4 II I - PDF 無料ダウンロード

1 2.1 1,... 1.1..................................... 1.2......................................... 3.3........................... 3 1 6 1.1.................................... 6 1.2...................................... 7 1.3...................................... 9 1.4.......................................... 13 2 I 16 2.1...................................... 16 2.2...................................... 18 2.3............................... 21 2.4 R d C........................................ 25 2.5.......................................... 28 3 I 34 3.1........................................ 34 3.2................................. 38 3.3........................................ 39 3.4.................................... 4 1 27 1 A B 2 5 9. 2 [e-mil] obuo@mth.kyoto-u.c.jp, [URL] http://www.mth.kyoto-u.c.jp/ e obuo 1

4 II 42 4.1...................................... 42 4.2..................................... 44 4.3................................ 46 5 I 49 5.1................................... 49 5.2........................................ 53 5.3........................................ 57 5.4...................................... 58 5.5....... 63 6 II 67 6.1................................... 67 6.2..................................... 71 6.3..................................... 73 6.4...................................... 74 6.5 ( ).................................... 76 7 78 7.1............................... 78 7.2................................ 83 7.3........................... 83 7.4 ( )......................... 87 7.5.................................... 92 7.6...................................... 94 8 98 8.1................................. 98 8.2................................... 11 8.3................................. 13 8.4..................................... 17 9 111 9.1.................................... 111 9.2................................. 115 9.3................................. 118 9.4 Γ- B-................................... 121 9.5 ( )...................... 124 9.6 ( )....................................... 126 9.7 ( )Γ- B-............................ 131 1 134 1.1............................... 134 1.2...................................... 137 1.3................................. 141 1.4................................... 151 11 157 11.1.................................... 157 11.2..................................... 163 11.3...................................... 167 11.4.................................... 172 11.5 ( )......................... 179 12 183 12.1..................................... 183 12.2........................................ 186 12.3................................ 19 12.4............................... 197 12.5 ( )............................... 2 2

13 21 13.1...................................... 21 13.2................................. 25 14 29 ( ) ( ).2 [ ] [Rud] Rudi, W.: Priciples of Mthmticl Alysis-3rd Editio McGrw-Hill Book Compy. [ ] I,II [ ] [ 1] [ 2] [ ] [ ] 1 exp, si, cos... [ 1] [ 2] 3.3.3.1 P, Q P = Q P Q Q = P P Q P = Q P = Q x,... x... x,...... x 3

1 1x,...... x,, 1 P def. Q P Q P def. = Q.3.2 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 }. x X, y Y (x, y) X Y X Y (x, y), (x, y ) X Y (x, y) = (x, y ) x = x y = y.3.3 X, Y ϕ x X x Y ϕ(x) ϕ X Y ϕ X Y ϕ : X Y x ϕ(x) x ϕ(x) ϕ : X Y, A X ϕ(a) Y ϕ(a) = {ϕ(x) ; x A} ϕ(a) ϕ A (imge) ϕ(x) = Y ϕ (surjective) ϕ : X Y, B Y ϕ 1 (B) X ϕ 1 (B) = {x X ; ϕ(x) B} ϕ 1 (B) ϕ B (iverse imge) y ϕ(x) ϕ 1 ({y}) 2 ϕ (ijective) (oe to oe) 4

ϕ y ϕ(x) ϕ 1 ({y}) ϕ 1 (y) ϕ 1 : y ϕ 1 (y) (ϕ(x) X) ϕ ϕ ϕ bijective Z ψ : Y Z x ψ(ϕ(x)), X Z ϕ ψ ψ ϕ.3.4 () X = {x 1, x 2, x 3 }, Y = {y 1, y 2 }, ϕ(x 1 ) = y 1, ϕ(x 2 ) = y 2, ϕ(x 3 ) = y 2 ϕ : X Y ϕ(x) = Y ϕ ϕ 1 ({y 2 }) = {x 2, x 3 } ϕ (b) X = {x 1, x 2 }, Y = {y 1, y 2, y 3 }, ϕ(x 1 ) = y 1, ϕ(x 2 ) = x 2 ϕ : X Y ϕ(x) = {y 1, y 2 } = Y ϕ ϕ 1 ({y 1 }) = {x 1 }, ϕ 1 ({y 2 }) = {x 2 } ϕ (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(x) = Y ϕ ϕ 1 ({y j }) = {x j }, j = 1, 2, 3 ϕ ϕ ϕ 1 : Y X ϕ 1 (x j ) = x j, j = 1, 2, 3.3.1 X, Y ϕ : X Y, ψ : Y X { ϕ, ϕ 1 x X ψ ϕ(x) = x = ψ ψ, ψ 1 = ϕ y Y ϕ ψ(y) = y 5

1 (13.1 ) 1.3.6 1.1 1.1.1 R R N, Z, Q N = {, 1, 1 + 1(= 2), 1 + 1 + 1(= 3),...}, ( ) Z = { b ;, b N}, ( ) Q = {/b ; Z, b N, b }. ( ) >, <,, +, { {,, = +,, =,.,. = {,,,., +, (bsolute vlue), (positive prt), (egtive prt) 1.1.1 (i) x R\{} x >. (ii) c, x R cx = c x (iii) x, y R x y x ± y x + y. (iv) x = x + x, x = x + + x. (v) x + = ( x + x)/2, x = ( x x)/2. 1.1.2 x, y R, m N\{} (i) x m y m = (x y) m 1 k= xk y m 1 k. (ii) x r, y r x m y m mr m 1 x y, x m y m mx m 1 (x y) 1 2 m(m 1)rm 2 x y 2. 1.1.2 R 2 +, R = R {, + } + 6

± R { } < +, R {+ } <. + (positive ifiity), (egtive ifiity) R = R {, + } ± R {+ } + = + =, R { } = + ( ) = + =, < + = =, ( ) = ( ) = ( ) = ( ) =. R / = /( ) =. < + = + 1,, / 2 1.1.2 (O1) (O6), b, c R 1.1.3, b R (, b) R, [, b] R (, b) = {x R ; < x < b}, [, b] = {x R ; x b}, (, b) (ope itervl) [, b] (closed itervl) [, b) R (, b] R [, b) = {x R ; x < b}, (, b] = {x R ; < x b}. (, b), [, b], [, b), (, b] (itervl), b 1.1.4 A N ϕ : {x N ; 1 x } A A fiite set A ifiite set 1.2 1.2.1 A R, m R (U) A [, m]. m A (upper boud) (L) A [m, ]. m A (lower boud) 7

m A (U) m A (mximum) mx A m A (L) m A (miimum) mi A A A A A (bouded) ( ± ) { { A, mx A = A = { }, mi A = A, A = { }. A R mx A, mi A 1.2.1 A R mx A, mi A 1.2.2 ( ) A R, m R, A = { A} m = mx A m = mi( A), m = mi A m = mx( A). 1.2.2 ( ) A R mx A, mi A A 1.1.4 1.2.3 A = {, b} mx A = b, mi A = b 1.2.3 1.2.2 1.2.3 1.3.1 1.2.3 1.2.5 1.2.3 u Z, = A Z [, u] mx A l Z, = A Z [l, ] mi A : 1.2.2 mx A A R (1) l A [, u], m = mx(a [l, u]) m = mx A. 1.2.4 l A A = A [l, u] 13.1.2). mx A 1.2.2 mx A (1). 1.2.3 u Z l Z u R l R 1.3.1) 1.2.4 1.2.3 (1) 1.2.5 ( ) d N\{}, x Z x = qd + r (q, r) Z {,.., d 1} ( ) 1.2.3 1.2.5 r x d r = d d x (divisor) x d d (multiple) p N\{} 1 p p (prime umber) x, y Z 1 8

x, y (reltively prime) 1.2.5 d 1 = x, d = d d 1 d d +1 =, 1, 2,.. d 1 d +1 d 1. d N 1 d N+1 = N N x d d N 3 1.2.6 ( ) 1.2.5 x, d, q q x/d [q, q + 1) 1.2.7 ( ) S {1, 2,, 2} + 1 S 1.3 1.3.6 () x x < (b) x 2 = 2 (),(b) 1.3.9, 2.3.6 1.3.1 A R, m R 2 m A (supremum) sup A (U) A [, m], m A 1.2.1 (S) x < m (x, m] A. 2 m A (ifimum) if A (L) A [m, ], m A 1.2.1 (I) m < x [m, x) A. m (S) m m A m = x < m x R (S) m (I) m m A m = m < x x R (I) 1.3.1 A R sup A, if A 3 Euclid(Euκλειδηζ), 365 B.C.? 275 B.C.? 9

1.3.2 ( ) A R, m R, A = { A} m = sup A m = if( A), m = if A m = sup( A). A A [, m] A [ m, ], x [, m) (x, m] A x ( m, ] [ m, x) A. m A (U),(S) 1.3.1 m A (L),(I) m = sup A m = if( A) 1.3.3 ( ) A R, m R () m = mx A m = sup A A (b) m = mi A m = if A A () = : m = mx A 1.3.1 (U) (S) x < m m (x, m] A (S) () = : m = sup A A m A 1.3.1(U) m = mx A. (b): 1.2.2, 1.3.2 () 1.3.5 1.3.4, b R, < b (, b). x x (, b), ( =, b = ), + 1, ( <, b = ), x = b 1, ( =, b < ), ( + b)/2, ( <, b < ). 1.3.5 ( ) < b, I= (, b) I I = [, b] sup I = b, if I =. 1.3.3 mx I b I mx I = b, mi I I mi I =. 1

