 3.2.4 ( ) (a) n a n b n c n a n a, c n a b n a. (b) n a n b n. a n = b n, a n = b n. : (a):ε > a n a n 1 N, n n 1, a n B(a, ε), c n a n 2 N, n n 2, c n B(a, ε), n 3 N, n n 3, b n [a n, c n ]. n n 1 n 2 n 3, b n [a n, c n ] B(a, ε). (b):(a) ( ) a n a, b n b \( )/ a n + b n a + b, {a, b} = {, } (3.4) a n b n ab, { a, b } = {, } (3.5) a 1 n a 1, a n a n (3.6) : a, b R a = b = ε >. (1) n 1 N, n n 1, a n a < ε, (2) n 2 N, n n 2, b n b < ε, (3.4): (1) (2) ε ε/2 n n 1 n 2. (a n + b n ) (a + b) a n a + b n b < ε/2 + ε/2 = ε. (3.5): b n M (, ) b n. (1) ε ε (2) 2M ε ε 2( a +1) n n 1 n 2 a n b n ab (a n a)b n + a(b n b) < ε 2M M + a ε 2( a + 1) ε. (3.6): (1) ε ε a 2 2 n 3 N, n n 3, a 2 < a n. 29

30 n n 1 n a n a = 1 a n a a n a < 2 a ε a = ε ( ) (a n ), (b n ) \( )/ s n = n a j (b j b j 1 ) j=1 a R, = b < b 1 <... < b n a n a s n b n a. b n = n s n /b n a n : ε >, n 1 N, n n 1, a n a < ε/2. a n a n a C [, ) a n a a = 1 b n n (b j b j 1 )a j=1 n n 1 s n a 1 n b n = (b j b j 1 )(a j a) b n j=1 1 n 1 1 (b j b j 1 ) a j a + 1 n b n b n j=1 j=n 1 (b j b j+1 ) a j a 1 n 1 1 (b j b j 1 )C + 1 n (b j b j 1 ) ε b n b j=1 n 2 b n1 1C b j=n n 1 }{{} (3) n 1 lim n (3) =. n 2 N, n n 2, (3) < ε/2. n n 1 n 2 a < ε. \( )/ s n bn I R f n : I R n = 1, 2,.... x I f(x) = lim n f n (x) R f : I R (a n ) lim n 1 n a n = lim n 1 n max 1 j n a j = 3 + ε 2.

31 a n a R s n /b n a ( ) n 1 t n = (a j+1 a j )b j (i) a n b n = s n + t n. (ii) a R, = b < b 1 <... < b n a n a s n /b n a t n /b n. a n a t n /b n j=1 ( ) ( ) a = ε > (3) n 3 N, n n 3, a n > 1 ε (3.4) ( a = b = a = b. b n 3.2.1). c R b n n N a n + c a n + b n. (3) 1/ε 1 ε c n n 3 1 ε (3) a n + c a n + b n. a n + b n. (3.5)( a = b = ): a = b =. a =,b > (3) ε bε/2 n 4 N, n n 4, b 2 < b n. n n 3 n 4 a n b n > b 2 2 bε = 1 ε. a n b n (3.6)( a = ) a = (3) n n 3 < a 1 n < ε. a = \( )/ 31

32 ( ) a n a def = lim n a n (a) a n a [a, ]. a < a n (3.7) a = a n (3.8) (b) a n a [, a ]. a > a n (3.9) a = a n (3.1) ) ( ) 6.5.1, (l n ), (r n ) l n r n (n =, 1, 2,..), (l n ), (r n ), r n l n c [l, r ] lim l n = lim r n = c. n n (1) l l n r n r (l n ) (r n ) c, c R lim l n = c, n lim r n = c. n, l n = r n + (l n r n ) ( 3.2.5) c = c (1) c [l, r ]. \( )/ 32

33 3.3.1 ( ) s m,n R (m, n N) m, n N s m+1,n s m,n, s m,n+1 s m,n lim m lim n s m,n = lim n lim m s m,n ( ) (a) (a n ) (b) (a n ) lim a n = sup a n. (3.11) n n lim a n = inf a n. (3.12) n n (a): (1) (3.11) (1) l N, m N, n m, a l a n. a = sup a n a n a n (2) b < a, m N, n m, b < a n. b < a sup l N, b < a l. l (1) m n m b < a l a n (2) (b): (3) (3.12) (3) l N, m N, n m, a l a n. (a) \( )/ I R f : I R a I f I (a n ) lim n a n = a lim n f(a n ) = f(a). (3.13) I C(I), C(I R) 33

34 f(a) f(a n) a n a A R A : a 1, a 2 A, a 1 x a 2 = x A. (3.14) ( ) I R, f C(I R), a, b I, f(a) s f(b) s = f(c) c [a, b] f(i) f(b) s f(a) a c b : a = b f(a) = s = f(b) c = a a < b b < a [a n, b n ] I, n =, 1,... : [a, b ] = [a, b]. [a n, b n ] c n = (a n + b n )/2 [a n+1, b n+1 ] = { [a n, c n ], s < f(c n ), [c n, b n ], f(c n ) s, 34

35 (1) f(a n ) s f(b n ). n = f(a) s f(b) (1) n s < f(c n ) f(a n+1 ) = f(a n ) f(c n ) s f(a n+1 ) = f(c n ) s s < f(c }{{} n ) = f(b n+1 ), {}}{ f(b n ) = f(b n+1 ). a n b n (n =, 1,...), a n, b n, b n = a n + 2 n (b a) c I lim a n = lim b n = c. n n, s = f(c). f(c) f = lim f(a n ) (1) s (2) lim f(b n ) f = f(c). n n \( )/ (3.13) n a n a (a n ) (a n ) f : R R (i) f lim n f(a n ) =, lim n f(b n ) = (a n ), (b n ) (ii) f lim n f(a n ) =, lim n f(b n ) = (a n ), (b n ) f f : [, 1] [, 1] f(c) = c c [, 1] f C(R) a R f(a + b) = f(b) b R 35

36 3.4 ( ) 3.4.2: :l = inf A, r = sup A (l, r) A [l, r] A [l, r] l, r (l, r) A x (l, r) a 1 < x < a 2 a 1, a 2 A ( 2.2.7) (3.14) x A. : = A R R\U(A) (3.14) max(r\u(a)) 2.2.5) R\U(A) = [, m) m R U(A) = [m, ]. m = sup A \( )/ ( 3.5.2) ( ) I R f : I R (a) f C(I R) f(i) (b) f C(I R) f 1 C(f(I) I) f 1 ). y = f(x) f(i) y = f 1 (x I I f(i) : (a): : : (b): f f f 1 : f(i) I 1.3.3). I (a) f 1 \( )/ 36

