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2 (, ),, :, ( 5% ),,, (CIP) :,4 8 ISBN O7 CIP (4) http: www hep edu c http: www hep com c ,
3 ( ),,, :,, ; ; ; 58 ( ),,,, ( ),,, (A, B ),,,, ; ; ; ;,,,,,,
4 ,, 4
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8 p q ( p, q, q ),,,, =, =, = +,,,, 3, = y =, > y :, > y :,, +, y,, y + y, - y, - y - y >,, > y,
9 3,,,, 999 9? = = 3 3 = 999 9, = , p ( p, q, q ) q? p (, ), p < q q : r, p = q + r, 9, r q -, ( : p, q, (i) p < q, =, r = p; ( ii) p = q, =, r = ; (iii) p > q,,, < 9, ( + ) q > p, q p, r = p - q ) p q = + r q, r q < r = q + r, 9, r q -, r q = + r q, r q <, p = q + r, r = q + r, p q = + + r q r - = q + r, 9, r q -, ( )
10 p q = r q,r q <, p q = r {,,,, q - }, ( ), p q,, : ( ) f ( ) = + f ( ) = + = - +, >, < <, f ( ) < f ( ) + +, = ,, ( + ) = C k k - k, ( 3) k = =, ( 3 ) = m, = m +, ( + ) m + m ( + ) m = C k m k m - k, k = = ( + ) ( + ) m = ( + ) C k m k m - k m k = m m = C k m k m - k + + k = k = C k m k + m - k 3
11 m m - = m + + C k m k m - k + + k = k = m m = m + + C k m k m - k + + k = k = C k m k + m - k + m + C k - m = m + + ( C k m + C k - m ) k m + - k + m + k = m = m + + C k m + k m + - k + m + k = m + = C k m + k m + - k, k = m k m - k + + m + C k m + C k - m = C k m +, 3 5, i 3,,,, i ( i =,,, ), > -, ( + ) ( + ) ( + ) ( 4) = = = =, > -, (Beroulli) ( + ) + ( > - ) N + ( 5) = ( 5) = m, ( + ) ( + )( + m ) m, i i > - ( i =,,, m ) = m +, + m + >, = m, ( + ) ( + ) ( + m ) ( + m + ) ( m ) ( + m + ) = m + m + + ( m ) m m +, m + i ( i =,,, m ) ( 5) = = = =, > -, ( + ) + ( > - ) 4 ( Cuchy ) :,,,,,,,, i = i i i i = i ( 6) i = 4
12 t, t i = 5 p,, mn + ( i + t i ) = i i = i i i = i = + t i i i = - 4 i i = i = i i i : i = + t i, i = i, i i = ( ) m - m = ( - ) ( m - + m - + m m - ) ( 7) ( ) p < ( + ) p + - p + p + (3 ) : < ( + ) p ( 8) - + k p < p k = p + < k = ( ) ( - ) ( m - + m - + m m - ) = m + m - + m m - - m - - m m - - m = m - m ( ) ( 7) = +, =, m = p +, k p ( 9) ( + ) p + - p + = ( + ) p + ( + ) p p, ( p + ) p < ( + ) p + - p + < ( p + ) ( + ) p, p < ( + ) p + - p + p + (3 ) =, p, p < p + p + < p + p ( 9 ) = m, m - + m k p < mp k = p + < k = = m +, ( 8), ( ) ( m + ) p + p + < ( + ) p k p, = ( m + ) p + - m p + p + + mp + p + ( ) 5
13 ( m + ) p + p + < ( m + ) p + k p m + = k p k = m k = = ( m + ) p + - m p + p + + mp + p + m p m - + k p k = m = k p k = ( 9 ) = m + ( 4 ) 6 cr + (R + ) : c - c?, c = + c - c c + c c - c - c 8 p : p, p p, p = u v, u, v,, v, p = u v, p, v, u v, u v ; u = pv, v u, v =, p 6
14 , ( -, ], ( -, ), [, + ), (, + ), ( -, + ) - +,, +, -, S, M S ( ) : M S ( ) S, > M ( < M ) S (), M S () ( ) 3, ( ) : S = (, ), ( ), S S S, S,, () (), : S R, () () (i) S,, S ; (ii) <, S, >, S (i) S,, S ; (ii) >, S, > -, S ( ) - ( ) <, ( ) 4 : S, S, S ; S, S 7
15 k, k + k, k =,,, k S, k + S, = k k S,, S S? S?,,? S ( ), S ( ) (), ( ) S ( ) S () S = ( - ) - =,,, sup S =, if S = - sup S =, = k,, ( - ) k - (i), ( - ) - ; (ii) >, k k >, - < - k, sup S = k if S = - sup S, if S S S sup SS, sup S ; S if SS, if S, () ()? : S R, L, S, < L, S : S R, M >, S, > M, S, S = { - ( - ) =,, } L, = k > L, - ( - ) = - k < L 8
16 , M > M > ; S ; M > M, M, 3 S? S, S, S, S : (i) S, > ; (ii) <, S, S = = k, k + k = + ( - ) N + = k +, k ( k + ) = k +, k = k k k + >, k, + = 5 S, sup S = 5, if S =, 5S, sup S = (i), (ii) = + ( - ) ; k + k if S =,, - = ( - ) ( ) k + k + - = >, k N + k > log k + + k + k + k + k k +, -, - <, k + k + + < +, k + 9