: 1.3.2 sup I = b 1.3.1 (U),(S) (U): I [, b] (S): x < b x < b. 1.3.4 = ( x, b). ( x, b) (x, b] I (x, b] I. 1.3.6 (AC) (the xiom of cotiuity) : (AC) A R sup A 13.1.1 1.3.7 (AC) (AC ) (AC ) A R if A : (AC) (AC ): 1.3.2 A A (AC) sup( A) 1.3.2 if A sup( A) (AC ) (AC): 1.3.8 sup if () A R sup A R, if A R { sup A b A [, b], (b) b R b if A A [b, ]. (c) { sup A < + A, if A > A. (d) = A [l, u] l u. if A sup A. : (): 1.3.2 sup A sup A { +, A sup A =, A =, { }, sup A = A (U),(S) m = sup A = A { } A =, { }. A R A, { } A R A R (AC) m = sup(a R) m = sup A 1.3.1 (U), (S) (U): A A = A R. = m A R m = sup(a R) m. (S): x < m A R, x <. A, x <. m = sup A sup A 11

m = sup A, m = if A (b): 1.3.2 sup A : (U) : b < m (S) (b, m ] A. A [, b]. (c): (b) (d): 1.3.2 A R, γ R\{}, β R γa + β = {γ + β ; A} γ (, ) sup(γa + β) = γ sup A + β, if(γa + β) = γ if A + β, sup( γa + β) = γ if A + β, if( γa + β) = γ sup A + β. 1.3.3 = A (, ] sup{1/ ; A} = 1/ if A, if{1/ ; A} = 1/ sup A. 1/ = 1.3.4 (sup if ) A, B R (i) A [, b] b B = sup A sup B. (ii) A [b, ] b B = if B if A. (iii) = A B = if B if A sup A sup B. (iv): (i) (iii) A = [, 1], B = (, 1). 1.3.5 ( ) A R A U(A), L(A) sup A = mi U(A), if A = mi L(A) 1.3.9 ( 4 ) Z Z (AC) m def = sup Z R. m 1 < m z Z, m 1 < z 1.3.1 (S). z m < z + 1 Z. 1.3.1 (U) (AC ) Z Z x x < z z Z 1.3.1 () A Z mx A A Z mi A (b) ( ) R q ( 1, ], [q, q + 1) q Z q 5 Q 1.2.6 4 Archimedes (287 212 B.C.) 5 [] 12

(): 1.2.2 mx A A = m Z, A [, m] 1.2.3 = mx A (b):( 1, ] 2 b, c ( 1, ] Z, b < c 1 = ( 1) > c b 1 ( 1, ] Z 2 (1) A def. = Z (, ] mx A (2) m = mx A 1 < m. m ( 1, ] Z (1) () A : A (, ] A : z Z, z., z A. (2) m 1, m+1 m < m+1 A. m A 1.3.11 D R < b (, b) D D R (dese) R R 1.3.4 1.3.12 ( ) Q R : < b (b ) 1 <, + 1 < b N\{} 1.3.1 m (, + 1] m Z m (, b), m/ (, b) Q. 1.3.6 ( ) x R N = 1, 2, x p q < 1 Nq p Z q {1,, N} 6 N + 1 x j = jx jx [, 1) (j =, 1,.., N) 1.3.12 x p/q Diophtus 1.4 1.4 6 Peter Gustv Lejeue-Dirichlet (185 1859) 13

1.4.1 D f : D R (fuctio) 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) } 1.4.1 ( ) D 1.4.1 1.4.2 D R, f : D R f (o-decresig), (icresig) x, y D, x < y = f(x) f(y) f x, y D, x < y = f(x) < f(y). f ( strictly icresig) f f (o-icresig), (decresig) f f f ( strictly decresig) (mootoe fuctios) 1.4.2 (i) (ii) ( ) ( ) 1.4.3 D f : D R, E D sup f, if f E E sup f = sup f(e), E mx f(e), mi f(e) mx E f, mi if E f, mx E E if E f = if f(e). f, mi E f f sup E f sup f(x) x E E D {f(x) ; x E} f E (bouded) E ( ) f E sup f <, E f E sup f <, E f E if E f D f D ( ) 14

1.3.8 sup, if 1.4.4 sup if A f : A R, β R () if A f sup A f. (b) sup A f β A f() β. (c) if A f β A f() β. : 1.3.8 1.4.3 A f : A R, γ R\{}, β R sup A (γf+β) = γ sup A f+β, if A (γf+β) = γ if A f+β, sup A ( γf+β) = γ if A f+β, if A ( γf + β) = γ sup A f + β. 1.4.4 A, B f : A R,g : B R (i) A f() g(b) b B sup A f sup B g. (ii) A f() g(b) b B if g if f. (iii) A B B A if B g if A g sup A g sup B g. (iv) A = B f() g() if A f if A g, sup A f sup A g. 1.4.5 A f : A R, g : A R if A f + if g if A A (f + g) sup A (f + g) sup A f + sup g.. A 1.4.6 ( ) B b B A b R sup ( b B A b ) = sup sup A b, b B if ( b B A b ) = if b B if A b. 1.4.7 ( ) A,B f : A B R (i)(sup sup ) (ii)(if if ) sup (,b) A B if (,b) A B f(, b) = sup sup f(, b) = sup sup f(, b). f(, b) = if b B A if b B A sup b B A (iii) ( sup if if sup ) sup if f(, b) if A b B f(, b) = if f(, b). A b B if A b B f(, b). 1.4.8 ( ) {, 1} A B A B [, 1], f(, b) = b sup if f(, b) = < 1/2 if A b B sup b B A f(, b). 1.4.9 ( ) 1.4.7 (, b ) f, f(, b ) f(, b ) f(, b), (, b) A B sup if f(, b) = if A b B sup b B A f(, b) = f(, b ). 15

2 I 2.1 = 1, 2,... 1/ 1/ ε 1/ε > 1/ε 1/ < ε 2.1.1 ( ) m N N [m, ) (sequece) (rel sequece) ( ) =m, ( ) m, ( ) m m = R, ε (, ) ( ε, + ε) ( R ), B(, ε) = (1/ε, ] ( = ), [, 1/ε) ( = ). B(, ε) ( ) ( ) (limit) : ε (, ) B(, ε). (2.1) ε (, ) m N m B(, ε). 7 (2.1) lim =, lim =, (2.1) R ( ) R (coverges to ) (2.1) = ± ( ) ± (diverges to ± ) 1 (2.1) ε (, ) ε (, ) ε 7 m N m 16

2 R, m N lim = lim m+ = 3 C ( ) (2.1) B(, ε) B(, Cε) ( ) (2.1) ( ): (2.1) ε ε Cε ( ) ( ) (2.1) : ( ) ε ε ε/c (2.1) 2.1.2 ( ) () 1/ 1/( ) = 1/. (b) c > c, 1/. (): B(, ε) 1/ B(, ε) 1/ B(, ε) (b): () ε (, ) N > 1 cε. N c > 1 ε. 2.1.3 () p N\{} lim p = lim p =. (b) r > 1 lim r =, lim r =. (): p 2.1.2 (b): r 1 + (r 1) 2.1.2 2.1.1 R, 2.1.2 ( ) lim = R = 2.1.3 D R R R D ( ) 2.1.4 ( 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 () = 2, k 2 () = 2 + 1. K 1, K 2 ) R (),(b) (), (b) K i i ki (). 2.1.5 = ( 1) 2.1.6 ε (, ) B(, ε) R, =, = [ ε, + ε], [1/ε, ], [, 1/ε] (2.1) B(, ε) B(, ε) 17

2.1.7 ( ) lim R, lim R lim = if sup = if sup m+, lim = sup if = sup if m+. m m m m m m m N, R (i) if lim lim sup, (ii) m m lim < < (iii) < lim < (iv) < lim < (v) lim < = < (vi) lim = lim = lim = 2.1.8 ( ),, R F (, 1,..., ) = + 1 1 + 1 2 + 1... 2 + 1 1 + 1 Z, N\{} ( = 1, 2, ) P, Q Z ( = 1,, 1, 2,...) : P 1 = 1,P =, Q 1 =, Q = 1, P = P 1 + P 2, Q = Q 1 + Q 2 1. : (i) Q < Q +1, (ii) P Q 1 P 1 Q = ( 1) 1, (iii) P /Q = F (, 1,, ). (iii) ((ii) ) 2.1.9 ( ) x R x = x x 1 = x 1 + 1 x x > 1, = 1, 2, ( N x N Z x N+1 ) = x P, Q Z ( = 1,, 1, 2,...) 2.1.8 (i) x 1,...x x = F (,..., 1, x ), x P Q < 1. Q 2 (ii) x Q N x N Z x = F ( 1,..., N 1, x N ). (iii) x Q P N x lim Q = x. (iv) x x p q < 1 p/q K 5 q 2 x Q x p q < 1 p/q (Hurwitz ) Kq 2 2.2 2.2, b, c R, ( ), (b ), (c ) 2.2.1 ( ) () if >. sup <. (b) R sup <. : (): sup, if 1.3.2 b R, m, m, b < ( = b = 1 b = 1 ) =, 1,.., m 1 mi{, 1,..., m 1 }. m def. = mi{, 1,..., m 1, b}. < m if (b): () 18

2.2.2, b b b b. : b < c R, b < c < ( 1.3.4)., b b b < c <. 2.2.3 ( ) ( ) R :, b R,, b 2.2.2 b b. 2.2.4 ( I), c b c b. : ε > N, c B(, ε). N b [, c ] B(, ε). 2.2.1 ( II) b, b. b 2.2.2 R m N if sup m m 2.2.3 = A R A ( ), (b ) sup A, b if A 2.2.5 ( ), b b () + b + b, {, b} = {, } (b) b b, {, b } = {, } (c) 1 1, : ():, b R ε > N < ε, b b < ε. N ( + b ) ( + b) + b b < 2ε. 2.1.1 3 = b = = b. c def. = if b > ( ). N + b + c. N > 2c, + c 1 2. + b. (b):, b R M def. = sup b <. ε > N < ε, b b < ε. N b b ( )b + (b b) ε M + ε = (M + )ε. 2.1.1 3 = b = : = b =. =,b > 19