37 ( ) m N\{} f : [, ) [, ), f(x) = x m f 1 : [, ) [, )( ) f 1 (x) = x 1/m x 1/m x m m x 2 x x a n, b n, a, b > a n a, b n b a n b n ab, : : : p n = a n b n, q n = a n bn a n = p n q n x x a n b n a b. ab a b = a, b pn n = ab b q n a = b. \( )/ (i) a (, ) lim n a 1/n = 1. (ii) (a n ), a R lim n a n+1 /a n = a = lim n a 1/n n = a. (iii) ( ) (a n ) lim n ( n j=1 an j ) 1/n = supn 1 a n a b 2 ab a + b 2 ab + a b a > b >,a n, b n > (n 1) a n = a n 1+b n 1 2, b n = a n 1 b n 1 (i) a n b n a n 1 b n 1 2, (n 1). (ii) a n, b n. (iii) lim n a n, lim n b n a >, b, c R, f(x) = ax 2 + bx + c, S = {x R ; f(x) = } S b 2 4ac b 2 4ac S = {s, s + }, s ± b± b 2 4ac 2a. x [s, s + ] f(x) >, x (s, s + ) f(x) < ( ) I R a < b, f : I R, x n I (n N) x n+1 = f(x n ) 37

38 (i) x f(x ) f [x, b) n x n x n+1 x n f(x n ). x (c, m) f(x) < x m (a, b) x n x (a, m] (ii) x f(x ) f (, x ] I n x n x n+1 x n f(x n ). x (a, m) f(x) > x m (a, b) x n x [m, b) (iii) f lim n x n = x I x = f(x ) ( ) b, c >, s = b + b 2 + c, f(x) = 2bx + c (i) x < s, x = s, x > s x < f(x), x = f(x), x > f(x). (ii) x n > (n N) x n+1 = f(x n ) lim n x n s ( ) I R (± ), f : I R x n I (n N) x n+1 = f(x n ) (i) (x 2n ), (x 2n+1 ) g = f f (ii) f s = f f(s) s I x n s (iii) I = [, ) f(x) = 1/(ax + b) (a, b > ) (x n ) ( ) ( ) (i) 2 Q. (ii) x, y Q, y x + y 2 Q. (iii) R\Q R. a, b R, a < b, (a, b) Q. x (a, b) n x + 2/n (a, b). 3.5 ( ) 3.5.1(a) 5 a n, a I, a n a, a n a (1) lim n f(a n ) = f(a) K 1, K 2 N K 1 = {n N ; a n < a}, K 2 = {n N ; a n > a} K 1 K 2 = N. K i = {k i () < k i (1) <..} (2) K i lim n f(a ki (n)) = f(a), (2) I (1). f (f ) i = 1 i = 2 a k1 (n) a f δ ε, δ ε 38

39 (3) l N, m N, n m, f(a k1 (l)) f(a k1 (n)) (4) lim f(a k1 n (n)) = sup f(a k1 (n)). n (1) (5) sup f(a k1 (n)) = f(a). n s. f n f(a k1 (n)) f(a). s f(a). s < f(a) n (6) f(a k1 (n)) s < f(a). (6) f(i) s f(i). s = f(b) b I (6) (7) f(a k1 (n)) f(b) < f(a). (7) f a k1 (n) b < a. a n a. \( )/ ( ) 3.5.1(a) : ε >, a I (1) a I I a 1 I a 1 < a x (a 1, a) f(a) f(x) < ε. (2) a I I a 2 I a < a 2 x (a, a 2 ) f(x) f(a) < ε. (1),(2) δ = (a a 1 ) (a 2 a) x (a δ, a + δ) I f(x) f(a) < ε (1) (2) a I b < a b I f(b) = f(a) x (b, a) f(x) = f(a) (1) f(b) < f(a) f(b) < f(a) ε < f(a) ε (, ε) f(i) f(a 1 ) = f(a) ε a 1 I f a 1 (b, a). x (a 1, a) f(a) f(x) f(a) f(a 1 ) = ε < ε. (1) \( )/ 39

40 3.6 R d C d A 1,..., A d A 1,..., A d a j A j (j = 1,.., d) A 1 A d = {(a j ) d j=1 ; a j A j, j = 1,.., d.} (3.15) A 1,..., A d a j j j A j = A (j = 1,..., d) A d : A d = {(a j ) d j=1 ; a j A, j = 1,.., d.} (3.16) R d ((3.16) A = R ) d- d-. def = (,..., ), c R R d x = (x j ) d j=1, y = (y j ) d j=1 cx Rd, x + y R d cx = (cx 1,..., cx d ), x + y = (x 1 + y 1,.., x d + y d ). x y R x [, ) x y = d x i y i, x = x x. i=1 x y, x, R d d, B R d { x ; x B} R B ( ) 4

41 x, y R d (a) c R cx = c x, (b) x + y 2 = x 2 + 2x y + y 2, (c) x y x y, (d) x + y x + y. (a): (b): (x j + y j ) 2 = x 2 j + 2x j y j + yj 2 j = 1,..., d (c): x = x, x > c = x y x 2 (c) R cx y 2 (a),(b) = c 2 x 2 2cx y + y 2 (x y)2 = + y 2 x 2 (d): x + y (b) = ( x 2 + 2x y + y 2) 1/2 (c) ( x x y + y 2) 1/2 = x + y. \( )/ R 2 x = (x 1, x 2 ) x = x 1 + x 2 i (3.17) C (3.17) x 1 R x Re x (3.17) x 2 R x Im x x 1 x 2 i z x Re x = x + x, Im x = x x, 2 2i C (x 1 + x 2 i) + (y 1 + y 2 i) = (x 1 + y 1 ) + (x 2 + y 2 )i, (3.18) x 1 + x 2 i = x x 2 2. (3.19) (x 1 + x 2 i)(y 1 + y 2 i) = (x 1 y 1 x 2 y 2 ) + (x 1 y 2 + y 1 x 2 )i. (3.2) {x C ; Im x = }, {x C ; Re x = }, R R C i i i 41

42 x, y C (a) x + y = x + y, xy = x y, (b) x 2 = xx, (c) xy = x y, (d) x 1/x = x/ x 2, 1/x = 1/x, 1/x = 1/ x. \( )/ x, y C, n = 1, 2,.. x, y r n 1 x n y n = (x y) x k y n 1 k (3.21) k= x n y n nr n 1 x y. (3.22) x, y R ( 1.4.1) \( )/ N R d n a n R 2 = C. a n a R d (a n ) n=, (a n ) n, lim a n a = n a n a R d lim a n = a, a n a n {a n ; n N} (a n ) 1 a n N, n = 1, 2,.. N [n, ) 42

43 z C z < 1 z n. z n = z n. a, b R d, (a n ), (b n ) R d ( ) a = (a j ) d j=1, n a n = (a n,j ) d j=1 R d (a1) a n a j = 1,.., d lim n a n,j = a j. (a2) a n a Re a n Re a Im a n Im a a n a (b1) (a n ) j = 1,.., d (a n,j ) n= (b2) a n Re a n, Im a n (c) : (a1): max a n,j a j a n a 1 j d 1 j d a n,j a j. (a2):(a1) d = 2 (b1): max a n,j a n 1 j d (b2):(b1) d = 2 1 j d a n,j. (c): (a), (b) \( )/ ( ) a n a, b n b a n + b n a + b, a n b n a b. a n b n ab, a n b 1 n b n, b ab 1 : R d \( )/ 43