17 if S = S = + si N +, sup S, if S 3 = 6 k +, 6 k +,, 6 k + 5( k =,, ), = 6 k + ( k =,,3, ) S S = + (6 k + ) 3 kn + ; = 6 k + 5( k =,, 3, ) S = - (6 k + 5 ) 3 kn+ S, S, sup S = +, if S = - 3 S = { y y = +, }, if S, sup S S, sup S = + S,, y = +, S sup S = + M >, (M >, > M - ), + > M, S, sup S = + if S = : (i) (ii) + < + ys, y = + ( ) ; >,,, - < <, if S = 4, A R, : sup( + A ) = + sup A, if( + A ) = + if A, + A = { + A} sup( + A ) = + sup A sup A, : (i) A, sup A ; (ii) >, A, > sup A - : (i) A, + + sup A ; (ii) >, A, + > + sup A - sup( + A) = + sup A
18 if( + A) = + if A 5 A, B, AB = { y A, yb}, : sup AB = sup Asup B sup ABsup A sup B, A, sup A, yb, ysup B,, y, ysup A sup B, sup Asup B AB, y sup A sup Bsup AB sup ABsup Asup B, > ( < ), A, > sup A -, y B, > sup B -, y AB, y > ( sup A - ) ( sup B - ) sup AB y > ( sup A - ) ( sup B - ), = sup Asup B - ( sup A + sup B)+ > sup Asup B - ( sup A + sup B + ) A, B, sup A, sup B, sup A + sup B + >, = (sup A + sup B + ) sup ABsup A sup B sup AB = sup Asup B ( 9 ) 8 >,,, : = sup{ r r, r < }, >, if{ r r, r < }, < >, : (i) (ii) r <, r, r ; <, r, r <, < r < r,, (i) ( ii), < <, log <,, r, log < r <, < r < < <
19 3 : y,, y, y, y = f( ) : ( ) D M,, ( ) D, ym,, ( 3) ( ) f, y,,, R D( ) = [, ] R ( ) =,,, q, = p q ( p, qn +, p, q ),, =, (, ), 3 : y = f( u ), ud u = g( ), E, ( fg) ( ) E = { g( )D}E, y = f( ), D, f D f ( ), = f - ( y), yf ( D) f - f - ( f ( D) ) = D,, 4 ( >, ),,
20 = sup { r < r r }, >, if { r < r r }, < <, f ( ) =,,,, g( ) =, >, fg gf? y = f( u ), ud, u = g( ), E, E = { g( )D}E, E, f g E E () f( u) = y = f ( g( ) ), E, u,, u, D = R, g( ) =, >, E = { > }, = { g( ) D}E = E, f g fg, ( ) g( u ) = u, D = { u u > }, f ( ) = g f gf rcsi ( si,, E = R,, E = { f( )D}E =, ) =, R?, f - ( f( ) ) =, D( f - f )? R, rcsi ( si ) = rcsi y, - rcsi y, rcsi( si ) -, k- < k+, - - k<, si = si ( - k), rcsi( si ) = rcsi ( si( - k ) ) = - k, kz 3
21 k+ 3 < k+, - < ( k + )-, si( ( k + )- ) = si, rcsi( si ) = rcsi( si[ ( k + )- ] ) = ( k + )-, kz rcsi( si ) = - k, k- < k+, ( k + )-, k+ kz 3 < k+, f - ( f ( ) ) =, D f -, y f ( D), D f ( ) = y ; y[ -, ], R, si = y -,, rcsi( si ) =, -,, 8 kg, 4 kg, kg, kg s, 6 s, 3 s, 5 s,,,,, 4 kg, G( t) t 6 s, t < 6, G( t) = 4 - t; t = 6 s,, 6 kg 6t < 9, G( t) = 6 - ( t - 6 ) ; t = 9 s,, kg 9t 5, t > 5,, 4 G( t) = G( t) = - ( t - 9 ) G( t) = t, t < 6, 6 - ( t - 6 ), 6t < 9, - ( t - 9 ), 9t 5, 5, 5 < t
22 - - f ( ) = +, f ( ) = f ( f (f ( ) ) ) f f, f f, f ( ) = f ( ) = f( f ( ) ) = f ( f( f ( ) ) ) = = + = + ; + 3 +, +, f + ( ) = f( f ( ) ) = = + ( + ) 5
23 y = ( ( ) ) ( ) =,,, >, ( ) = -,,, >, { ( ) } = { - }, - 3, (( ) ) = < > 3, ( ) >, ( ( ) ) =, (( ) ) = 4, 3,, < > 3 rcsi + rccos =, =, <, = rcsi,= rccos, si =, cos =, cos = -, si = -, si(+ ) = si cos + cos si = = <, <, < + <, + =, rcsi + rccos =, < - < 5 D( ) R ( ) D ( R ( ) ) R ( D( ) ) D( R ( ) ) R ( ) D, D R 6
24 (, ), = p q, p, q, R p = q q, D R p q = D q =, (, ), R( ) =, D( ) =, D( R ( ) ), [, ] R ( D( ) ) D( ) {, }, R ( ), R( ) = R ( ) =, R ( D( ) ), R ( 5 ) y = [ ] : ( ) >, - < ( ) <, ; < - ( ) >, - <, - < ( ) <, - <, <, < - 4 (), : f ( ), D M >, D, f ( ) > M, ( ) : f ( ), D,,, sup f( )if f( ) : D D = sup D f ( ) (i) D, f ( ) ; (ii) <, D, f ( ) > ( >, D, f( ) > - ) 7