> b > b/2, b > b/2. b. (c): R : ε > N < ε ( /2), < ε /2. N 1 1 = < 2ε 2. 2.1.1 3 = : = ε > > 1/ε < 1 < ε. = 2.2.6 ( ), (b ) s = j=1 j(b j b j 1 ), t = 1 j=1 ( j+1 j )b j R, = b , m N, m, < ε/2. ( ) C = sup 1 < = 1 b j=1 (b j b j 1 ) m s b 1 b m 1 (b j b j 1 ) j + 1 b j=1 1 m 1 (b j b j 1 )C + 1 b b j=1 (b j b j+1 ) j j=m (b j b j 1 ) ε 2 b m 1C + ε b }{{ 2. } (2) m lim (2) =. l N, l, (2) < ε/2. m l s b < ε. 2.2.4 ( ) lim 1 = lim 1 mx 1 j j = 2.2.5 2.2.6 R s /b 2.2.6 ( ) ( ), (b ) s = 1 b 1 + 2 b 2... + 1 b 1 j=m R, b b R s / b 2.2.7 ( ) ( ), (b ) = lim, = lim, b = lim b, b = lim b ( 2.1.7 ) (i) {, b} = {, } lim( + b ) + b. (ii) {, b} = {, } + b lim( + b ). (iii) (b ) + b lim( + b ) + b, + b lim( + b ) + b. (vi) b b R lim( + b ) = + b, lim( + b ) = + b. 8 Eresto Cesáro (1859 196), Leopold Kroecker (1823 91) 2

2.2.8 ( ) 2.2.6 lim lim s /b lim s /b lim. 2.2.9 ( ) ( ) m, 1 m+ m + lim / = if 1 / = : m 1 m, = mq + r (q /m ) q m + r. lim / m /m. 2.3 2.3.1 ( ) ( ) R () ( ) lim = sup. (b) ( ) lim = if : (): (1) lim (1) l N, m N, m, l. = sup = sup (2) b <, m N, m, b <. b < sup l N, b < l. l (1) m m b < l (2) (b): (3) lim (3) l N, m N, m, l. = if () 2.3.1 2.3.1 19 ) 9 2.3.1 2.3.1 2.3.1 (3) lim = if 2.3.2 (, b) R, f : (, b) R (i) < c < c b, c c lim lim f(c ) = if f(x). c<x<b 9 Augusti Cuchy, 1789 1857 f(c ) = sup f(x). (ii) c < c < b, c c <x<c 21

2.3.3 ( ) s m, R (m, N) m, N s m+1, s m,, s m,+1 s m, lim m lim s m, = lim lim m s m, 2.3.2 ( ), (b ) ( ) (b ) b ( =, 1, 2,..) b ( ), (b ) b b ( ), (b ) = b + ( b ) ( ), (b ) 2.3.4 4.2.1 2.3.3 I R f : I R I f I ( ) lim = lim f( ) = f(). (2.2) I C(I), C(I R) 1 (2.2) lim = f( ) f() ( ) ( ) 2 δ ε (2.2) ε > δ > x I ( δ, + δ) f(x) f() < ε. 4.1.6 ( ) 2.3.4 ( ) I R f C(I R) (if I f, sup f) f(i). I : f(i) [if I f, sup I f] 2.3.4 f(i) if I f, sup I f : (1), b I, f() s f(b) c I, s = f(c). = b f() = s = f(b) c = < b b < [, b ] I, =, 1,... : [, b ] = [, b]. [, b ] c = ( + b )/2 [ +1, b +1 ] = { [, c ], s < f(c ), [c, b ], f(c ) s, 22

(2) f( ) s f(b ). = f() s f(b) (2) s < f(c ) f( +1 ) = f( ) f(c ) s f( +1 ) = f(c ) s s < f(c }{{} ) = f(b +1 ), {}}{ f(b ) = f(b +1 )., b, b = + 2 (b ). 2.3.2 ( ), (b ) c c I. f(c) f = lim f( ) (2) s (2) lim f(b ) f = f(c). s = f(c) (1) 2.3.4 19 1817 2.3.4 1 1821 2.3.4 2.3.4 f : R R (i) f lim f( ) =, lim f(b ) = ( ), (b ) (ii) f lim f( ) =, lim f(b ) = ( ), (b ) f 2.3.5 f : [, 1] [, 1] f(c) = c c [, 1] 2.3.6 f C(R) R f( + b) = f(b) b R 2.3.5 ( ) I R f : I R () f C(I R) f(i) (b) f f 1 : f(i) I ). 11 : (): : 2.3.4 :, I,, lim f( ) = f() 2.3.3 K 1, K 2 N K 1 = { N ; < }, K 2 = { N ; > } 1 Berrd Bolzo (1781 1848) 11 () 2.3.3 δ ε δ ε 2.3 23

K 1 K 2 = N. K i = {k i () < k i (1) <..} (1) K i lim f( ki ()) = f(), I 2.1.4 lim f( ) = f() f (f ) i = 1 i = 2 k1 () f (2) l N, m N, m, f( k1 (l)) f( k1 ()). 2.3.1 lim f( k1 ()) = s f(), s def. = sup f( k1 ()) s < f() f( k1 ()) s < f(). f(i) s = f(b) b I k1 () b <. s = f(). (b): f f 1.4.2 f 1 : f(i) I.3.1 1.4.2). I () f 1 2.3.6 ( m ) m N\{} x x m ([, ) [, )) ( 2.3.5 ) x x 1/m 1/m m m 2 2.3.7 (i) (, ) lim 1/ = 1. (ii) ( ), R lim +1 / = = lim 1/ =. (iii) ( ) lim ( j=1 j ) 1/ = sup 1. 2.3.8 ( ) ( ) lim +1 / lim 1/ lim 1/ lim +1 /. 2.3.9 > b >, b ( 1) = 1+b 1 2, b = 1 b 1 lim, lim b 2.3.1 >, b, c R, f(x) = x 2 + bx + c, S = {x R ; f(x) = } S b 2 4c b 2 4c S = {s, s + }, s ± b± b 2 4c 2. x [s, s + ] f(x) >, x (s, s + ) f(x) <. 2.3.11 ( ) I R < b, f : I R, x I ( N) x +1 = f(x ) (i) x f(x ) f [x, b) x x +1 x f(x ). x (c, m) f(x) < x m (, b) x x (, m] (ii) x f(x ) f (, x ] I x x +1 x f(x ). x (, m) f(x) > x m (, b) x x [m, b) (iii) f lim x = x I x = f(x ). 24

2.3.12 ( ) 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) x +1 = f(x ) lim x s 2.3.11. 2.3.13 ( ) I R (± ), f : I R x I ( N) x +1 = f(x ) (i) (x 2 ), (x 2+1 ) g = f f (ii) f s = f f(s) s I x s (iii) I = [, ) f(x) = 1/(x + b) (, b > ) (x ) 2.3.14 ( ) ( ) (i) 2 Q. (ii) x, y Q, y x + y 2 Q. (iii) R\Q R., b R, < b, (, b) Q. x (, b) x + 2/ (, b). 2.3.5() ε >, I (1) I I 1 I 1 < x ( 1, ) f() f(x) < ε. (2) I I 2 I < 2 x (, 2 ) f(x) f() < ε. (1),(2) δ = ( 1 ) ( 2 ) x ( δ, + δ) I f(x) f() < ε (1) (2) I b < b I f(b) = f() x (b, ) f(x) = f() (1) f(b) < f() f(b) < f() ε < f() ε (, ε) f(i) f( 1 ) = f() ε 1 I f 1 (b, ). x ( 1, ) f() f(x) f() f( 1 ) = ε < ε. (1) 2.4 R d C 2.4.1 d A 1,..., A d A 1,..., A d j A j (j = 1,.., d) A 1 A d = {( j ) d j=1 ; j A j, j = 1,.., d.} 25

A 1,..., A d (direct product) A j = A (j = 1,..., d) A d j j j R d d- d- c R R d x = (x j ) d j=1, y = (y j) d j=1 cx Rd, x + y R d cx = (cx j ) d j=1, x + y = (x j + y j ) d j=1 x y R x [, ) x y = d x i y i, x = x x. i=1 x y (ier product), x (Euclide orm), R d d B R d sup x . (ii) c R cx = c x (iii) x y x y. (iv) x y x±y x + y. 2.4.2 A R d (dimeter) dim(a) = sup x y A x,y A dim(a) < 2.4.2 R 2 2.4.1 (, b)(, b ) = ( bb, b + b) (2.3) R 2 (complex field) C 12 C (complex umber) i = (, 1) z = (, b) C z = + bi (2.4) + i, + bi, bi i 13 (2.3) ( + bi)( + b i) = ( bb ) + (b + b)i. (2.4) R z (rel prt) Re z b R z (imgiry prt) Im z bi z (cojugte) z Re z = z + z 2, Im z = z z, 2i 12 Willim Hmilto (185 65) 13 i 14 i 26

{z C ; Im z = }, {z C ; Re z = } (rel xis), (imgiry xis) R R C 2.4.3 x, y C (i) x + y = x + y, xy = x y, (ii) x 2 = xx, (iii) xy = x y, (iv) x 1/x = x/ x 2, 1/x = 1/x, 1/x = 1/ x. 2.4.4 x, y C, m N\{} (i) x m y m = (x y) m 1 k= xk y m 1 k. (ii) x r, y r x m y m mr m 1 x y, x m y m mx m 1 (x y) 1 2 m(m 1)rm 2 x y 2. 2.4.5 α C, N ( ) ( ( α ) α ) = 1, 1 ( ) ( ) α = α(α 1) (α + 1)/!. (i) α 1 1 + ( ) ( α 1 = α ). (ii) ( ) α +1 ( + 1) + ( ( α ) = α α ) (. (iii) α ) = ( 1) ( ) α+ 1. x ( 1, ) α C (1 + x) α x < 1 (1 + x) α = ) x ( α = 2.4.6 ( ) c C C d x = (x i ) d i=1, y = (y i) d i=1 cx, x, x + y Cd cx = (cx j ) d j=1, x = (x j) d j=1, x + y = (x j + y j ) d j=1 x y C x [, ) x y = d i=1 x iy i, x = x x C = R 2 C d = R 2d C d R 2d 2.4.1 2.4.7 ( ) x, y C, x < 1, y < 1 x y 1 xy < 1 2.4.8 ( ) x C, 1..., + 1 x +... + x = x 1 2.4.3 m N N [m, ) R d (sequece) R 2 = C (complex sequece) ( ) =m, ( ) m, ( ) m m = R d lim = ( ) R d lim =, lim =, { ; N} ( ) 27