44 3.6.1 ( ) x C, a a 1... a n, a + a 1 x a n x n = x z C, n N (1 z) n j=1 (1 + z2j 1 ) = 1 z 2n, z < 1 lim n n j=1 (1 + z2j 1 ) = 1/(1 z) ( ) A R d diam(a) = sup x y A x,y A diam(a) < 44

45 4 ( ) a n s n,a n, a n s n = n a j (4.1) j= s n a n (4.1) s = lim n s n (s C, s = ± ) s a n, n= a n (4.2). s C (4.1) s = ± (4.1) 1 (4.1) (4.2) (4.2) 2 (a n ) a n n (a n ) a n = a + n j=1 (a j a j 1 ) ( ) a n p n N, q n N { } (n N) p n q n, lim p n =. n lim n q n j=p n a j =. p n = q n = n, p n = n, q n lim a n = lim n n a j =. j=n 45

46 n = s n = n j= a j s n s q n j=p n a j = s qn s pn s s = ( ) z C z n z < 1 = 1 1 z, z 1 n= n 1 1 z n = (1 z) z j. z < 1 z n j= \( )/ z 1 z n = z n \( )/ ( 3.3.1) a n s = s [, ]. (a) s n = a a n. (b) s <, a n. s n = a a n ( 3.3.1) s (a) (b) \( )/ a n 46

47 4.1.5 ( ) a n, b n A = (a) c 1, c 2 C a n, B = n= n= b n c 1 A + c 2 B = (c 1 a n + c 2 b n ), A = a n, n= n= A a n. (4.3) n= (b) ( ) a n, b n R, a n b n ( n N), a n b n A < B. (a):(4.3) N N N (b): a m < b m m N B A = (b n a n ) b m a m >. n= \( )/ (a) a n a n < = a n. (b) a n a n < Re a n <, Im a n < (4.4) = Re a n, Im a n (4.5) a n, (a): a ± n = ( a n ± a n )/2 a n = Re a n + i Im a n. (4.6) (1) a n = a + n a n, a n = a + n + a n. 47

48 s n = n a j, s ± n = j= n a ± j, t n = j= n a j j= (2) s n = s + n s n, t n = s + n + s n. a n < (2) t n s ± n s ± (2) n = s n (b): (4.4) : Re a n Im a n a n Re a n + Im a n (4.5) : (a) (4.6) : \( )/ ( ) a n a n < a n a n a n = a n ( 5.1.2) ( ) a n, b n C, c n def = n j= a jb n j : A = a n, B = b n, C = c n A,B C C = AB j n, k n n 2 j n n+1 2, k n = n j n j n, k n. def. δ n = n c l l= j n a j j= k n k= b k = j,k j+k n a j b k j n a j j= k n b k k= } {{ } (1) δ n j + k n j, k 48

49 (1) j j n, k k n, (2) j n < j, (3) j j n, k n < k k (3) k n (1) (2) j n j δ n = n j=j n+1 n j a j b k k= } {{ } (2) j n 1 + S = a n, S = b n j= a j n j k=k n +1 b k } {{ } (3) def. S n = a a n S, S n def. = b b n S S n S,S n S.. (2) (3) n j=j n+1 j n 1 j= n j a j b k (S n S jn )S, a j k= n j k=k n+1 b k S(S n S k n ). δ n \( )/ (a),(b) (a) (b) (b) (a) (a) (b)

50 4.3.1 ( ) (a n ), (b n ) a n (a) ( ) b n n a n b n. (b) ( ) r [, 1), n a n+1 ra n. (c) ( ) n a n b n b n+1. (a): m N, j m, a j b j. n m s n = s m 1 + n a j s m 1 + j=m n b j. j=m b j (s n ) (b) m N, j m, a j ra j 1... r j m a m. n m s n = s m 1 + n a j s m 1 + a m j=m (s n ) j=m n j=m r j m }{{} 1/(1 r) (c): m N, j m, a j b j b j+1. n m s n = s m 1 + n a j s m 1 + j=m n (b j b j+1 ) s m 1 + b m. j=m } {{ } =b m b n+1 (s n ) \( )/ { n p =, p = 1, <, p = 2, 3,... n=1 p = 1 s n = n j=1 j p s 2n s n = 2n j=n+1 j 1 n (2n) 1 = 2 1. s n s n p 2. 1 n 1 n + 1 = 1 n(n + 1) 1 2n 1 2 2n. p ( 4.3.1) n p \( )/ 5.

51 4.3.3 (a n ), a n (a) (a 2n + a 2n+1 ) a n (b) ( ) d = 1, a n = ( 1) n a n, a n a n+1 ( n N) a n : (a): s n = n j= a j (1) s 2n+1 = n (a 2j + a 2j+1 ). j=,s 2n+1 (2) s n lim n s n = lim n s 2n+1. s 2n+1 = s 2n + a 2n+1, lim a 2n+1 =. n s 2n lim s 2n = lim s 2n+1. I( 3.1.5) n n (2) (b): (a) s 2n+1 (1) a 2j + a 2j+1 s 2n+1. a 2j 1 + a 2j, a 2j+1 s 2n+1 = a + n (a 2j 1 + a 2j ) + a 2n+1 a j=1 s 2n+1 \( )/ ( 1) n p = 1, p 2 (n + 1) p p = p \( )/ (b) a n (i) m 2n m j=2n a j a 2n. (ii) m 2n + 1 a 2n+1 m j=2n+1 a j (x n ) R d (a) (b) (c) (a): r [, 1) n x n+1 x n r x n x n 1. (b): n= x n+1 x n <. (c): (x n ) 51

52 4.3.3 ( ) f : R d R d r [, 1) f x, y R d f(x) f(y) r x y. f f(x) = x x R d x R d x n R d, n = 1, 2,.. x n = f(x n 1 ) (x n ) x = lim n x n ( ) a m,n C (m, n N) (i) n= m= a m,n = m= n= a m,n ( ). (ii): (i) n m= a m,n m n= a m,n n= m= a m,n = n= a m,n. m= ( ) 2 q N, D {, 1.., q 1} (x n ) n 1 E D n x n = q 1 (i) (x n ) n 1 D\E x = n=1 x n/q n x [, 1) ( x q ). (ii) (x n ) n 1 x D\E [, 1) 4.4 a n x C f(x) = a n x n (4.7) n= ) r (, ], a n+1 lim n a n = 1 r (4.8) (4.7) x < r a n, r (4.8) (a) p Z, a n = (n + 1) p, r = 1. (b) a n = 1 n!, r =. b n = a n x n b n < x < ρ < 1 ρ r b n+1 = a n+1 x x b n a n r < ρ. n b n+1 /b n < ρ ) 52

53 (a): (b): a n+1 a n a n+1 a n ( ) p n + 2 = 1 = 1 n = 1 n + 1 = 1. \( )/ ( ) (4.7) x < r (a) < r < r, p N n p a n r n <. n= x < r : f (x) def = na n x n 1, n=1 g(x) def = n a n x n 1. (4.9) n=1 (b) x, y C, x, y r < r (c) x n, x C, x n, x < r, x n n x f(x) f(y) g(r) x y. (4.1) f(x n ) n f(x). (4.11) (d) a n R, a 2, a 3,... x, y R, y < x < r f (y) f(x) f(y) x y f (x). (4.12) a 2 = a 3 =... = (a): r < r 1 < r r 1 ( ) n r = lim n np r 1 n n p (r/r 1 ) n 1, n p a n r n a n r n 1. ( 4.3.1) (b): f(x) f(y) = a n (x n y n ) a n x n y n n= n= (3.22) x y n a n r n 1 = g(r) x y. n= (c): x < r < r r. x n n x n 1 N, n n 1, x n < r. n n 1 f(x) f(x n ) (4.1) 53 g(r) x x n n.