25 3 ( ),,, 3 4, R,, D ()? D f ( ) : M, D, f ( ) > M ; D f ( ) : L, D, f ( ) < L ; D f ( ) : M >, D, f ( ) > M, 7 : A, B, sup( A + B) = sup A + sup B ( 4 ), f, g D, sup{ f ( ) + g( ) }sup D D f( ) + supg( ), ( 4 ) D ( 4 ) (4 )?? (4 )( 4 ), { f( ) + g( ) D} { f ( ) D} + { g( ) D}, ( 4 3) (4 3 ), f ( ) =, g( ) = -, D = [, ], { f( ) + g( ) D} = {}, { f( ) [, ] } + { g( ) [, ] } = [ -, ], { f( ) + g( ) D}{ f ( ) D} + { g( ) D} { f ( ) + g( ) D} f g, ; { f ( ) D} + { g ( ) D} D (4 3 ) (4 ), 3 ( -, + )? f( ) = si, =, [ k, ( k + )], kz ( -, + ) 4 ( )? 8,,
26 ,, ( ) 5 I f I I, f I? ( 3) 6 f I? f I f I, ; f I, I, <, f( )f( ) ; f I 3, 4 I, 3 < 4, f( 3 )f( 4 ) y = l (, + ),, y = l (, + ) : M, > e M, l > M y = l (, + ) : L, ( < < e L ), l < L y = l f ( ) = cos =, U ( ) = M >, =, > M, U ( ), f ( ) = > M, f ( ) = 3 D, D f, f?, D = [, ] f ( ) =,, -, f D f ( D), f f,,, 9
27 [, ], <,,, f ( ) =, f ( ) = -, f ( ) > f ( ), f, <,,, f( ) = -, f ( ) =, f ( ) > f ( ), f 4, f I ( ),, f I f I 4 : f( ), D,,, 3 D, < < 3, [ f ( ) - f ( ) ] [ f ( ) - f ( 3 ) ] > ( 4 4) [ ] f ( ),,, 3 D, < < 3, (4 4 ) f ( ) - f ( ) <, f ( ) - f ( 3 ) <, [ ],, 3 D, < < 3, f ( ) ( 4 4),f,, D, <, f ( ) f ( ), 3, 4 D, 3 < 4, f ( 3 ) f ( 4 ) :,, 3, 4,,, 3, < < 3, f ( ) f ( ), f ( )f ( 3 ) ( f ( )f ( ), f( )f( 3 ) ), [ f ( ) - f( ) ] [ f( ) - f( 3 ) ] ( 4 4 ) f 3, ( 4 4 ),, 3 D, < <, f ( ) > f ( ) > f ( 3 )f ( ) < f ( ) < f ( 3 ), f 5 : y = f( ) ( - < < + ) = = ( > ), f ( ) y = f( ) =, R, f ( + ( - ) ) = f ( ) f ( - ) = f ( + ) ( 4 5) f ( - ) = f ( + ), ( 4 6) = f ( ) ( 4 6) = f ( - )
28 = f ( + - ) = f ( - + ) ( 4 5) = f ( ), R f ( ) ( - ) ( ) : f ( ) = + si R, R, >, f ( ) - f ( ) = - + si - si si <, ( ) = - + cos si - > - - ( - ) =, si - [, + ) f [, ], [, + ) : m ( ) = m ( )M ( ), if f ( y), M ( ) = sup f ( y) y y ( ) f ( ) = cos, [, + ) ; ( ) f( ) =, [ -, + ) ( ) m( ) = ( ) m( ) = M ( ) = cos,, -, < < + ; M ( ), < +, -,, < < + ;, -,, < < +, R, : ( ) ( ) m{, } = ( ) ; ( ) mi{, } = ( )
29 < f g D M ( ) = m{ f ( ), g( )}, m( ) = mi{ f ( ), g( )}, D M ( )m( )?, M ( ) = ( f( ) + g( ) + f ( ) - g( ) ) f ( ), g( ), f ( )g( ) f ( ) - g ( ) y = u = u u = f ( ) - g( ),, M ( ) m( ) 3 f ( ) = - +, : f( - ), f ( + ), f ( ) +, f 4 f + -, - +, +, - +, 5 y = [ ] :, f ( ), f ( ), f( f ( ) ) + -, - +, = + +, f ( ) f ( ) = + + ( ), 5, 3 y ( 35 ) ; y = + 5, = 3, 3,, 5 () y, y ( y = [ + 5 ], > ) 6 y = f( ), : ( ) y = - f ( ) ; ( ) y = f ( - ) ; (3 ) y = - f ( - ) ; ( 4) y = f ( ) ; (5 ) y = sg f ( ) ; (6 ) y = [ f ( ) + f( ) ] ; (7 ) y = [ f ( ) - f( ) ] 7 f g, : ( ) ( ) = m{ f ( ), g( ) }; ( ) ( ) = mi{ f( ), g( ) } 6 (6 ), (7 ) 8 f, g h, f ( ) g( ) h( ), R : f ( f( ) )g( g( ) ) h( h( ) ) R, f ( )g( ), f,
30 f( f ( ) )f( g( ) ) ; f ( y ) g ( y ), y = g ( ), f ( g ( ) ) g ( g ( ) ) ; f ( f ( ) )g( g( ) ), g( ) g( g( ) )h ( h( ) ) h ( ) (, h ( ) ) 9 f g (, ), 7 ( ) ( ) (, ) ( ) = mi{ f ( ), g( )}, (, ), <, f( )f( ), g( )g( ), mi{ f ( ), g( )}f ( )f ( ), mi{ f ( ), g( )}g( )g( ) mi{ f ( ), g( )}mi{ f ( ), g( ) }, mi{ f ( ), g( )} m{ f( ), g( ) } f [ -, ] ( ) : f [, ], f [ -, ] (), [ -, ], <, -, - [, ], - > -, f( - ) > f( - ) f, g D : ( ) if { f ( ) + g( ) }if f( ) + sup D D D ( ) sup D f( ) + if D g( ) ; g( )sup{ f( ) + g( )} D ( ) D, g( ) sup D 4, 7 g( ), f ( ) + g ( )f ( ) + sup D g( ) if { f ( ) + g( ) }if { f( ) + sup D D D = if f( ) + sup D D 4 ( ) D, f ( ) + if D sup D g( )f ( ) + g( ), f ( ) + if g( ) sup D g( )} g( ), { f ( ) + g( ) }, D 3