2.4.4 z C z < 1 z. z = z. 2.4.9 z C, N (1 z) j=1 (1 + z2j 1 ) = 1 z 2, z < 1 lim j=1 (1 + z2j 1 ) = 1/(1 z), b R d, ( ), (b ) R d 2.4.5 = ( j ) d j=1, = (,j ) d j=1 Rd () j = 1,.., d lim,j = j, Re Re Im Im, (b) ( ) j = 1,.., d (,j ) = Re, Im (c) : (): mx 1 j d,j j 1 j d,j j. (b): mx 1 j d,j 1 j d,j. (c): (), (b) 2.4.6 ( ), b b + b + b, b b. b b, b 1 b 1 b, b : 2.4.5 2.2.5 2.2.5 R d 2.4.1 ( ) d d M d C d2 A M d A A M d ( =, 1,...) A, B, A, B M d (i) AB A B. (ii) A A, B B A +B A+B, A B AB. (iii) B, B B B B 1 B 1. 2.5 2.5.1 ( ) R d (d = 2 ) m, N, m ( ) (prtil sum) s m, = j (2.5) j=m 28

(2.5) (s m, ) =m series ( ), m m = (2.5) lim s m, R d, =m m (2.5) s,m = s, s m 1, m N ( ) ( ) = + j=1 ( j j 1 ) 2.5.2 z C z < 1 z = (1 z) 1. 1 z = (1 z) 1 j= zj 2.4.4. z 2.4.4 2.5.1 ( ) A A < 1 I A A = (I A) 1 2.4.1 I A = I 2.5.3 ( ) R d l : N N { } l() lim l() j =. j= l() =, l() lim = lim j =. = s = j= j s s j= l() j = s l() s s s =. j= 2.5.4 ( ) ( ), (b ) () s = +... + ( N) (b) ( ) b b. b 29

(c) ( ) r [, 1), +1 r (): s (b): m N, j m, j b j. m s = s m 1 + j s m 1 + j=m b j. j=m b j (s ) j=m (c) m N, j m, j r j 1... r j m m. m s = s m 1 + j s m 1 + m j=m j=m r j m }{{} 1/(1 r). (s ) 2.5.2 ( ) ( ), (b ) b b +1 2.5.5 =1 p p = 1 p = 2, 3,.. s = j=1 j p p = 1 s 2 s = 2 j=+1 j 1 (2) 1 = 2 1. 2.5.3 s s p 2 s s 2 +1 1 = k= 2 k+1 1 j=2 k j p k= 2 k+1 1 j=2 k 2 pk } {{ } =2 (p 1)k 1/(1 2 (p 1) ) <. s 2.5.4 2.5.3 ( ) =1 p (p = 2, 3,..) ( 2.5.2) 2.5.6 ( ) R d A =, B = b c R A, ca = c, A + B = ( + b ). (2.6) (2.6) 2 c C A =. (2.7) 3

N N N 2.2.2 2.4.6 2.5.4 ( ), b R, b ( N), b A =, B = b A < B 2.5.7 ( ) R d () d = 1 = + + = +. ( ). (b) d 1 ( ) = (,j ) d j=1 < j = 1,.., d = j = 1,.., d,j < =,j = ( = =,j)d j=1. C < = Re <, Im <. Re, Im = Re + i Im. < (coverge bsolutely) (): : = + + N N, N ± N N 2.2.5 2 : ± = + (2.6) 2 (b): 1 :,j,1 +... +,d 2.5.4 2 : 3 : 2.4.5 31

2.5.8 ( ) R d x C, x < r lim +1 / = 1/r = x lim x =. 2.5.6 b = x b < b +1 /b = +1 x / x /r < 1 b +1 /b < x /r. 2.5.4) 2.5.5 (),(b) 2.5.7 () (b) (b) () () (b) 2.5.6 C, r 2.5.8 (i) = p (p N), r = 1. (ii) = 1/!, r =. (iii) = ( α ) (α C, 2.4.5 ), r = 1. 2.5.7 (x ) R d () (b) (c) (): r [, 1) x +1 x r x x 1. (b): = x +1 x <. (c): (x ) 2.5.8 ( ) f : R d R d r [, 1) f (cotrctio mp) x, y R d f(x) f(y) r x y. f f(x) = x x R d x R d x R d, = 1, 2,.. x = f(x 1 ) (x ) 2.5.7 x = lim x 2.5.9 ( ), b, x C, c = j= jb j f(x) = x, g(x) = b x h(x) = c x h(x) = f(x)g(x) def. δ = 2 i= c ) ( ix ( i j= jx j k= b kx k) ( 2 ) ( δ j=+1 2 ) ( jx j b 2 ) ( kx k + j= 2 ) jx j k=+1 b kx k. 2.5.1 ( ) m, C (m, N) m= m, = m= = m, ( ). (ii): (i) (i) = = m, = m= m, m m= m, = m= = m,. 2.5.9 2.5.11 ( ) 2 q N, D {, 1.., q 1} (x ) 1 E D x = q 1 (i) (x ) 1 D\E x = =1 x /q x [, 1) ( x q ). (ii) (x ) 1 x D\E [, 1) [, 1) ( R ) 32

15 [, 1) ϕ : N [, 1) [, 1) = {ϕ(m)} m N ϕ(m) 2 ϕ(m) = =1 x m,/2 y = 1 x {, 1}, y = =1 y /2 y [, 1) y {ϕ(m)} m N ( y ϕ(m) m y m x m,m. [, 1) = {ϕ(m)} m N 2.5.9 R d ( ), (), ( 2 + 2+1 ) (b) ( ) d = 1 = ( 1) +1 ( N) : (): s = j s 2+1 = ( 2j + 2j+1 ) = s 2 + 2+1. s s 2+1 s s 2+1 s 2+1 s 2 I( 2.1.4) s lim s 2+1 (b): () s 2+1 2j + 2j+1 s 2+1 2j 1 + 2j, 2j+1 s 2+1 = + s 2+1 ( 2j 1 + 2j ) + 2+1 1 2.5.12 f(x) = = x (, x C) (i) f(±x) f(x)+f( x) 2 = = 2x 2, f(x) f( x) 2 = = 2+1x 2+1. (ii) ( ) f(x) f(x) f(x) = f(x) 2.5.13 1 ( 1) m (m = 1, 2,..) 2.5.14 2.5.9(b) 2+1 k=2+1 k k=2 k 2, k= k 2.5.15 ( ) α R m > 1 < α p q < 1 q p Z, m q N\{} α 16 α = k= ( 1)k 1 2 k! (i) k=+1 ( 1)k 2 k! 2 (+1)!. (ii) p = k= ( 1)k 2! k!, q = 2! m > 1 < α p q < 1 q. m 15 Georg Ctor (1845 1918) 16 Joseph Liouville 189 1882 33

3 I 18 f(x), i, e, si, cos ( 3.2.2) 3.1 2.5.8 3.1.1 ( ) x C exp x = x! x exp x (expoetil) e def. = exp(1) = 2.71828... exp x e x ( e x 3.4.1 3.1.2 ( 3.1.3 ) x C s (x) = m= x m ( m!, e (x) = 1 + x ) e (x) = m= p m, x m m! (3.1) p, = p 1, = 1, p m, = ( 1 ) 1 ( ) ( ) 1 2 1 m 1, (m 2). exp x s (x) x +1 exp x, (3.2) ( + 1)! ( ) x +1 exp x e (x) ( + 1)! + x 2 exp( x ) (3.3) 2 f (x) s (x) e (x) x, y C, x, y r, C, exp x exp y sup f (x) f (y) x y exp r, (3.4) 1 exp exp, f ( ) exp. (3.5) 34

(3.1): e (x) 2 = = ( 1) ( m + 1) ( x m m! ) ( 1 1 ) ( 1 2 ) ( 1 m 1 ) x m m! }{{} p m, m= m= (3.2): exp x s (x) = m=+1 x m m! (2.6) m=+1 x m m! = x +1 m= x m ( + m + 1)! ( + m + 1)! = ( + m + 1) ( + m) ( + 2) ( + 1)! (m + 1)! ( + 1)!. }{{}}{{}}{{} m+1 m 2 (3.3): exp x s (x) x +1 ( + 1)! m= x m (m + 1)! } {{ } exp x exp x e (x) exp x s (x) + s (x) e (x). (1) e (x) s (x) x 2 2 exp( x ). (3.2) (3.3) (1) s (x) e (x) = m=2 m 2 p m, = ( 1 m 1 ) pm 1,. 1 p m, = 1 s (x) e (x) ( 1 m 1 (1 p m 1, ) + m 1 1 p m, 1 p 2, }{{} =1/ m=2 ) p m 1, = x m m! (1 p m,). ( 1 m 1 ) (1 p m 1, ) + m 1 2 +... + (m 1) + = x m m! (1 p m,) 1 2 m=2 (m 1)m. 2 x m (m 2)! = x 2 2 (3.4): exp x = lim f (x) 1.1.2 m=2 x m 2 (m 2)! } {{ } =exp( x ). 35

( ) x m y m mr m 1 x y. s (x) s (y) m=1 x m y m m! (3.1) e (x) e (y) ( ) x y m=1 r m 1 (m 1)! } {{ } exp r (3.5): r = sup r <.. exp exp (3.4) exp r, f ( ) exp f ( ) f () + f () exp (3.4) exp r + f () exp (3.2),(3.3). (! 3.1.1 lim x ) k ( ) k!( k)! 1 x k = x k k! exp( x) (x C) x (, ) x/ k xk k! exp( x) 17 3.1.2 lim /(!) 1/ lim ( 1 + 1 ) = e. 3.1.3, e ( ) exp( ) 1, e ( ) exp( ) 1 (3.3). 3.1.4 ( ) e Q e = p/q (p, q N) (3.2) x = 1 p! qs (1)! qe/( + 1). 3.1.5 ( ) x [, ) (i) e (x) e +1 (x). (3.1). (ii) exp x e (x) s (x) e (x) (exp(x) x 1)/. (3.2) e (x) exp x s (x) 3.1.2 m 2 1 p m, (m 1)/. 3.1.6 ( ) x > l (x) = (x 1/ 1) (i) x > l (x) = x 1/ l (1/x). (ii) x > e l (x) = x, e (x) = (1 + x ). (iii) x 1 l (x) l +1 (x). (iv) < x 1 l (x) l +1 (x). (v) x > l(x) = lim l(x) 3.1.7 ( ) (i) X exp X = = 1! X ( 2.4.1 ) X exp X (ii) 3.1.2 3.1.3 ( ) exp : C C 17 Siméo Deis Poisso (1781 184) 36