54 (d): (1.8) a n n = 1, 2,... ((1.8) n = 1 a 1 ) \( )/ ( ) (4.7) x < r ( < r ) x m C < x m < r, f(x m ) = (m N), x m m. a n (n N). : a = f() (4.11) = lim m f(x m) =. f(x) = xf 1 (x), f 1 (x) = a n+1 x n. f 1 (x) x < r x m n= f 1 (x m ) = f(x m) x m =, m N. a 1 = f 1 () (4.11) = lim m f 1(x m ) =. a n = (n = 2, 3,...) \( )/ ( ) (4.7) (a) f(±x) f(x) + f( x) 2 = a 2n x 2n, n= f(x) f( x) 2 = a 2n+1 x 2n+1. (4.13) n= (b) a n f(x) f(x), f(x) = f(x). : (a): f(x) + f( x) (4.3) = (4.13) (b):(4.3) a n (1 + ( 1) n )x n = 2 a 2n x 2n. \( )/ a, a 1 R, a n (n 2) (4.7) x ( r, r) ( < r ) f [, r) a 2n+1 (n 1) f ( r, r) (4.7) x < r ( < r ) (i) f a 2n+1 = (n N). (ii) f a 2n = (n N). 54

55 4.4.3 ( ) x, y R, y < x (i) n ( ) x+y n 1 2 x n y n n x y 2 (xn 1 + y n 1 ), ( n = 1, 2 ) (ii) a n R, a 3, a 4,.. x < r f ( ) x+y 2 f(x) f(y) x y = a 3 = a 4 =... f (x)+f (y) 2. 55

56 5 I ( ) 18 f(x), i, e, sin, cos ( 5.2.2) 5.1 (exponential) exponere (ex +ponere = + ) a > } a {{ a} = a n n n e x e x ( ) x C exp x = n= x n n! (5.1) x exp x e def. = exp(1) = (5.1) \( )/ exp x e x ( e x

57 5.1.2 ( I) exp : C C (a) x, y C exp(x + y) = exp x exp y ( ), (5.2) exp x = exp x, (5.3) exp x, x R exp x >, (5.4) exp x = exp(re x). (5.5) (b) ( ) x, y C, x, y r exp x exp y x y exp r. (5.6) (c) ( ) x, x n C, x n x exp x n exp x. (5.2): a n = xn, b n! n = xn n! n n (1) c n def = j= a j b n j = 1 n! j= n! j!(n j)! xj y n j = 1 n! (x + y)n. exp(x + y) (1) = (5.3): (5.1). (5.4): x C c n = a n b n = exp x exp y. 1 = exp (5.2) = exp( x) exp x, exp x. x R exp x 2 (5.5) exp x (5.2) = (exp x 2 )2 >. exp x 2 = exp x exp x (5.3) = exp x exp x (5.2) = exp (x + x) = exp(2 Re x) (5.2) = (exp(re x)) 2. (5.4) exp(re x) >. (5.5) (b): a n = 1/n! f(x) = g(x) = exp x. (4.1) (c): \( )/ (5.4) x R exp x > exp : R (, ) (5.7) 57

58 5.1.3 ( II) (5.7) (a) ( ) x, y R, x > y (x y) exp y < exp x exp y < (x y) exp x. (5.8) 5.7, (b) ( ) x n x R x n x = exp x n exp x. exp( ) =, exp( ) = 5.7 exp x 1 (a): x = x + y, ỹ = y + y x > ỹ, x ỹ = x y. (4.12) (x y) exp ỹ < exp x exp ỹ < (x y) exp x. exp( y ) (b): a R : a = : m N, n m, x n >. n m (1) exp x n (5.8) > 1 + x n (n ). a = : x n exp x n = 1/ exp( x n ) (n ). \( )/ (5.8) (x y) exp x + y 2 < exp x exp y < (x y) exp x + exp y. (5.9) 2 58

59 5.1.4 ( ) x C s n (x) = n m= x m ( m!, e n(x) = 1 + x ) n n a, a n C, a n a exp x s n (x) x n+1 exp x, (n + 1)! (5.1) exp x e n (x) x 2 exp x 2n (5.11) (5.1): exp x s n (x) = s n (a n ) exp a, e n (a n ) exp a. (5.12) m=n+1 x m m! (4.3) m=n+1 x m m! = x n+1 m= x m (n + m + 1)! (n + m + 1)! = (n + m + 1) (n + m) (n + 2) (n + 1)! (m + 1)! (n + 1)!. }{{}}{{}}{{} m+1 m 2 (5.11): exp x s n (x) x n+1 (n + 1)! m= x m (m + 1)! } {{ } exp x. (1) z C 1 + z 1 + z exp z. (2) exp x n 1 x = n exp x ( x n s (5.1) 1 n) x 2n 2 exp ( ) x. n exp x = ( n. exp n) x ( exp x e n (x) = exp x ) n ( 1 + x n ( n ) n) 3.6.6, (1) n 1 exp n exp n x x n 1 x n (2) x 2 exp x. 2n (5.12): x, y C, n N, x, y r } s n (x) s n (y) x y exp r (5.13) e n (x) e n (y) 59

60 s n (5.13) (5.6) e n (5.13) e n (x) e n (y) (1) ( n 1 + r ) n 1 x y ( n ) n n 1 x y n exp n r n x y exp r. a n a r [, ) a n a r. s n (a n ) exp a s n (a n ) s n (a) + s n (a) exp a (5.13) a n a exp r + s n (a) exp a (5.1). e n \( )/ exp : R R n lim n (n!) lim ( 1/n n n n) = e a n, an n e n (a n ) exp( a n ) 1, e n ( a n ) exp(a n ) 1 (5.11) ( ) e Q e = p/q (p, q N) (5.1) x = 1 pn! qs n (1)n! qe/(n + 1) ( ) x exp x e n (x) x2. ( (5.1) 2n e n (x) exp x s n (x).) 6

61 ( ) x C ch x, sh x ch x = exp(x) + exp( x), sh x = 2 exp(x) exp( x), 2 ch x, sh x exp x = ch x + sh x. (5.14) (a) ( ) ch x = x 2n (2n)!, sh x = x 2n+1 (2n + 1)!. ch : R R [, ), (, ]. sh : R R. (b) ( ) a n, a C,a n a ch a n ch a, sh a n sh a. ch (± ) =, sh (± ) = ± a n R, a n ±. ch x 1 sh x (a): f(x) = exp x ch x, sh x (f(x) ± f( x))/ (b): \( )/ 61