31 sup D ( ) sup g ( ) = - if D D f( ) + if g( )sup { f ( ) + g ( )} D D if { f ( ) + g ( )} + if { - g( ) }if D D D if { f ( ) + g( ) }if D D { - g( ) }, 4, f( ) + sup D f( ), g( ) ( ) if D if D 3 f, g D, : ( ) if D f( )if D g( )if { f ( ) g( ) }; ( ) sup{ f ( ) g( ) }sup D D D f ( )sup D g( ) ( ) f, g D, if f ( ), if g( ) f ( ), if D g( ) > g ( ),, if f ( ) >, D f, g, D D f ( )if g( ) f ( ) g( ), D if { f( )if g ( )}if { f( ) g( )}, D D D if f ( )if g( )if { f ( ) g( )}, D D D if { f ( ) } = if f ( ) ( > ) D D f, g, ( ), f ( ) = - ( - ), g( ) = -, D = [, ], f, g, if f ( ) =, if g( ) = - D D, if f ( ) g( ) = if D D ( ) ( ) 4 ( - ) = - 6, 4 (, + ) f R, (i), ( ii), 4 ( ) f ( ) = si + ; ( ) f ( ) = - -, <, 3, >
32 ( ) f (, + ), f R, f ( -, ], R F( ), F ( )(, + ) f ( ) (i) f,, F ( ) = si +, >,, =, si - <, R, F ( - ) = - F ( ), F ( ) (ii) F ( ) = ( ) ( i) F ( ) = (ii) F ( ) = si +,, - si, < 3, >, - -,, - + -, - <, 3, < - 3, >, - -,, - 3, < - 5 f R h, : f [, + h], f R f [, + h ], M >, [, + h], f ( ) M R, mz, = mh +, [, + h ] f h, R, f R 6 f I f( ) = f( mh + ) = f( ) M, 5
33 : M = sup I f ( ), m = if I f( ) sup f ( ) - f ( ) = M - m, I, : ( ), I, f( ) - f( ) M - m ; ( ) >,, I, f ( ) - f ( ) > M - m - ( ), I, f( )M, f ( )m, f( ) - f( )M - m, f( ) - f( )M - m, f( ) - f( ) M - m ( ) M = m, f ( ) I, M > m, < M - m M = sup I f ( ), >,, I, m = if I f( ), M - < f ( ), f( ) < m +, M - m - < f( ) - f( ) f ( ) - f ( ) sup f ( ) - f ( ) = M - m, I ( A) :, : m, ( + - ) < < ( - - ) m m - < = < m S, S, S 6
34 3 f ( ) = si, R, 4 f ( ) R, g ( ) R, f ( g( ) ), g( f( ) )? 5 : rct + rccot = sg ( ) 6 : ( ) y = + + c; ( ) y = A, B R, : ( ) A, B < ; ( ) >, A, yb, y - < : sup A = if B (B) ( ) : + k - r = ( ) >, : E = { r r < + k - = + k! k = r = - r < + k! < 3 k = < 7, r }, sup E, if E 3 A, B, : if AB = if A if B AB = { y A, yb} 4 f ( ) (, + ), f ( ) R, f( ) : ( ) f ( ) = e ; ( ) f( ) = l 5 y = f( ), D 6 : y <, rct + rct y = rct + y - y 7 A, B, S = AB, : ( ) sup S = m{sup A, sup B}; ( ) if S = mi{if A, if B}, 7
35 =, : ( ) - N : = >, N, > N, - < ( ) : = U( ; ), U( ; ) > N : R,, : ( ) >, NN +, N, - ( ) U( ; ), U(, ), { },,,, - N,, >,, ; NN + ; > N - <,,, N, 8
36 (, ) 3 =, { } : { } { - } -, - N? - N =, : >, - - G( ) <, G( ) < N, > N G( ) <, - < : 9 ( ) G( ) ( ) ; ( ) G( ) < N ; (3 ), > N, N = m{ N, N } 3 4, 3-3 = 3, N = 3, G( ) =, < 9 < N = 9 N = 9, N = m 3, 9 4 q = ( q < ), G( ) = h, h = q, q - < h G( ) = -, - - < N = 5 h = ( > ), < N = - - N? { } {( - ) } : >, NN +, N, - { } R, - ( ) =, = 4, NN +, = m +, N, 9
37 , { } ( ) = ( - ) =, =,, - = > = -, =,, - ( - ) = >, = mi{ +, - }, N +, - {( - ) } = k = k( k + ) ( k + ),, = k = = = = k = k( k + ) ( k + ) k( k + ) - ( k + ) ( k + ) ( + ) - ( + ) ( + ) - ( + ) ( + ) - N, = = N = m 3 3 +, 4 = (3 - ) ( > 4 ) = 3,, > N, <
38 > 4, G( ) = 3 3 = (>, c > ) c k = [] +, k = c c ( k c) k c = >, k = [] +, k = c c k = ( k c) k c = + h( h > ), N = ( k c) k = k ( + h) ( - ) h! k = ( - ) h h +, > N k c < ( - ) h k k k k <, c = 4 < c, c m, m + < c c! = ( c > ), k!! <, c >, c! = ccc m c m + c - c, c >, k = [ c] +, k - c < k, > k, 3
39 M =, > N, c! = ccc ( k - ) c k c ck - ( k - )! c = M, c k ( k - )!, >, N > > N = m{ N, k}, c <, M c! M <! = ( c > ) 5 si A R, si si - A A, ( A < ) = A, >, N >, > N, N+ 5 7, N+ 4 4, > N,, N >, si - A > si A = ( 7 ) 4 : >, N, > N, 7 : =, k, + k = = : >, N, > N, - < : + k - < =, = =, >, N >, > N, - < - -, = = = = =, =, >, N >, > N, - = - <, = 3
40 = =, =,, = ( - ), =, ( - ) ( ) { }, ( ) { }, M, N +, M (3 ) > < < = > ( < ), N N +, > N, > (4 ) { }{ }, N, > N, (5 ) { }{ }, N, > N, { c } { c }, c = c, m m + m - m ( ) =, k = m, k k + k - k - m + + +, k > m, km, m, k ( ) =, = ( > ) (3 ) =, ) = ; ) > ( =,, ), m = log (4 ) = ( >,, k ) k 33