() x, y C exp(x + y) = exp x exp y ( ). x R exp x >. (b) ( ) x, y R, x y exp x = exp x, exp x = exp(re x), exp x. (3.6) (x y) exp y exp x exp y (x y) exp x, ( x = y). x exp x R (, ) (c) ( ) ( ) R exp exp. exp( ) =, exp( ) = x exp x R (, ) (): z = x+y+(xy/) z x+y, e (x)e (y) = e (z ). exp x exp y (3.3) = lim e (x)e (y) = lim e (z ) (3.5) = exp(x + y). (3.6) 2.5.12(ii) exp x exp x = (exp(x/2)) 2 2.4.3 = exp(x/2) 2 2.4.3 = exp(x/2)exp(x/2) = exp(x/2) exp(x/2) = exp ((x + x)/2) = exp(re x). x C 1 = exp = exp( x) exp x. exp x. x R exp x 2 exp x = (exp x 2 )2 >. (b): r (1) r exp r 1 r exp r, r exp( r) 1 exp( r) r. ( r = ) r r /! = exp r 1 (3.4) y = r exp r. =1 exp( r) (exp(x y) 1) exp y = exp x exp y = (1 exp( (x y))) exp x (1) r = x y (c): R : (3.5) = : m N, m,. m (1) exp 1 + ( ). = : exp = 1/ exp( ) ( ). x exp x R (, ) 37

3.1.8 ( ) 2.5.9 z, w C exp(z + w) = exp z exp w = z /!, b = w /!, x = 1 3.1.9 ( ) ( 3.1.7) M d d d (i) X, Y M d, XY = Y X exp(x + Y ) = exp X exp Y. (ii) X M d exp X (exp X) 1 = exp( X). (iii) X, X M d, X X exp X exp X. 3.2 3.2.1 ( ) x C ch x, sh x ch x = exp(x) + exp( x), sh x = 2 exp(x) exp( x), 2 ch x, sh x (hyperbolic cosie), (hyperbolic sie) () ( ) ch x = exp x = ch x + sh x. (3.7) x 2 (2)!, sh x = x 2+1 (2 + 1)!. ch : R R [, ) (, ] sh : R R (b) ( ), C, ch ch, sh sh. ch (± ) =, sh (± ) = ± R, R, (): f(x) = exp x ch x, sh x (f(x) ± f( x))/2 2.5.12 (b): 3.1.3 3.2.1 x, y C ch (x + y) = ch xch y + sh xsh y, sh (x + y) = ch xsh y + ch ysh x. y = x ch 2 x sh 2 x = 1. 3.2.2 ( ) x C cos x, si x exp(ix) + exp( ix) cos x = ch (ix) =, 2 exp(ix) exp( ix) si x = sh (ix)/i = 2i cos x, si x (cosie), (sie) (3.7) exp ix = cos x + i si x ( ). 38

() ( ) cos x = (ix) 2 (2)! = ( 1) x 2, si x = 1 (2)! i (ix) 2+1 (2 + 1)! = ( 1) x 2+1. (2 + 1)! (b) ( ), C, cos cos, si si. 3.2.2 x, y C cos(x + y) = cos x cos y si x si y, si(x + y) = cos x si y+cos y si x. y = x cos 2 x + si 2 x = 1. 3.2.3 cos, si 3.2.4 cos x 1 k= cos kx = 1 cos x cos(+1)x 2 + 2(1 cos x), si x+si x si(+1)x k= si kx = 2(1 cos x). 3.2.5 r < 1, x R 1 r cos x 1 2r cos x+r 2 3.3 = = r cos x, r si x 1 2r cos x+r 2 = =1 r si x. 3.3.1 ( ) exp : R (, ) 3.1.3 log : (, ) R (logrithm) () 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)/y. (d) ( ) ( ), [, ] log log, log( ) =, log() = (): (b): ( 3.1.3) exp(log(xy)) = xy = exp(log x) exp(log y) = exp(log x + log y). exp : R (, ) (c): ( 3.1.3) (d): (c) 39

3.3.1 γ = k=1 1 k log γ (, ) γ (i) 1 1 +1 log(+1) log 1 (ii) 1 γ γ +1 1 1 +1 3.3.2 > 1,p = j= (1 + j) p = (1 + ) (i) () (c) () (b) (c) () <. (b) log(1 + ) <. (c) p x ( 1, 1) x 1+ x log(1 + x) x (b) (c). 3.3.3 ( ) 3.1.6 l(x) log x 1 x. (ii) 3.3.4 ( ) (E ) E = 1, ( 1) k E k k= (2 2k)!(2k)! = ( 1) (i) E (2)!r 2 ( 1), r > ch r = 2 18 (ii) z C, z < r 1/ch z = = ( 1) E z 2 /(2)!, 1/ cos z = = E z 2 /(2)!. 2.5.9. E E 1,.., E 5 1, 5, 61, 1385, 5521. 3.3.5 ( ) (b ) b = 1, b k k= (+1 k)!k! = ( 1) (i) b!r ( 1), r > e r = 1 + 2r 19 (ii) z C, < z < r z/(e z 1) = = b z /!. 2.5.9. (iii) b 2+1 = ( 1) =2 b z /! B = ( 1) 1 b 2 ( 1) ζ(2k) (k = 1, 2,.., 3.4.5 ) B 1,.., B 5 1/6, 1/3, 1/42, 1/3, 5/66. Jkob Beroulli (1759 89) (? 178) 3.4 3.4.1 ( ) [, ) x C x { x exp(x log ), > =, = () x, y C x+y = x y ( ). x = x, x = Re x. (b) ( ) < b x [, ) x (, ] (/b)x x 1 (b ) b x x (b/)xb x 1 (b ), (/b) x b x 1 (b ) x b x (b/) x x 1 (b ). (3.8) 18 r = log(2 + 3) = 1.3169... (ii) < z < π/2 19 r = 1.256... (ii) < z < 2π 4

x y (x y) y log x y (x y) x log. (3.9) 1 e e x = exp x 2 ( 5.4.1) (3.8) ( 5.4.5) 3.4.1 (): ( 3.1.3) (b): x ( 3.1.3, 3.3.1) ( 3.4.1) 3.4.1 (3.8),(3.9) 3.4.2, x, y R ( x ) y = xy 3.4.3 x < 1, x 1 lim x /! =, 3.4.4 ( ) (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. 3.4.2 =1 p p 1 p > 1 p 1 p 1 p = 1 ( 2.5.5) p > 1 p = 2, 3,.. ( 2.5.5) 3.4.5 ( ) s C, Re s > 1 =1 s 1 ζ- 2 3.4.6 ( ) =1 p (p > 1) ( 2.5.2) 3.4.7 ( ) (i) f : N\{} C, f (),(b) : () m, N\{} f(m) = f(m)f(). (b) p: r=1 f(pr ) <. =1 f() < =1 f() = p: (1 + r=1 f(pr )). (ii) =1 s = 1 p:, Re(s) > 1. ). 1 p s (1737 ) ζ- ζ- 19 ζ(s) 1/2 (1859 ) ζ(s) π : lim / log = 1 π 1896 21 3.4.8 ( ) p: p 1 =. ( 3.4.7) s 1 ( ) p: 1 + 1 p 1 =. 3.3.2 2 Berhrd Riem (1826 66) 21 Crl Friedrich Guss (1777 1855), Adrie-Mrie Legedre (1752 1833), Jcques Hdmrd (1865 1963), de L Vlée Poussi (1866 1962) 41

4 II 4.1 f(x) = x 2 x R x 2 2 ( (± ) 2 = ) f x f(x) l 4.1.1 d, k N\{}, A D R d, f : D R k f d = 2, k = 2 R d (d = 1 = ± ), l R k (k = 1 l = ± ) A ( ) f( ) l. f A l lim f(x) = l x x A x A, x f(x) l A = D f l x f(x) l lim f(x) = l x 4.1.2 (i) R lim exp x = exp ( 3.1.3), x exp( ) def. =, exp( ) def. =. (ii) [, ] lim log x = log ( x 3.3.1), log( ) def. =, log() def. =. 4.1.3 A D R d, R d (d = 1 = ± ) f i : D R k, x lim f i (x) = l i (i = 1, 2) : x A () ( ) l i R k f i,l i f 1 C\{} l 1 lim (f 1 + f 2 )(x) = l 1 + l 2 (4.1) x x A lim (f 1 f 2 )(x) = l 1 l 2 (4.2) x x A lim (1/f 1 )(x) = 1/l 1. (4.3) x x A k = 1, l i R {l 1, l 2 } = {, } (4.1) { l 1, l 2 } = {, } (4.2) (b) ( ) k = 1, l 1 = l 2 R f : D R A f 1 f f 2 x lim f(x) = l 1. x A 42