62 5.2.2 ( ) x C cos x, sin x exp(ix) + exp( ix) cos x = ch (ix) =, 2 exp(ix) exp( ix) sin x = sh (ix)/i = 2i cos x, sin x, (sine) (5.14) (a) ( ) exp ix = cos x + i sin x ( ). (5.15) cos x = sin x = 1 i (ix) 2n = (2n)! ( 1) n x 2n, (2n)! (ix) 2n+1 (2n + 1)! = ( 1) n x 2n+1. (2n + 1)! (b) ( ) a n, a C,a n a cos a n cos a, sin a n sin a. \( )/ x, y C ch (x + y) = ch x ch y + sh x sh y, sh (x + y) = ch x sh y + ch y sh x, (5.16) cos(x + y) = cos x cos y sin x sin y, sin(x + y) = cos x sin y + cos y sin x. (5.17) (5.16), (5.17) y = x ch 2 x sh 2 x = 1, cos 2 x + sin 2 x = 1 (5.18) cos, sin cos x 1 n cos kx = 1 cos nx cos(n + 1)x +, 2 2(1 cos x) k= n sin kx = k= sin x + sin nx sin(n + 1)x. 2(1 cos x) r < 1, x R 1 r cos x 1 2r cos x + r = r n r sin x cos nx, 2 1 2r cos x + r = r n sin nx. 2 n= ( ) (E n ) n E = 1, n k= ( 1) k E k (2n 2k)!(2k)! (i) E n (2n)!r 2n ( n 1), r > ch r = 2 6 n=1 = (n 1) 6 r = log(2 + 3) = (ii) < z < π/2 62

63 (ii) z C, z < r ch z = ( 1) n E nz 2n (2n)!, 1 cos z = n= n= E n z 2n (2n)!. (5.19) E n. E 1 = 1, E 2 = 5, E 3 = 61, E 4 = 1385, E 5 = 5521, ( ) exp : R (, ) log : (, ) R (a) x (, ) exp log(x) = x, x R log exp(x) = x. log 1 =, log e = 1. (b) x, y (, ) log(xy) = log x + log y. (c) ( ) x y > x y x log x log y x y. (5.2) y (d) ( ) (a n ), a [, ] a n a log a n log a, log( ) =, log() = logx 1 63

64 (a): (b): ( 5.1.2) exp(log(xy)) = xy = exp(log x) exp(log y) = exp(log x + log y). exp : R (, ) (c): ( 5.1.2) (d): (c) \( )/ (5.2) (5.8) (5.9) (5.2) 2 x y x + y log x log y x y xy. (5.21) γ n = n 1 k=1 log n γ (, ) γ k (i) n 1 1 n+1 log(n + 1) log n 1 n (ii) n 1 γ n γ n+1 1 n 1 n a n > 1,p n = n j= (1 + a j) p n n= (1 + a n) (i) (a) (c) (a) (b) (c) (a) a n <. (b) log(1 + a n) <. (c) p n x ( 1, 1) x 1+ x (ii) a n (b) (c). log(1 + x) x 1 x. 64

65 ( ) a [, ) x C a x { a x exp(x log a), a > = (5.22), a = (a) x, y C a x+y = a x a y ( ). a x = a x, a x = a Re x. (b) ( ) < a < b x > 1 xa x 1 (b a) < b x a x < xb x 1 (b a), < x < 1 xa x 1 (b a) > b x a x > xb x 1 (b a), (5.23) x < x a x log b > a ax b x > x b x log b. a x < y, a, 1 (x y)a y log a < a x a y < (x y)a x log a. (5.24) e log e = 1. (5.22) e x = exp x. 2 (5.23) x < log b a b a b log b b a (5.23) x < a a x a x 1 (b a) > a x b x > x b x 1 (b a) (5.25) x > (a): ( 5.1.2) (b): (5.23) (1) c > x > 1 x(e c 1) < e cx 1 < xe c(x 1) (e c 1), (1) < x < 1 x(e c 1) > e cx 1 > xe c(x 1) (e c 1), x < x c > 1 e cx > x ce cx. (1) c = log(b/a) a x (5.23) (1) x > 1 x = y + 1 (y > ) ( ) ( ) = (y + 1)e cy (e c 1) e c(y+1) + 1 = e cy (y(e } c {{ 1 } ) + e c 1) e c(y+1) + 1 > cye cy (e cy 1) (5.8) >. >c ( ) ( ) = e}{{} c+cy +1 (y + 1)(e c 1) > y(ce c (e c 1)) (5.8) >. >e c (1+cy) 65

66 < x < 1 1/x > 1. x > 1 (1) x 1/x c cx 1 x (ecx 1) < e c 1 < 1 x ec(1 x) (e cx 1). < x < 1 (1) x < y = x > (5.8) cy cye > e cy 1 > cy. e cy x < (1) (5.24) a > 1 : x log a y log a a x a y = exp(x log a) exp(y log a) (5.8) < (x log a y log a) exp(x log a) }{{} =a x = (x y)a x log a. (5.24) (5.24) < a < 1 : a > 1 (x y)(1/a) y log(1/a) (1/a) x (1/a) y (x y)(1/a) x log(1/a). a x+y (5.24) \( )/ { n p =, p = 1, <, p > 1. n=1 p 1 : n p n 1 p = 1 ( 4.3.2) p 2 : n p n 2 p = 2 ( 4.3.2) 1 < p < 2 : (1) 1 n 1 n+1 1 2n 2. ( ) p 1 ( ) p (5.23) ( ) p 2 ( 1 1 (p 1) n n + 1 n n 1 ) (1) p 1 n n p. ( 4.3.1) n p \( )/ a >, x, y R (a x ) y = a xy a, b, p, q >, p q = 1 ab ap p + bp q ( ) x < 1, x 1 lim n n xn /n! =, 66

67 5.4.4 ( ) (i) f C(R R) x, x R f(x + x ) = f(x) + f(x ) f(x) = f(1)x. (ii) g C(R (, )) x, x R g(x + x ) = g(x)g(x ) g(x) = g(1) x (a n ) s C, Re s > 1 a n n=1 n s a n 1 ζ- 67

68 6 II ( ) 6.1 f(x) = x 2 x a R x 2 a 2 ( (± ) 2 = ) f x a f(x) l d, k N\{}, A D R d, f : D R k f d = 2, k = 2 a R d (d = 1 a = ± ), l R k (k = 1 l = ± ) A (a n ) a n a f(a n ) l. f a A l lim f(x) = l x a x A x A, x a f(x) l A = D f a l lim x a f(x) = l x a f(x) l (i) a R (ii) a [, ] lim exp x = x a lim log x = x a exp a, a R,, a =,, a =. log a,, a =,, a =. a (, ), 68