41 k (5 ) = ( c > ) c c (6 )! = ( c > )! (7 ) = (8 ) =, >, >, = 3 { k }{ }, k ( ) k < k + ; ( ) k k, { }{ } { k } { } { }, { }, { } { k } k, k? { k } k k { k } { }, k = k k k < k + ; k k, > { } { k }, k -?, >, N >, > N, - N =, >, -, N =, >, -, N = k -, k > k -, k -, { } { k k - { k }, k - 34
42 + S = ( > ), S = + + +, S = = =, + S =, - = = ( - ) , =, + + = + +, + = ( ( 8) ), = 3 { } : N k( < k < ), N 35
43 =, N + < k <, k < + < k, < + N < k + N - < < k N =, + N =, 4 : >,, k, >, = log = k < log log k =, log = = log, >, k, log log k log =, >, N >, > N, < log <, < < =, >, N >, > N, < < < <, log = - log, >, 36 log = - k log k =, >,, k, c >,
44 log k,, k c, 5 ( Stolz) : c!,! log k c! ( ) { }, { y }N +, < +, =, y + - y =, + - y = : = y + - y + - -,,, y = y + - y - + -, >, N >, N ( > N ) : < y = y - + ( - + ) ( - - ) = y - + ( - + ) ( ) + ( - + ) ( - - ) = = y N + ( N + ) ( N + - N ) + + ( - + ) ( - - ) = y N + N ( N + - N ) ( - - ) + ( - N ), ( > ),, y - y N - N + N + ( N + - N ) ( - - ) y N - N + - N 37
45 y N - N + =, N >, > N, N = m{ N y + - y = + - yn - N, N }, > N < y - < + = =, =, y = + + +, + 6 : =, p + p + + p = p + p + ( p ) y = p + p + + p, = p +, + >, y + - y = + - ( + ) p ( + ) p + - p + =, = =, ( + ) p p + + C p + p p + y + - y ( + ) p = + - ( p + ) p + + = p +, p + p + + p = p + ( 33 ) p + 5 { }{ }, { } { } ( ) { }, {, { } = c, = ( c - ), { },, { }, { }, : =, = 38
46 9,,, m m, : m = m{,,, m } = m{,,, m }, = m m = m,, m =, m = 3, e +, + = e ( 3 ) e, l = log e +, + +, + < e < + + ( 3 ) + +, 9 3 ( Cuchy ) { } : >, N,, m > N ( > N, p) - m < ( + p - < ) ( 3 3),,, ( ) ( 3 3),, 39
47 ? : { } >, N >,, m > N, - m :, N, N, m, m, = : =, N,, > N, - = = 4 ( ) : = >, N, > N, - < ( ) : = >, U ( ; ) { } (3 ) : (4 ) : = { k } { }, k k = { } { k } { }, { k } { } >, N,, m > N, - m < (5 ) : (6 ) : c, { } { } = =, c = ( ) : { }, >, N, > N, - ( ) :
48 >, U( ; ) { } (3 ) : { k } { }, { k }{ } { k } { }, k k =, { } { k } { }, k =, k (4 ) : { } >, N,, m > N, - m (5 ) { } { } + + +, = + < + - = -, (3 ), + + -, < = < + = e, = e, = e > - 6, + = + 6 ( =,, ) { }, + - = =
49 = ( + ) ( 3 - ) ( 3 4) < 3, = + 6 < 3,, < 3, + = + 6 < 3, { } (3 4 ) + >, { }, > 3, > 3, =,3, (3 4 ),, + <,, { }, = 3, 3, =, = + 6, = 3, 3 : ( ) + < l + = 3 < ; ( 3 5) ( ) { c } = l + < e < + (3 5 ); (3 5 ) ( ) ( ) ( 3 ), + < e < + + +,, l + < < ( + )l +, + < l + < ( ) { c } : ( 3 5), c + - c = = + + l - l( + ) + - l + { c } : (3 5 ), kn +, l( k + ) - l k < k, < k =,, 3,,, 4
50 l ( + ) < + + +, < l( + ) - l < l { c },, c = c = c ( Euler), = l + c +, ( 3 6), ( 3 6 ) l 4 =, =, + = ( =,, ), + =, + = = +, +, = 68, = - 68,, = 68, { } + - = = - ( - - ) ( + ) ( + - ), - <, 3 - >, 4-3 <,, k - k - <, k + - k >, { } + = ( = 68), <, + = + = ; >, + = + < + > + = ; = > 68, k - >, k <, { } { k }{ k - } + - = = ( 3 7)
51 = = ( + + ) ( - ) +, <, + - > ; >, + - < k <, k - >, { k } ( ), { k - } ( ), k k =, k - k = + = = k -, = k, + = +, = +, = = = 68 { k - }{ k }, = = 68 = 68 5 ( Kepler) = qsi +, q, q < q <, + = qsi +, =,,, ( 3 8) :{ }, (3 8 ), pn +, 44 - = q si - si = q si - q - ; 3 - q - q - ; cos q - - q - + p - + p - + p
52 q ( q + p q ) - = q ( - q p ) - q - q - q - - =, { } - q < q <, + p - q - q -, q - - q >, N >, > N, pn + = + p - q - - q { }, = l si - si l - l, si = si l, ( 3 8), l = si l +,, <, l, { } ( 39 ) 8 : { } ( ),? = sup{ } (if{ } ) { }, = sup{ },, >,, - <, { }, >, - < < + : >,, >, - <, 45