: A ( ) lim = f i ( ), f( ) 2.2.5, 2.2.4, 2.4.6 4.1.4 ( ) 4.1.1 g : f(d) R m, () x 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 x A : A ( ) lim = () f( ) l. (b) g f( ) l. 4.1.5 p, q, x > (i) [, ] lim x p = p, lim x p = p, x x p def. =, p def. =. (ii) lim x xp e qx =. (iii) lim x x p log x =, lim x p log x =. x (i) x ±p = exp(±p log x) ( 4.1.2), ( 4.1.4) (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). 4.1.1 < x x x, x 1/x, x xx, 4.1.2 f(x) = p j= jx j,g(x) = q j= b jx j ( p >, b q > ) lim x f(x) g(x). (ii) p R lim x {(x + 1) p x p }. 4.1.6 4.1.1 4.1.1 () x lim f(x) = l. (b) ε > δ > x A x A B d (, δ) = f(x) B k (l, ε). {x R d ; x < δ} ( R d ), B d (, δ) = (1/δ, ] (d = 1, = ), [, 1/δ) (d = 1, = ). (4.4) B k (l, ε) B d (, δ) 43

: (b) δ-ε : () = (b): (b) ε >, δ >, A δ def. = {x A B d (, δ) ; f(x) B k (l, ε)} =. A 1/ B d (, 1/). f( ) B k (l, ε) f( ) l. () () = (b): A, ε > (b) δ > B d (, δ). (b) f( ) B k (l, ε). f( ) l ( 2.4.5) 4.1.7 4.1.6 lim x f(x) = l R d δ > x A sup{ f(x) ; x A B d (, δ)} <. 4.1.6 (b) ε = 1 δ {f(x) ; x A B d (, δ)} B k (l, 1). 4.2 I R 2.3.3 4.2.1 ( ) d, k N\{}, D R d, f D R k f d = 2, k = 2 D f (cotiuous) (4.5) lim f(x) = f(). (4.5) x x D D ( ) f( ) f(). ( 4.1.1 ) f D f D 22 D D R k - D C(D), C(D R k ) 4.2.2 {exp, ch, sh, cos, si} C(C C) (3.5), 3.2.1, 3.2.2. 22 D 44

4.2.3 4.2.1 f L [, ), α (, 1] f C(D). x, y D f(x) f(y) L x y α. f ( α = 1 23 x, D, x x. x α ( 4.1.2). f(x) f(). 4.2.1 4.2.1 ():f C(D R k ). (b): D f C(D B d (, δ) R k ) δ > 4.2.2 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 ) 4.2.3 f : R d R, α R t >, x R d f(tx) = t α f(x) f α α f (i) α > sup x =1 f(x) < f (ii) α = f α < f lim x f(x) ( f x = ) 4.2.4 1,.., d N, r > f : R d R f(x) = x 1 1 x d d / x r (x ), f() = f 1 +... + d > r 1 +... + d r (x = ) 4.2.5 ( ) x Q, x R\Q f(x) = 1, f : R R 4.2.4 D R d () f 1, f 2 D f 1 + f 2 D f 1, f 2 f 1 f 2, f 1 /f 2 D f 1 /f 2 f 2 () (b) ( ) D R d, f : D R k, g : f(d) R m f g f() g f : () 4.1.3, (b) 4.1.4 23 Otto Hölder (1859 1937), Rudoluf Otto Sigismud Lipschitz (1832 193) 45

4.2.5 ( ) 1,.., d N, c C m : C d C (moomil) m(x) = cx 1 1 x d d. (polyomil) p 1, p 2 D = {x C d ; p 2 (x) } r(x) = p 1 (x)/p 2 (x) (rtiol fuctio) r C(D C). : x = (x i ) d i=1 C x x i 4.2.4() 4.2.6 ( ) r >, C ( N), x D def. = {x C ; x < r} : f(x) = x = () < s < r, p N p s <. (b) f C(D C). = (): s < t < r t lim p (s/t) = 2.5.8. p (s/t) 1, p s t. ( 2.5.4) (b): D < s < r s x < s x < s. ( ) x 2.4.4 s 1 x. f(x) f() = (x ) = = x ( ) x s 1. = } {{ } () f 4.2.6 x C\{}, x 4.2.6 exp, si, cos (e x 1)/x, si x/x, (1 cos x)/x 2. 4.3 < b, f : (, b) R k x c (, b) f(x) x c 4.3.1 < b, f : (, b) R k 46

< c b lim x c f(x) x<c c (left limit) f(c ) c = b f(c ) x c lim f(x) x c c c c (right limit) f(c+) c = f(c+) x c lim f(x) x c < c b f(c ) = f(c) f c (left cotiuous) c = b c c < b f(c+) = f(c) f c (right cotiuous) c = c A (, b) f c A f A (, b) 4.3.2 ( ) 4.3.1 k = 1 sup f(x), f, () < c b f(c ) = <x<c if f(x), f. <x<c if f, (b) c < b f(c+) = f(x), c<x<b sup f(x), c<x<b f. 2.3.2 4.3.3 c R 1 (c, ) c 1 [c, ) c 4.3.1 g (x) ( N,x [, 1)) x x f(x) = = g (x) f(1 ) = = g (1 ) ( ) 4.3.4 ( ) 4.3.1 c (, b), l R k lim f(x) = l x c x c f(c+) = f(c ) = l, f(c+) = l, f(c ) = l, k = 1, l = ± ( < c < b ) (c = ) (c = b ) 47

: = : =: c =, b 4.3.1 c, x (, b),x c, x c lim f(x ) = l K 1, K 2 N K 1 = { N ; x < c}, K 2 = { N ; x > c} K i = {k i () < k i (1) <...} K 1 lim f(x k1 ()) = f(c ) = l, K 2 lim f(x k2 ()) = f(c+) = l. I( 2.1.4) lim f(x ) = l. 4.3.5 ( ) 4.3.1 c (, b) f(c+) = f(c ) = f(c), ( < c < b ) f c f(c+) = f(c), (c = ) f(c ) = f(c), (c = b ) : ( 4.3.4) l = f(c) 48

5 I 3 f(t) = t(cos t, si t), t ) 17 24 y = f(x) (y )/( x ) x 25 (y )/( x ) dy/dx 26 y 5.1 5.1.1 ( ) I R f : I R k (f : I C k = 2 ) x I f (x) x ( ) (differetil coefficiet) f (x) = y x lim D x,y (f), D x,y (f) = y x,y I f(y) f(x). (5.1) 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 (differetible) f x I x f (x) f (derivtive) ( ) 5.1.1 f (x) (x, f(x)) x f (x) f(x) x 5.1.2 ( ) (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 1.1.2 = p 1 x k y p 1 k px p 1. (ii):x, y (, ), y x log x log y ( 3.3.1) 1 x 1 y log x log y x y k= y x ( ) 1 x, ( ) 1 x. 24 Reé Descrtes (1596 165), Pierre de Fermt (161 1665) 25 Issc Newto (1642 1727) 26 Gottfried Leibiz(1646 1716) 1 x 1 y. 49

5.1.3 ( ) () : f(x) = x x = y >, y < D,y (f) = 1, 1. f () (b) : f(x) = x p, D,y (f) = yp y = yp 1, (y ). 5.1.4 ( ) 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). 5.1.5 I R f, g : I R k x I () ( ) 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 : (),(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 5.1.4 y x g(y) 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 5.1.1 I R f, g : I R k x I f g x I (f g) (x) = f (x) g(x) + f(x) g (x), 5.1.2 I R f i : I C (i = 1,.., ) x I (i) f = f 1 f x I x f = i=1 f 1 f i 1 f i f i+1 f. (ii) p : C C f = p(f 1,..., f ) f x I f q : C 2 C f = q(f 1,..., f, f 1,..., f ) 5

5.1.6 ( ) 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 5.1.7 ( ) r (, ], C ( N), z D def. = {z C ; z < r} : f(z) = z = z D (1) f (z) def. f(w) f(z) = w z lim = z 1 ( ). w z w z =1 z = x + iy (x, y R ) z D f(z) x, y (2) x f(z) = (1/i) y f(z) = f (z). z D z < s < r s w z w < s 2.4.4 ( ) w z z 1 (z w) 1 2 ( 1)(z w)2 s 2. w z f(w) f(z) z 1 ( w z = w z w z =1 =1 z 1 ) ( ) w z ( 1) s 2. =1 } {{ } 4.2.6 (1) (1) z = x + iy y x f x (z) = f (z) x y 1 f i y (z) = f (z) 5.1.8 x R (i) c C (e cx ) = ce cx. (ii) (, ) ( x ) = x log. (iii) (ch x) = sh x, (sh x) = ch x, (cos x) = si x, (si x) = cos x. : (i): e cx = = c x! 5.1.7 (e cx ) = =1 c x 1 ( 1)! = c 51 = c x! } {{ } =e cx.

(ii): (i) c = log (iii): ch, sh, cos, si exp 3.2.1, 3.2.2 (i), 5.1.5 5.1.8 5.1.7 5.1.3 (, ),x R ( x ) = x log ( 5.1.8). 3.4.1 x y 5.1.4 ( ) X exp(x) 3.1.7 (i) t R d dt exp(tx) = X exp(tx). (ii)x = Y t R exp(tx) = exp(ty ). 5.1.9 ( ) I R f : I R k, l R k (k = 1, l = ± ), x I (),(b) () f x f (x) = l (b) ϕ : I R k y I f(y) f(x) = l(y x) + ϕ(y). (5.2) lim y x ϕ(y)/ y x =. (5.3) y x,y I : () = (b): ϕ : I R (5.2) ϕ(y) = f(y) f(x) l(y x) ϕ () (5.3) (b) () = (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 () 5.1.1 ( ) J R x J, J x f g(x) f g x (f g) (x) = f (g(x)) g (x). g f I R k g : 5.1.9 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 52

(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) 5.1.9 lim ϕ(y)/(y x) =. y x y x,y J (3) x J\{x}, x x (4) lim ϕ 2 (g(x ))/ x x =. g(x ) = g(x) ϕ 2 (g(x )) = ϕ 2 (g(x)) =. (4) g(x ) g(x) g x lim g(x ) = g(x). ϕ 2 (g(x )) x x = ϕ 2(g(x )) g(x ) g(x) } {{ } (2) g(x ) g(x) x x } {{ } (1). (4) 5.1.11 ( ) 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) 5.1.8 = ce c log x 1 x = cxc 1. 5.1.5 g : I (, ) x I log g x (log g) (x) = g (x)/g(x) 5.1.6 f j : I R (j = 1, 2) (i) I = R, f 1 (x) = si m x, f 2 (x) = si m x (m, N\{}). (ii) I = (, ), f 1 (x) = x x, f 2 (x) = x xx. 5.1.7 I R f : I R k x I f(x) ( 1 f(x) ), (f(x) 1 f(x) ) f(x) f (x) 5.2 5.2.1 ( ) 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 5.1.1 ) (f ) f f (2) 53