69 6.1.3 A D R d, a R d (d = 1 a = ± ) f i : D R k, x a lim f i (x) = l i (i = 1, 2) : x A (a) ( ) l i R k f i,l i f 1 C\{} l 1 lim (f 1 + f 2 )(x) = l 1 + l 2 (6.1) x a x A lim (f 1 f 2 )(x) = l 1 l 2 (6.2) x a x A lim (1/f 1 )(x) = 1/l 1. (6.3) x a x A k = 1, l i R {l 1, l 2 } = {, } (6.1) { l 1, l 2 } = {, } (6.2) (b) ( ) k = 1, l 1 = l 2 R f : D R A f 1 f f 2 x a lim f(x) = l 1. x A : A (a n ) lim n a n = a f i (a n ), f(a n ) 3.2.5, 3.2.4, \( )/ ( ) g : f(d) R m, (a) x a lim f(x) = l (b) lim g(y) = l y l x A l R m m = 1 l = ± lim g f(x) = l. x a x A : A (a n ) lim n a n = a (a) f(a n ) l. (b) g f(a n ) l. \( )/ p, q, x > (i) a [, ] lim x p = a p, lim x p = a p, p def. p def. =, =. x a x a (ii) lim x x p e qx =. (iii) lim x p log x =, x lim x p log x =. x (i) x ±p = exp(±p log x) ( 6.1.2), ( 6.1.4) 69

70 (ii) m N (p, ) (qx) m /m! e qx x p e qx m!q m x p m. (i) x p m. x p e qx. (iii) y = log x x ( y ) x ( y ) x p log x = exp( py)y (ii), x p log x = exp(py)y = exp( p y ) y (ii) (a) x a lim f(x) = l. x A (b) ε > δ > x A B d (a, δ) = f(x) B k (l, ε). \( )/ {x R d ; x a < δ} (a R d ), B d (a, δ) = (1/δ, ] (d = 1, a = ), [, 1/δ) (d = 1, a = ). (6.4) B k (l, ε) B d (a, δ) a a : (b) δ-ε : (a) = (b): (b) ε >, δ >, A δ def. = {x A B d (a, δ) ; f(x) B k (l, ε)}. a n A 1/n a n B d (a, 1/n) a n a. f(a n ) B k (l, ε) f(a n ) l. (a) (a) = (b): a n A, a n a ε > (b) δ > a n a n a n B d (a, δ). (b) n f(a n ) B k (l, ε). f(a n ) l \( )/ ( 3.6.9) x a lim f(x) = l R d x A {f(x) ; x A B d (a, δ)} δ > 7

71 6.1.6 (b) ε = 1 δ {f(x) ; x A B d (a, δ)} B k (l, 1). \( )/ < x x x, x 1/x, x xx, (i)f(x) = p j= a jx j,g(x) = q j= b jx j (a p >, b q > ) lim x f(x) g(x). (ii) p R lim x {(x + 1) p x p }. 6.2 I R ( ) d, k N\{}, D R d, f D R k f d = 2, k = 2 a D f a (6.5) lim x a f(x) = f(a). (6.5) x D D a n a n a f(a n ) f(a). ( ) f a D f D D D R k - D C(D), C(D R k ) a n x C f(x) = a n x n n= x < r D = {x C ; x < r } f C(D C). exp, ch, sh, cos, sin C(C C) (5.12), 5.2.1,

72 f L [, ), α (, 1] f C(D). x, y D f(x) f(y) L x y α. f ( α = 1 x, a D, x a x a. x a α ( 6.1.2). f(x) f(a). \( )/ D R d (a) f 1, f 2 a D f 1 + f 2 a D f 1, f 2 f 1 f 2, f 1 /f 2 a D f 1 /f 2 f 2 (a) (b) ( ) a D R d, f : D R k, g : f(d) R m f a g f(a) g f a : (a) 6.1.3, (b) \( )/ ( ) n 1,.., n d N, c C m : C d C m(x) = cx n 1 1 x n d d. p 1, p 2 D = {x C d ; p 2 (x) } r(x) = p 1 (x)/p 2 (x) r C(D C). : x = (x i ) d i=1 C x x i 6.2.4(a) \( )/ (a):f C(D R k ). (b): a D f C(D B d (a, δ) R k ) δ > q : R d [, ) (1) (3) q (1) c R, x R d q(cx) = c q(x). (2) x, y R d q(x + y) q(x) + q(y). (3) q(x) = x =. (1), (2) q C(R d ) 72

73 6.2.3 f : R d R, α R t >, x R d f(tx) = t α f(x) f α α f (i) α > f(x) x = 1 f (ii) α = f α < f lim x f(x) ( f x = ) n 1,.., n d N, r > f : R d R f(x) = x n 1 1 x n d d / x r (x ), f() = f n n d > r n n d r (x = ) ( ) x Q, x R\Q f(x) = 1, f : R R x C\{}, x exp, sin, cos (e x 1)/x, sin x/x, (1 cos x)/x a < b, f : (a, b) R k x c (a, b) f(x) x c a < b, f : (a, b) R k a < c b lim x c f(x) x<c c f(c ) c = b f(c ) x c lim f(x) x c a c < b lim x c f(x) x>c c f(c+) c = a f(c+) x c lim f(x) x c a < c b f(c ) = f(c) f c c = b c a c < b f(c+) = f(c) f c c = a c A (a, b) f c A f A (a, b) 73

74 6.3.2 ( ) c R (a) 1 (c, ) c (b) 1 [c, ) c 1 (c, ) 1 [c, ) c c ( ) c (a, b), l R k f(c+) = f(c ) = l, (a < c < b ) lim f(x) = l f(c+) = l, (c = a ) x c x c f(c ) = l, (c = b ) k = 1, l = ± : = : =: c = a, b c, x n (a, b),x n c, x n c lim n f(x n ) = l K 1, K 2 N K 1 = {n N ; x n < c}, K 2 = {n N ; x n > c} K i = {k i () < k i (1) <...} K 1 lim n f(x k1 (n)) = f(c ) = l, K 2 lim n f(x k2 (n)) = f(c+) = l. I( 3.1.5) lim n f(x n ) = l. \( )/ ( ) c (a, b) f(c+) = f(c ) = f(c), (a < c < b ) f c f(c+) = f(c), (c = a ) f(c ) = f(c), (c = b ) : ( 6.3.3) l = f(c) \( )/, ,

75 6.3.5 ( ) k = 1 (a) a < c b f(c ) R (b) a c < b f(c+) R ( 2.2.1) 6.3 ( ) (a) a < c b f(c ) = (b) a c < b f(c+) = sup f(x), a<x<c inf f(x), a<x<c inf f(x), c<x<b f(x), sup c<x<b f, f. f, f. (a) f a < c n < c, c n c (1) lim f(c n ) = sup f(x). n a<x<c f l N, m N, n m, f(c l ) f(c n ) lim f(c n) = sup f(c n ) sup f(x). n n a<x<c }{{}}{{} s t s < t sup t s < f(y) t y (a, c) y n c n < y < c. c n c s = t. \( )/ ( ) a < b, f : (a, b) R f g(x) = sup a<y x f(y) x (a, b) ( ) f : [, 1] R [, 1) (, 1] f : s = sup{x [, 1] ; f [, x] } : (i) s >, (ii) f [, s], (iii) s = ( ) g n (x) (n N,x [, 1)) x x f(x) = n= g n(x) g n (1 ), f(1 ) f(1 ) = n= g n(1 ) ( ) 75