53 , : = - = sup{ } ( + ( - ) ) =,,, < ( ) { }, : ( ), ; m ; = sup{, +, }, = if{, +, } ( ) { }, {,, m (3 ) { }{ }, (4 ) { } = ( ) :, sup{, +, }if{, +, }, ( ) { }, M >,, - M M, - M M, { }, { } + = sup{ +, +, }sup{, +, } = { } { }, m, m + m + m (3 ), =,, (4 ) [ ] =, =, >, N, > N - < < + > N - +,, = = [ ] = =, =, > N - < < +, - < = =, >, N, < +, ( 4 ) : 46
54 3 ( ) ( = 3) ; ( ) ( = ) ; e (3 ) ( ) ( = ) : ( ) q lg = ( q < ) ; ( ) = ( ) ; (3 ) =! ( ) q = lg ( ) = + h, h > (3 ) >, N, > N, 3 =, : <, M =!, M! < ( ) = ( = ) ; ( ) > ( =,, ), = ( ) N =, >, N, > N, - < N - + N N N,, > N N - > N = m{ N, N } = =, = ( - ), { }, = <, 5 ( ) =, 47
55 + + +,, ( ) =,, >, = =, ( 5 8 ( ) ) = i > ( i =,,, ), , =, = = = = ( ) 4 : + ( ) + + = ; ( ) = ( > ) ; ( : =, i = ( i ) ) (3 ) = ; (4 ) = ;! (5 ) + (6 ) = e; ( : 3 ( ) = +! = ; ) + (7 ) = ( > ), (8 ) ( = ; - - ) = d, = d 5 : { }, { }, 48
56 ( - ) =, { }, { } 6 { } : M,, A = M : { }{ A } { A },, { A } ( ), >, N, > N, p, A + p - A <, + p - + p < + p - = + p - + p p p p - + p p p <, { } ( ), { } 7 >,>, = { },, +, + = +, =,, : + = + =, = { } { } :, +, + = + = + + =, = l + = +, l = l +, l 49
57 l =, = 8 > >, = - + -, = - -, =, 3, : { }{ } 9 { }, { } : ( ) = ( - ) ; ( ) = si ; (3 ) = ( ) =, N, = N +, m = N +, - m ( ) =, N, = N +, m = N +, : ( ) S =, - m =, S = m{, }, T = mi{, }, =,, = m{, } ; ( ) T = mi{, } ( A) : = = 5-3, 3 ( ) = S, 4 5 ( ) = + = ( >, > )
58 = 5, : ( ) = = = ( ) =, if{ }sup{ } 6 = A, : ( + C + + C k k + + C ) = A 7 { } m + m +, = if, (B) ( ) = ; ( ) = i = i 3, y =,, y = + y - 3 =, 4 { }, 5 =, ( =, 3, ) y =, < { c }, c =, =, { } = =, = + + +, i >, ( i =,,, ), = : - + +, ( - - ) = 5 =
59 ( ) f( ) = A >, M >, > M, f( ) - A < + ( ) f( ) = A >, M >, < - M, f( ) - A < - (3 ) f ( ) = A >, M >, > M, f ( ) - A < () () + (3) - (4 ) f( ) = A >, >, < - <, f( ) - A < f( ) = A >, >, < < +, f( ) - A < f( ) = A >, >, - < <, f( ) - A < f ( ) = A f ( ) = f ( ) = A + -,,,, 9,,, :,, ( Weierstrss ) : = f ( ),,, f ( ) L, L f ( ) 5
60 -,, f ( ) -,? - >, ( ) >, < - <, f ( ) - A <, ;, f ( ), - f ( ) = A ( ) f ( ) : ( - ), ( ) f ( ) - A : f ( ) - A = ( ) - (3 ) >, U ( ; ), ( ) M, ( ) U ( ;) < (4 ) >, = mi, M, < - <, f ( ) - A 3 f ( )A + f( )A : >, >,, < - <, f ( ) - A : f ( ) A? >,, U ( ; ), f ( ) U ( A ; ) f ( )A : + >, M >, > M, f ( ) - A : >, M, U( + ) = { > M}, f ( ) U( A ; ) 53
61 - : + - ( ) = - 3 ( ) ( - ) : f ( ) = + - ( ) = ( + ) ( - ) ( - ) ( - ) = + ( - ) ( ) f ( ) - ( - 3 ) ( ) -, ( ) : ( ) = f ( ) + 3 = + ( - ) + 3 = ( - ) = , (3 ) = < - <, ( ) : =, < - <, < 5, - = - ( - ) > 3 4, < = 3 (4 ) f ( ) + 3 = <, - < 3 = mi, 3, < - <, f ( ) - ( - 3 ) < 54 - M : = -
62 = = + + ( + - ) ( + - ) -,, <, ( + - ) < - M,, M ( + - ) = <, ( - ) 8 > 8, - < - >, M = 8, < - M, 8 3 si < si +, si, > > M ( M > ), si f ( ) A, : >, M >, > + M, si =, M >, N +, = + > M, si + 4 si = si + f ( ) = - [ ] f( ) + - f( ) = >,, 55
63 f( ) <, < - <, + + f ( ) = - [ ] = -,, <, - < - <, f( ) = + ( - ) = f ( ) = - [ ] = - ( - ), - 5 f ( ) = f ( ) = - ( - + ) =,,, (, + ), f ( ) (, + ), A, f ( ) A >, >, U ( ; ), f ( ) - A A =, =, > ( < ),, U + ( ;) ( U + ( ; )?) f ( ) - = = A, = A, > (< ), U ( ; ), f ( ) - A = A > A = f( ) =, f ( ) + ( 47 ) 7 f ( ) = A, f + + = A f( ) = A >, M >, > M, f ( ) - A < + y =, =, >, < y < M, 56
64 f y - A <, f + = A 8 : R( ) R( ) =, [, ] ( =, ) R ( ) = (, ), : q, = p q ( p, qn +, p q ),,, (, ) = R( ) =, >, >, < - <, R( ) - < U ( ;), R ( ) =, R ( ) < ; p U ( ; ), q, R ( ) =, >, p q q U ( ; ), q < p >,, q q q,,,,, k, >, U ( ; ) = mi{ -, -,, k -,, - }, U ( ; ) (, ), R( ) <, >, >, U ( ;),, R ( ) <, R( ) = ( ) f ( ), ( ) f ( ) = A > ( < ), r < A ( r < - A ), U ( ), U ( ) 57