(secod derivtive) f m (m-th derivtive) 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), ( ) d m (x). (5.4) dx m N D m (I) I m C m (I) f D m (I) f (),..., f (m) C(I) f 27 D (I) f : I R k C (I) = C(I) ( 4.2.1 ) 5.1.4 D m+1 (I) C m (I) D m (I). f D m 1 (I) x I (f (m 1) ) (x) f (m) (x), (5.4) 5.2.2 f 1 f C (R). 5.1.2 5.1.5 f D 1 (R) f 1 f C (R). 5.2.1 f : I C f C (I) : (i) I = R, f(x) = x, ch x, sh x, cos x, si x ( > ). (ii) I = (, ), f(x) = log x, x c (c C). 5.2.3 f, g D m (I) () ( ) 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), ( ). (5.5) (c) ( ) f, g : I C x I g(x) f/g D m (I). (d) f, g C m (I) (),(b),(c) f + g C m (I), fg C m (I), f/g C m (I). : (): ( 5.1.5) (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) 5.1.5 (1) 27 C m (I) D m (I) 54

(2) (fg) (m) = m 1 r= ( m 1 ) (3) ( m 1 r 1 r + ( ) ( m 1 r = m ) r 2.4.5 (2) = f () g (m) + ) (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 = (5.5). (c): A m, B m A m B m m 5.1.5 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) 5.1.5 = f g fg g 2, f g fg, g 2 (b) D m 1 (I). A m 1 B m 1 f g fg, g 2 (f/g) D m 1 (I). f/g D m (I) (d): (),(b),(c) 5.2.2 ( ) 2 q (x) = (x 2 1) m q (m) (i) (x 2 1)q (1). P (x) = 1 (x) = 2xq (x). (ii) (x 2 1)q (+2) 2! q() (x)+2xq (+1) (x) (+1)q () (x) = (x) (ii) (1 x 2 )P (x) 2xP (x) + ( + 1)P (x) = 5.2.3 H (x) = ( 1) exp(x 2 /2) d dx exp( x 2 /2), x R, =, 1, 2... (i) H (x) ( 28 ) x 1. (ii)h (x) = xh (x) H +1 (x) = H 1 (x) 2 (iii) H (x) xh (x) = H (x). 5.2.4 ( ) H (x) ( 5.2.3) H 1 (x) = x = 1 H (x) = j=1 (x c j), (c 1 <... < c ) (i) (iii) (i) ( 1) j H (c j ) >, j = 1, 2,..,. (ii) H +1 (c j ) j j 5.2.3 H (x) = xh (x) H +1 (x). (iii)h +1 (x) + 1 5.2.4 ( ) I, J R J g f I R k () 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 ). : ():m m = m 1 m 1 5.1.1 28 Chrles Hermite (1822 191) 55

(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): () m = 4.2.4(b) 5.2.5 ( ) r (, ], C ( N), z D def. = {z C ; z < r} : f(z) = z = f(z) z D m f (m) (z) f (m) (z) = ( 1) ( m + 1) z m ( ). (5.6) =m f (m) () = m! m. (5.7) 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 5.1.7 f(z) z z D f (z) = z 1, =1 ( ). f (z) z D ( 4.2.6) f(z) (z D) (5.6) (5.6) z = (5.7) y x x y 5.2.5 r >,, b C ( N), z D def. = {z C ; z < r} : ( x f(z) = = z, g(z) = = b z ) m ( f() = m x) g() ( m N) D f = g 5.2.6, b, c, z C, c N, z < 1 F (z) = = ( + 1) ( + 1)b(b + 1) (b + 1) z!c(c + 1) (c + 1) z(1 z)f + (c ( + b + 1)z)F bf = F, F z F (z) 56

5.3 f (c, f(c)) x < c x > c f(x) (c, f(c)) 5.3.1 ( ) < b <, (, b) I [, b] R, f : I R k (f : I C k = 2 ) x I\{} f (x) x (left differetil coefficiet) f (x) = y x lim D x,y (f), D x,y (f) = y<x,y I f(y) f(x). y x k = 1 f (x) = ± f (x) (f(x)), ( ) d f(x). dx f (x) R k f x (left differetible) ( k = 1 f (x) = ± ) x I\{b} f +(x) x (right differetil coefficiet) f +(x) = y x lim D x,y (f) y>x,y I k = 1 f +(x) = ± f +(x) (f(x)) +, ( ) d f(x). dx + f +(x) R k f x (right differetible) ( k = 1 f +(x) = ± ) 5.3.2 ( ) 5.3.1 c (, b), l R k (),(b) () f ±(c) = l (b) f (c) = l. : x (, b)\{c} F (x) = f(x) f(c) x c F c f (c), f +(c), f (c). ( 4.3.4) f ±(c) f (c) f(x) = x f ±() = ±1 f x = ( 5.1.3) 5.3.1 5.3.1 c I (i) f c( ) f c (ii) f c( b) f c 57

( 5.3.2) { exp( 1/x), x >, 5.3.3 f(x) = f C (R). x. ():m N (1) p m x > f (m) (x) = p m (1/x)f(x). (1) 5.3.2 (2) f D m (R), f (m) () =. m (2) (2) m m = m 1 f (m 1) () =. x < f (m 1) (x) = (f (m 1) ) f (m 1) (x) () = lim =. x x x< (f (m 1) ) f (m 1) (x) (2) +() = lim x x x> = lim (1/x)p m 1 (1/x) exp( 1/x) x x> = lim y yp m 1(y) exp( y) =. 5.3.2 f (m) () =. (2) 5.3.2 5.3.3 (1) 5.3.3 5.3.3 f(x) x ( r, r) (r > ) 5.2.5 5.3.4 < r < R < g C (R d [, 1]) g(x) = 1 x r g(x) = x R. 5.3.5 p N\{}, x, x < f p (x) = x p, f p C p 1 (R), f p D p (R) 5.4 58

5.4.1 ( ) < c, f(c) f, f(c) = mx f, B(c,ε) D B(c, ε) = {x Rd ; x c < ε}. f(c) = mi B(c,ε) D f ε > c, f(c) f,, f, f 5.4.2 c f f(c) = mx D f c f c f f(c) = mi D f c f 5.4.3 < b, f : (, b) R, c (, b) f f c f (c) =. : c f x < c < y f(x) f(c) x c f (c) =. c f f(y) f(c) y c. x c, y c 5.4.1 ( ) < b, f : (, b) R, c (, b) f f c f +(c)f (c). 5.4.1 5.4.4 ( I) < < b <, f C([, b] R) mi [,b] f, mx [,b] f 5.4.4 5.5 ( 5.5.5) 5.4.1 : l = (f(b) f())/(b ), F (x) = f(x) lx c (, b), F (c) = F C([, b]) I 5.4.4 c 1, c 2 [, b], F (c 1 ) = mx [,b] F, F (c 2 ) = mi [,b] F. 29 Michel Rolle (1652 1719) 59

F (c 1 ) = F (c 2 ) F c (, b) F (c) = F (c 1 ) > F (c 2 ) F (b) F () = f(b) f() l(b ) =, {c 1, c 2 } {, b}. c {c 1, c 2 }\{, b} c (, b). c F F F (c) = ( : 5.4.3). 5.4.5 () < < b <, f D 1 ([, b]), f [, b] f ()(b ) f(b) f() f (b)(b ). 3.1.3, 3.3.1 (b) < ) f (y) = xy x 1. c (, b) b x x = x c x 1 (b ). x [1, ), x (, 1] y y x 1 5.4.2 c, c 1,.., c R c + c 1 2 +.. + c 1 + c +1 = c + c 1 x +... + c x = (, 1) 5.4.1 5.4.3 f D 1 ((, )), lim f (x) = c R lim (f(x + 1) f(x)) = c x x 5.4.4 ( ) 2 q (x) = (x 2 1) m q (m) (i) m 1 q (m) ( 1) = q (m) (1) =. (ii) 1 m q (m) ( 1, 1) m P 5.2.2 ( 1, 1) 5.4.5 ( ) < . (b): g(b) g(). c (, b) ( ) : g (c) f(b) f() g(b) g() = f (c) g (c). 5.4.6 ( ) < b, c [, b], f, g : (, b) R, lim x c f(x) = lim x c g(x) =. x c g(x) lim x c f (x)/g (x) x c f(x)/g(x) 3 lim x c x c 3 Guillume Frçois Atoie de L Hôpitl (1661 174) 6

5.4.6 ( ) < b, (, b) I [, b] R, f : I R, f C(I) D 1 ((, b)) () f I (, b) f. (b) f I (, b) f. (c) f I (, b) f =. ( ) < s < t lim x b f(x). x<b (1) f I ( ) f (, b) ( ). f D 1 ((, b)) f C([s, t]) D 1 ((s, t)) 5.4.1 (2) < s < t . (2) c (s, t) f (c) > ( ) (2) = (1) f (, b) < s < t h t (x) = ct d/2 exp c > ( x 2 2t x R d \{} 61