76 6.4 I ( ) ( I) I = [l, r] ( < l < r < ), f C(I R) c, c I x I f(c ) f(x) f(c). f min I f = f(c ), max I f = f(c) I f C(R R), lim x f(x) = f T >, f : R R x R f(x + T ) = f(x) f 6.5 ( ) I ( 3.3.2) ( I) I = [l, r] ( < l < r < ), a n I (n =, 1,..) c I a n a k(n) lim n a k(n) = c [l n, r n ] I, n =, 1,... [l, r ] = [l, r]. c n = (l n + r n )/2 { [l n+1, r n+1 ] = n (1) a k [l n, r n ] [l n, c n ], a k [l n, c n ] k, [c n, r n ], a k [l n, c n ] k. k k(n) l n, r n, r n l n = 2 n (r l) n ) c [l, r] 76

77 (2) lim n l n = lim n r n = c. (1) (2) lim n a k(n) = c. \( )/ ( ) ( ) c c f(i) R( ) a n I, n =, 1, 2,... f(a n ) R c I (a n ) (a k(n) ) a k(n) c R = lim f(a k(n) ) f = f(c). n R < R f \( )/ 7 Dugac, P: Elément d analyse de Karl Weierstrass, Archive for History of Exact Sciences 1, 1973,

78 7 ( ) 3 f(t) = t(cos t, sin t), t ) 17 y = f(x) (y )/( x ) x (y )/( x ) dy/dx y ( ) I R f : I R k (f : I C k = 2 ) x I f (x) x ( ) f (x) = y x lim D x,y (f), D x,y (f) = y x,y I f(y) f(x). (7.1) y x k = 1 f (x) = ± f (x) (f(x)) d, dx f(x), df dx (x). f (x) R k ( k = 1,f (x) = ± ) f x f x I x f (x) f ( ) y = f(x) (c,f(c)) y = f (c)(x c)+f(c) f (x) (x, f(x)) x f (x) f(x) x 78

79 7.1.2 ( ) (i) p N, x R (x p ) = px p 1. (ii) x (, ) (log x) = 1/x. : (i): y x, y x xp y p x y = p 1 x k y p 1 k px p 1. (ii):x, y (, ), y x log x log y ( 5.3.1) k= 1 x 1 y log x log y x y 1 x 1 y. y x ( ) 1, ( ) 1. x x \( )/ ( ) (a) : f(x) = x x = y >, y < D,y (f) = 1, 1. f () (b) : f(x) = x p, < p < 1 x = y > D,y (f) = yp = y yp 1, (y ) ( ) I R f : I R k x I f x : I\{x} y x D x,y (f) f(y) f(x) = D x,y (f)(y x). \( )/ I R f, g : I R k x I (a) ( ) f + g x I (f + g) (x) = f (x) + g (x). (b) ( ) f, g : I C fg x I (fg) (x) = f (x)g(x) + f(x)g (x). (c) ( ) f, g : I C, g(x) f/g x : (a),(b):y x (f/g) (x) = f (x)g(x) f(x)g (x) g(x) 2. D x,y (f + g) = D x,y (f) + D x,y (g) f (x) + g (x), D x,y (fg) = g(y)d x,y (f) + f(x)d x,y (g) f (x)g(x) + f(x)g (x). (c):g(x) g x y x g(y) 79

80 D x,y (1/g) = D x,y(g) g(x)g(y) g (x) g(x) 2. f 1 b f/g = f 1 g \( )/ ( ) D R d, f : D R x D x 1,.., x i 1, x i+1,.., x d i t f(x 1,.., x i 1, t, x i+1,.., x d ) t (x i ε, x i + ε) ε > f x i t = x i f x x i i f(x), xi f(x), x i f(x), f x i (x) x D i f(x) i f : x i f(x) x i ( ) r (, ], a n C (n N), z D def. = {z C ; z < r} : f(z) = a n z n n= z D f (z) def. = lim w z w z f(w) f(z) w z = na n z n 1 ( ). (7.2) n=1 z = x + iy (x, y R ) z D f(z) x, y x f(z) = 1 i yf(z) = f (z). (7.3) z D z < s < r s w z w < s (1) w n z n nz n 1 (z w) 1 2 n(n 1)(z w)2 s n 2. 8

81 w z f(w) f(z) w z na n z n 1 = n=1 ( ) w n z n a n w z nzn 1 n=1 n(n 1) a n s n 2. (1) w z n=1 } {{ } (7.2) (7.2) z = x + iy y x x y f x (z) = f (z). 1 f i y (z) = f (z) x R (i) c C (e cx ) = ce cx. (ii) a (, ) (a x ) = a x log a. (iii) (ch x) = sh x, (sh x) = ch x, (cos x) = sin x, (sin x) = cos x. : (i): e cx = c n x n n= n! (e cx ) c n x n 1 = (n 1)! = c c n x n. n! n=1 n= }{{} =e cx (ii): (i) c = log a \( )/ (iii): ch, sh, cos, sin exp 5.2.1, (i), \( )/ ( ) I R f : I R k, l R k (k = 1, l = ± ), x I (a),(b) (a) f x f (x) = l (b) φ : I R k y I f(y) f(x) = l(y x) + φ(y). (7.4) lim y x φ(y)/ y x =. (7.5) y x,y I 81

82 : (a) = (b): φ : I R (7.4) φ(y) = f(y) f(x) l(y x) φ (a) (7.5) (b) (a) = (b): (b) lim D y x x,y (f) = y x,y I ( y x lim l + φ(y) ) = l, y x y x,y I (a) \( )/ ( ) J R x J, J g f I R k g x f g(x) f g x (f g) (x) = f (g(x)) g (x). : φ 1 : J R, φ 2 : I R k (1) y J, g(y) g(x) = g (x)(y x) + φ 1 (y), y x lim φ 1 (y)/ y x =. y x,y J (2) z I, f(z) f(g(x)) = f (g(x))(z g(x))+φ 2 (z), y x lim φ 2 (z)/ z g(x) =. z g(x),z I (3) y x lim φ 2 (g(y))/ y x =. y x,y J (3) (3) f(g(y)) f(g(x)) (2) = f (g(x))(g(y) g(x)) + φ 2 (g(y)) (1) = f (g(x))g (x)(y x) + f (g(x))φ 1 (y) + φ 2 (g(y)) }{{} φ(y) (1), (3) lim φ(y)/(y x) =. y x y x,y J (3) x n J\{x}, x n x (4) lim n φ 2 (g(x n ))/ x n x =. g(x n ) = g(x) φ 2 (g(x n )) = φ 2 (g(x)) =. (4) g(x n ) g(x) n g x lim n g(x n ) = g(x). n φ 2 (g(x n )) x n x = φ 2(g(x n )) g(x n ) g(x) } {{ } (2) 82 g(x n ) g(x) x n x } {{ } (1).