65 f( ) > r > (f( ) < - r < ) (3 ) f ( ), f (4 ) f ( ), g ( ), U ( ; ) f ( )g( ), (5 ) h ( ) = A f( ) g( ) ( ) f ( ) = g( ) = A, U ( ; ) f ( )h( )g( ), f ( ), ( ) [ f( )g( ) ] = ( ) [ f( )g( ) ] = (3 ) f ( ) g( ) = f( ) g( ), f( ) g( ) ( ) f ( ) g( ) ( 3) g( ) ( g( ) ) ( 4) (, ) f ( ), f ( ) (, )? (, ) f( ),? f ( ), (, ), (, ), U ( ), f ( ) U( ), f( ), (, ), (, ) (, ) ; (, : (, ), = f( ) =, (, ), f ( ), f( )(, ) (, ) f ( ), (, ) : (, ) 58 f ( ) = q, = p q, p, q,, (, )
66 (, ), { k },, k k = p k q k, { q k }, { q k }, { k }, { k } U ( ), K >, k > K, k U ( ) { q k }, M >, k ( k > K ), f( k ) = q k > M, f( ), f( )(, )? ( 4) g( ), g( ) g ( ),, U ( ), f( ) U ( ), f( ), g( ) U ( f( ) ), g( ) = f( ) g( ) g ( ), g ( ) =, g( ) = ( - ),, g( ), g( ) = ( + ) ( + ) ) =, ( 5 + 5, y = 5 + 5, y y = 5 + 5, = y5-5 5 = ( + ) 5 ( y5 - ) y y - 5 = 5 ( y5 - ) 5( y - ) - y 5 -, + = y5 + 4, 5 = 5 ( y4 + y 3 + y + y + ) ( y - ) [4 - ( y 4 + y 3 + y + y) ] ( y - ) = - 5 ( y 4 + y 3 + y + y + ) y 3 + y + 3 y
67 , y, ( + ) = y - 5 ( y 4 + y 3 + y + y + ) y 3 + y + 3 y + 4 = - ( ) =, U ( ; ) ( ), t f ( t) = A t f (( ) ) = A ( 5) f ( t) = A, >, >, tu ( ; ), f ( t) - A < ( ) =, >, > ( < ), U ( ; ), ( ) - < : >, >, U ( ; ) f(( ) ) - A <, f (( ) ) = A ( 5), ( 5), +, -,, +, = =, 6 4 >,
68 + = : =, + = >, < + ( N + ) + < =, >, N, > N, - < > N +, [ ] + > [ ] > N, < + - < [ ] + < [ ] < +, + = 5 i m i i k > ( i =,,, ), = k ( k), = k +, = m i i k = k,,, = m i i ( 5 ) 5 f ( ) >, f( ) = A f ( ) = A, A = A > A > f( ) - A = f ( ) - A f( ) - + f( ) - A + + A - 6
69 9 ( ) : ( ) ( ) f ( 3 ), f ( ) = f ( 3 ) f ( ), f ( ) = f( )? f ( 3 ), >, >, < <, f( 3 ) - A <, y = 3, < 3 y <, f ( y) - A <, < y < 3, f ( y) - A <, ( ) : f ( ) = f ( ) =, f ( ) = f( 3 ) = A,, <,,, <, =, - f( ), f( ) f( ) = A ( ), f( ) = A ( 3 ) f( ) = A + ( ), f( ) = A ( 3 ) f( ) = A - ( ), f( ) = A ( 3 3) : + - f( ) = A { } U + ( ), f ( ) = A f( ) = A { } U - ( ), f ( ) = A ( 3 4) ( 3 5) ( ) f( ) U + ( ), f, f ( + ) = if U + ( ) f ( ) ; ( 3 6) 6
70 f, f ( + ) = sup U + ( ) ( ) f U ( ), f ( - ) = sup U( ) - f ( ), f ( + ) = if U + ( ) f ( ) ( 3 7) f ( ) ( 3 8) f U ( ) 3 f( ) = A >, >,, U ( ; ), f ( ) - f ( ) < f ( ) = A + >, M >, > M, > M, f( ) - f( ) < ( 3 9) (3 ) f U + ( ), f ( + ) (3 8 ), f U ( ), f( + ),,? f U ( ), f U + ( ) U - ( ), U + ( ), f ( ) f ( ), f ( ) f ( ) U + ( ),, f ( + ) = if U + ( ) f ( ) f ( ) ( ) : A, f ( ) A, f( )A >, >, U ( ; ), ( ) : (i) (ii), =, f ( ) { }, { } =, f ( ) - A =, f ( ) f ( ) f ( ) f( ) (3 ) (3 ) : 63
71 >, >,,, < - <, f( ) < - <, f ( ) - f ( ) f ( ) =,,,, f ( ), { }, = ; { }, = f( ) = =, f ( ) = =, f( ) f( ) =,, -,, f ( ) f ( )? >,,, U ( ), f ( ) - f ( ) >, =, >,, U + ( ; ),,, f( ) - f( ) = - ( - ) = + > > <, =, >,, U - ( ; ),,, f ( ) - f ( ) = - - = - - > > f( ) f ( ) ( ), f ( ) = 3 f ( ) [, ], (, ] ( =,, ), f( ) = f( ) = f ( )[, ], f - f( ) = f( ) ( f - f ) ( ) = ( f - f ) ( ), 64
72 =,, f ( )[, ] ( ), f -,,, >, { k } { }, k - k +, f ( ), f ( k )f( + ) > f ( ) f ( k k ) = f ( ) = f ( ), k, f ( )f( + ) > f ( ), = 4 f ( )[, ) f [, ), f ( ), [ ] - - f( ),, >, f ( ) U - ( ; ) [, - ] ( <, [, f [, ) - )f ( ) - ], f ( ) m{ f ( ), f ( - ) }, [ ] f [, ), f [, ),, f ( ) - 5 f ( ) U ( ) : { }, U ( ),, f( ) = A, f ( ) = A < + - < -, (3 ) :,, { } (3 ) f ( ) A, >, >, U ( ; ), f ( ) - A =, U ( ; ), f ( ) - A ; = mi, -, U ( ; ), f( ) - A ; 65