5.4.8 I R ε > I f, g x, y I f(x) f(y) g(x) x y 1+ε f 5.4.9 I R 5.4.6 u, F, G C(I) D 1 ((, b)) (),(b) () (, b) u = F u + G e F. (b) I u = e F (G + c) (c ). 5.4.1 ( ) u C (R) P (x) = x + 1 j= jx j = r j=1 (x b k) m k ( j, b k C, m k N\{}, b 1,.., b k, r k=1 m k = ) (),(b) () P (D)u def. = u () + 1 j= ju (j) =. (b) u x j e b kx (1 k r, j m k 1) 5.4.11 ( ), b, c,... R 3 b R 3 b = ( 2 b 3 b 2 3, 3 b 1 b 3 1, 1 b 2 b 1 2, ). (i) b = b, =. (ii) (b c) = ( c)b ( b)c ( ). (iii) ( b) c = (b c) = b (c ), ( b) = ( b) b =. (iv) ( b) (c d) = ( c)(b d) ( d)(b c), ( b) 2 = b ( b) 2. (iv) f, g D 1 (I R 3 ) (f g) = f g + f g. 5.4.12 ( ) x : t x(t) C 2 ([, ) R 3 ) x = x = k x 3 x (k ) (i) = x x x = x =. (ii) b = 1 k x 1 x x b =. (iii) 2 = k(x b+ x ). (iv) c = b, x b, c X, Y k 2 (1 b 2 )X 2 + k 2 Y 2 2 2 b X = 4. 5.4.12 x x 2 k > k < /2 x (ii) (iv) k > b < 1, b = 1, b > 1 k < (ii) x b x b b < 1 k > 2 k > k < 31 5.4.12 2 2 5.4.7 c R\{}, x > ( c) x ( 1 + c x) x f(x) = x log ( 1 + c x) ( 1 + c x) x = e f(x). f 31 Johes Kepler (1571 163), Tycho Brche (1546 161) 62

( ( ) x > ( c) log 1 + c ) > c x x + c. ( ) c > c < ( 3.3.1) 5.4.13 ( f (x) = log 1 + c ) + x 1 ( x 1 + c c ) ( ) ( x x 2 = log 1 + c ) c x x + c > 5.4.6 f 5.4.13 5.4.7 ( ) ( 5.4.14 c R lim 1 + c x = e x x) c 5.4.8 (log(1 + x) ) x ( 1, 1] log(1 + x) = ( 1) 1 x. x = 1 x ( 1, 1) x ( 1, 1) f(x) 5.2.5 f C (( 1, 1)) f (x) = =1 ( 1) 1 x 1 = 1 1 + x = (log(1 + x)). =1 5.4.6 f(x) log(1 + x) = c ( ) c = f() log(1 + ) =. 5.5 5.5.4 5.5.4 ( 5.5.5) 5.5.1 ( ) A R d A A A (closed) 5.5.2 ( ) I 1,..., I d R I = I 1 I d R d. R d (itervl) I 1,..., I d I 5.5.1 5.5.2 5.5.2 ( ) A R d A x R d A (dheret poit) A A A (closure) A, B R d (i) A (ii)a B A B. (iii)a B = A B, A B A B. A B A B (iv) A, B A B, A B 63

5.5.3 ( ) f : R d R k () (c) () f C(R d R k ). (b) F R k f 1 (F ) R d (c) G R k f 1 (G) R d 5.5.3 ( ) ( ) R d l() < l(1) <... ( l() ) ( ) (subsequece) 5.5.4 ( ) A R d (), (b) A () A (b) () A ( A ) (b) A A ( A ) 5.5.4 R d 5.5.4 (): A l A [ l, l] d. A (x k ) k d = 1 : [, b ] [ l, l], =, 1,... [, b ] = [ l, l]. c = ( + b )/2 { [, c ], x k [, c ] k, [ +1, b +1 ] = [c, b ], x k [, c ] k. x k [, b ] k k(), b, b = + 2 +1 l. 2.3.2 ( ), (b ) (x k() ) d 2 : d = 1 (x k ) k (x k,1 ) k d = 1 (x k ) (x k1 ()) 1 (x k1 (),1) (x k1 ()) 2 (x k1 (),2) d = 1 (x k1 ()) (x k2 ()) 2 (x k2 (),2) (x k2 ()) (x k2 (),1) (x k1 (),1) (x k2 ()) 1 2 (x kd ()) () : A < x x A 64

(x ) (x k() ) k() x k() (b): A () A A A (b) : (b) () (b) : x A lim x = x (b) (x ) lim x l() A (x l() ) (x l() ) (x ) lim x l() = x x A. A 1817 5.5.4 187 32 5.5.4 ( ) 5.5.4 5.5.5 ( ) A R d ε > x i (i = 1,..., N) A N i=1 B(x i, ε) A ( B(x, ε) = {y R d ; y x < ε}.) 5.5.6 f C(R d R k ) (i) F R k f 1 (F ) R d (ii) K R d f(k) R k 5.5.7 ( ) (i) K R f C(R R), f 1 (K) (ii) F R f C(R R), f(f ) 5.5.8 r R < (i) A r,r = {x R d ; r x R} A,R R A R,R (ii) {x R d ; r < x R} 5.5.9 K 1, K 2 R d K 1 + K 2 = {x + y ; x K 1, y K 2 } (i) K 1, K 2 K 1 + K 2 = {x + y ; x K 1, y K 2 } (ii) ( ) K 1, K 2 R d K 1 + K 2 5.5.1 I 5.4.4 5.5.5 ( II) K R d f C(K R) mi K f, mx K f : mx K f mi K f K, =, 1, 2,... f( ) sup K f ( 2.2.3 A = f(k)) ( ) ( l() ) l() K sup f = lim K f( l() ) f = f() f(k). 32 Krl Weierstrss (1815 97). Dugc, P: Elémet d lyse de Krl Weierstrss, Archive for History of Exct Scieces 1, 1973, 41 176. 65

1.3.3 mx K f = f(). 1874 I 5.4.4 5.5.4 5.4.4 5.5.5 5.5.1 T >, f : R R x R f(x + T ) = f(x) f 5.5.11 f C(R d R), lim x f(x) = f 5.5.12 q : R d [, ) 4.2.2 c 1, c 2 (, ) x R d c 1 x q(x) c 2 x 5.5.8 S = {x R d ; x = 1} 66

6 II 3 6.1 π = 3.14159... ( )/( ) π 33 π 1 π 34 6.1.1 ( ) () cos π 2 = π (, 2 3) π (b) z C, m, Z ( ( cos z + π 2 ) (, si z + π 2 )) = (cos z, si z), = 4m, ( si z, cos z), = 4m + 1, (cos z, si z), = 4m + 2, (si z, cos z), = 4m + 3. (6.1) cos, si 2π cos(z + 2π) = cos z, si(z + 2π) = si z. (c) cos, si [, 2π] x π/2 π 3π/2 2π cos x 1 1 1 si x 1 1 x R (cos x, si x) = (1, ) x 2πZ. (): (1) (, 6) si >. 2 + 2+1 = si x = = ( 1) (2 + 1)! x2+1 } {{ } 2.5.9 = ( 2 + 2+1 ), = x4+1 (4 + 1)! x4+3 (4 + 3)! = x4+1 (4 + 1)! ( 1 x 2 ). (4 + 2)(4 + 3) 33 Willim Joes (1675 1749) 34 1761 J. H. Lmbert, 1882 C. L. F. Lidem 67

x (, 6), N x 2 (4 + 2)(4 + 3) < 6 2 3 = 1, 2 + 2+1 >. x (, 6) si x >. (2) cos [, 6] (, 6) (cos) = si <. ( 5.4.6) (2) (3) cos = 1, cos 3 <. cos = 1 cos cos 2+1 + 2+2 = x 12 ( 1) 1 x 2 1 cos x = (2)! =1 }{{} 2.5.9 = ( 2+1 + 2+2 ) = x4+2 (4 + 2)! x4+4 (4 + 4)! = x4+2 (4 + 2)! ( 1 x 2 ) (4 + 3)(4 + 4) x 2 (4 + 3)(4 + 4) 12 3 4 = 1, 2+1 + 2+2. x 12 1 cos x 1 + 2 = x2 2 ) (1 x2 12 1 cos 3 3 2 ( 1 1 ) = 9 4 8, cos 3 1/8 <. () (3), cos 2.3.4 c (, 3), cos c =. (2) c 2c (b): cos π 2 = cos2 π 2 + si2 π 2 = 1. < π/2 < 3 < 6 (1) si π/2 >. ( cos π 2, si π 2 ) = (, 1). ( 3.2.2) ( cos z + π ) = cos z cos π 2 }{{ 2} = si z si π }{{ 2} =1 (, si z + π ) = si z cos π + cos z si π. 2 }{{ 2}}{{ 2} = =1 = 1 (6.1) z z π 2 = 1 (6.1) = ±1 (6.1) Z (6.1) (c): (6.1) [, π/2] [ mπ 2, (m+1)π 2 ] (m = 1, 2, 3) [, π/2] cos (2) (, π/2) si = cos >. ( 5.4.6) si [, π/2] 3.2.2 (= ) 68

6.1.2 ( ) () t, s R e it = e is t s 2πZ. (b) c R t e it [c, c + 2π) S 1 def. = {z C ; z = 1} (): e it = e is e i(t s) = 1 6.1.1 t s 2πZ. (b): ϕ(t) = e it (t R) e ic ϕ [, 2π) S 1 z e ic z S 1 S 1 c = c = () 6.1.1 ϕ() = ϕ(2π) (1) ϕ : [, 2π] S 1 z S 1, c [, 2π], z = e ic. z S 1 Re z [ 1, 1]. cos = cos 2π = 1, cos π = 1 ( 2.3.4) c + [, π], c [π, 2π], cos c + = cos c = Re z. 6.1.1 si c + si c. Im z Im z = 1 (Re z) 2 = 1 cos 2 c + = si c +. z = Re z + i Im z = cos c + + i si c + = e ic +. Im z z = e ic. (1) R (, ) 3.1.3 6.1.2 R [c, c + 2π) (c R ) C\{} 6.1.3 () z, w C e z = e w z w 2πiZ. (b) c R z e z {z C ; Im z [c, c + 2π)} C\{} () e z = e w e z w = 1. z w z w = = : e z = 1 ere z 3.4.1 = e z = 1, Re z = 3.1.3 x e x R ). Re z =, e z = 1 e i Im z = 1. 6.1.2() Im z 2πZ. z = Re }{{} z +i Im z 2iπZ. = = 6.1.2() (b): () w C\{} w/ w S 1 6.1.2 t [c, c + 2π), w/ w = e it. w = w e it = e log w +it. 69