83 (4) \( )/ ( ) x >,c C (x c ) = cx c 1. x c = e c log x = f g(x), f(y) = e cy, g(x) = log x. (x c ) = f (g(x)) g (x) = ce c log x 1 x = cxc 1. \( )/ I R f, g : I R k x I f g x I (f g) (x) = f (x) g(x) + f(x) g (x), I R f i : I C (i = 1,.., n) x I (i) f = f 1 f n x I x f = n i=1 f 1 f i 1 f if i+1 f n. (ii) p : C n C f = p(f 1,..., f n ) f x I f q : C 2n C f = q(f 1,..., f n, f 1,..., f n) a (, ),x R (a x ) = a x log a ( 7.1.8) a x a y g : I (, ) x I log g x (log g) (x) = g (x)/g(x) f j : I R (j = 1, 2) (i) I = R, f 1 (x) = sin m x, f 2 (x) = sin m x n (m, n N\{}). (ii) I = (, ), f 1 (x) = x x, f 2 (x) = x xx I R f : I R k x I f(x) ( 1 f(x) ), (f(x) 1 f(x) ) f(x) f (x) x, y C, x r, y r n = 2, 3,.. x n y n nx n 1 (x y) 1 2 n(n 1)rn 2 x y 2. n 1 n 1 (1) nx n x j y n j = (x y) (j + 1)x j y n 1 j. j= j= p(x, y) def = n 1 j= (j + 1)xj y n 1 j n xp(x, y) = nx n + jx j y n j, yp(x, y) = j=1 n (j + 1)x j y n j. j= 83

84 (1) n 1 x n y n = (x y) x j y n 1 j j= x n y n nx n 1 (x y) = (x y) ( n 1 ) x j y n 1 j nx n 1 j= ( ) n 2 = (x y) (n 1)x n 1 x j y n 1 j j= (1) n 2 = (x y) 2 (j + 1)x j y n 2 j. j= \( )/ ( ) I R f : I R k (f : I C k = 2 ) f x I f I I f : I R k D 1 (I), D 1 (I R k ) f D 1 (I) f D 1 (I) (f ) ( I ) (f ) f f (2) (second derivative) f m (m-th derivative) f (m), m = 1, 2,.., f () = f f (m) (x) (f(x)) (m), d m dx m f(x), d m f dx m (x), ( ) m d (x). (7.6) dx m N D m (I) I m, C m (I) f D m (I) f (),..., f (m) C(I) f D (I) f : I R k C (I) = C(I) ( ) D m+1 (I) C m (I) D m (I). f D m 1 (I) x I (f (m 1) ) (x) f (m) (x), (7.6) 84

85 7.2.2 f 1 f C (R) f D 1 (R) f 1 f C (R) f, g D m (I) (a) ( ) f + g D m (I) (f + g) (m) = f (m) + g (m). (b) ( ) f, g : I C fg D m (I) (fg) (m) = m ( m r r= ) f (r) g (m r), ( ), (7.7) ( ) m r ((1.11) ) (c) ( ) f, g : I C x I g(x) f/g D m (I). (d) f, g C m (I) (a),(b),(c) f + g C m (I), fg C m (I), f/g C m (I). : (a): ( 7.1.5) (b):m m = m 1 m 1 m 1 ( ) m 1 (1) (fg) (m 1) = f (r) g (m 1 r). r r= r m 1 f (r), g (r) D 1 (I) (1) (2) (fg) (m) = m 1 r= ( m 1 (3) ( m 1 r 1 ) ( + m 1 ) ( r = m ) r 1.11 (2) = f () g (m) + r ) (f (r+1) g (m 1 r) + f (r) g (m r) ). m r=1 {( ) m 1 + r 1 ( m 1 r )} f (r) (m r) (3) g = (7.7). (c): A m, B m A m B m m A 1 B 1. m 2 A m A m 1 B m 1 A m A m 1 B m 1 f/g D m 1 (I) D 1 (I). (f/g) = f g fg, f g fg, g 2 (b) D m 1 (I). g 2 85

86 A m 1 B m 1 f g fg, g 2 (f/g) D m 1 (I). f/g D m (I) (d): (a),(b),(c) \( )/ ( ) I, J R J g I f R k (a) g D m (J R), f D m (I R k ) f g D m (J R k ). (b) g C m (J R), f C m (I R k ) f g C m (J R k ). : (a):m m = m 1 m (1) (f g) = (f g)g. f D m 1 (I R k ) f g D m 1 (J R k ). g D m 1 (I R). (1) (f g) D m 1 (J R k ), f g D m (J R k ). (b): (a) m = 6.2.4(b) ( ) r (, ], a n C (n N), z D def. = {z C ; z < r} : f(z) = a n z n n= f(z) z D m f (m) (z) f (m) (z) = \( )/ n(n 1) (n m + 1)a n z n m ( ). (7.8) n=m f (m) () = m!a m. (7.9) z = x + iy (x, y R ) z D f(z) x, y C m =, 1, 2,.. ( x ) m f(z) = (1/i) m ( y ) m f(z) = f (m) (z). ( x ) m, ( y ) m x, y m 86

87 7.1.7 f(z) z z D f (z) = na n z n 1, n=1 ( ). f (z) z D ( 4.4.2) f(z) (z D) (7.8) (7.8) z = (7.9) y x x y \( )/ f : I C f C (I) : (i) I = R, f(x) = a x, ch x, sh x, cos x, sin x (a > ). (ii) I = (, ), f(x) = log x, x c (c C) ( ) 2n q n (x) = (x 2 1) n m q (m) n (i) (x 2 1)q n (1) (x) = 2nxq n (x). (ii) (x 2 1)q (n+2) n 1 2 n n! q(n) n (x) + 2xq n (n+1) (x) n(n + 1)q n (n) (x) =. n P n (x) = (x) (ii) (1 x 2 )P n (x) 2xP n(x) + n(n + 1)P n (x) = H n (x) = ( 1) n exp(x 2 /2) dn exp( x 2 /2), dx n x R, n =, 1, 2... (i) H n (x) n ( x n 1. (ii)h n(x) = xh n (x) H n+1 (x) = nh n 1 (x) 2 (iii) H n(x) xh n(x) = nh n (x) ( ) H n (x) ( 7.2.3) n H 1 (x) = x n = 1 H n (x) = n j=1 (x c j), (c 1 <... < c n ) (i) (iii) (i) ( 1) n j H n(c j ) >, j = 1, 2,.., n. (ii) H n+1 (c j ) n j n j H n(x) = xh n (x) H n+1 (x). (iii)h n+1 (x) n r >, a n, b n C (n N), z D def. = {z C ; z < r} : f(z) = n= a nz n, g(z) = n= b nz n ( ) m ( x f() = m x) g() ( m N) D f = g a, b, c, z C, c N, z < 1 F (z) = n= a(a + 1) (a + n 1)b(b + 1) (b + n 1) z n n!c(c + 1) (c + n 1) 87