73 = mi, - -, U ( ; ), f ( ) - A ; { }, < + - < -, f( ) = A R ( 55 ) =, f( ) - A f ( ) = A 7 : f R, f ( ) =, f ( ), T f, f ( ) < R, N +, f( ) = f(, R f ( ) =, >, M >, > M, = + T > M f + T ) = f( ) <,, f ( ) =, f ( ) f ( ) A, + { } U + ( ), f( ) - A, 4 si t =, rcsi + 66 =, = - cos = rct =, = e y ( + y) y = e
74 + +, + + < + + < + = ( < + ), = e = e?,, + = e + : f ( ) = + g( ) = + + +, < +,, < +, N + f ( ), g( ), f ( ) = + f( ) < + < g ( ), +,, = e + g( ) = e, - N - M + + = e = e, = si cos 3 y = - 3,, y, 3 si cos = y 3 = y si y - cos y + 3 si y - cos y + 3si y = y - cos y + 3 si y 67
75 = y si y si y cos y + 3 t ( + )t ( - ) + t,, ) t( + ) t( - ) + t = t - t - t t + t t ( - t 4 ) = ( - t t ) = 3, k+ ( k =, = ( - t 4 ) t - t t = ( - t 4 ) = + ( ) - y = -,, y, : = y ( + y) = y ( + y) y y + y ( + y) = e si ( ) ; ( ) si ; (3 ) si ; (4 ) si ; ( 5) - si si ( ) ; ( ) ( 4 ) 68
76 , ; ( 3)(5 ) ( ) si si, = ( ) si, si = ( si = si = ) (3 ) si, si = (4 ) = +, si (5 ) y =, y, si = y = +,, si si y y = 5 B, V = kb, k, t? t t,,, t, t,, - t, t,,, t B + k t ; t kb, t, t B + k t t, B + k t ;, = kt, B + k t = B + kt = B + k t = B e kt 69
77 ,, ( ) ( 58 ) 3 : cos cos cos cos = + cos cos cos cos si = si, si = 5 : f ( ) U ( ), f ( ) =, f +, -,, f ( ) =,, ( ) 3 f( ) = A f( ) - A 4 : f ( ), f ( ), f( ),, f g ( ) f ( ) g( ), =, f g, g f f ( ) = o( g( ) ) ( ) ( ) K L, U ( ) 7 < K f ( ) g( ) L,
78 f( ) f g g( ) f ( ) (3 ) g( ) f ( ) g( ) L, U ( ), f( ) = O( g( ) ) ( ) = A, f g =, f g, f ( )g ( ) ( ) :, si, t, rcsi, rct, ( 5 ) - cos (4 ) : f, g, h U ( ), (i) f ( ) g( ) ( ) f( ) h( ) = A, g( ) h( ) = A h( ) (ii) f( ) = B, h( ) g( ) = B ( 5 ) (5 ), f( ) = k ( < k < + ), ( 5 3) f ( ) +, -,, +, - 3 f U ( ), f f( ) = G >, >, U ( ; ), f ( ) > G f( ) = + G >, >, U ( ; ), f ( ) > G f( ) = - G >, >, U ( ; ), f ( ) < - G = ( ) G >, N, > N, > G > G < - G 7
79 +, -,, 4 f, g, f ( ) g( ) =, g f : >, A >, c >, >, +, A + c log =, = ( 5 4) + +, c A, log 5 y = f ( ) y = k +, f f ( ) k = + f ( ) - = [ f ( ) - k ] ( +, - [ f ( ) - k ] ) ( 5 5) f ( ) = ( f( ) =, f ( ) = ), + - f ( ) = ( = ),, y y - =, y y + =, y = f( ), t - si si 3, si, t? 3,,,, t = o( 3 ), si = o( 3 ), 7
80 o( 3 ) 3, t - si = 3 + o( 3 ), si, t, 3,? f ( ), G >, >, U ( ; ), f ( ) > G f ( ), G >, >, U ( ; ), f ( ) G, : f ( ) U ( ), G { }, =, f ( ) G, =, U ( ; ), f ( ) G, = mi, -, U ( ; ), f ( ) G, = mi, - -, U ( ; ), f ( ) G, f ( ), { }, { f ( ) }, ( - ) : ( ) ; ( ) l ; ( 3) e - e ( ) = ( - ) ( - ), = ; ( - ) ( + ) = ( - ) = 3, ( - ) ( ) y = -,, y, = ; l - = y l( + y) y = y l( + y) y =, 73
81 (3 ) e - e - = ee - - (y = - ) - = y e e y - ( z = e y - ) y = e z z l( + z ) = e z l( + z ) z = e, = y l( + y) y = ue l u =, : ( ) ; ( ) + - ; ( 3) si ( ) = 3 ( ) + - = = (3 ) =, + +, ( + - ) = =, si = si = =, 3, ( ) f ( ) = l, ( - ) f( ) = e ( > ) 74 ( ) N +,,
82 f ( ) + = + l = - y + = - y + = -, y l y = y ( (5 4 ) ) l y y l + ( > ) ( ) N +, f ( ) = e - = y y = y (y = y ) e y = y y + e y =, ( (5 4 ) ) ( > ) e - : 4, f( ), g( ), h( ), g( ) [ h( ) + f ( ) ] ( ) f( ) = o( g( ) ), g( )h( ), f ( ) = o( g( ) ), g( ) h( ), ( ), f ( ) g( ) h( ) + f( ) g( ) h( ) =, g( ) =, h( ) = g( ) + f ( ) g( ) = + = g ( )[ h( ) + f ( ) ] ( ),, 75
83 5 - y =, (, > ) y = > : y = f ( ) = (5 5 ) k k = + - = + -, y = f ( ) = = + = -, - = + ( f ( ) - k ) = + = =, + ( - - ) = + ( f ( ) - k ) = - = + ( - - ) - =, + y = f ( )y =, y = f ( ) y = -, < y = f 3 ( ) = - y = f 3 ( )y = - -, y = f 4 ( ) = - y -,, y = f 4 ( )y = - = y = ( 66 ) 7 : S, { } S, + ( ) 76 S, M >, S, > M M =, S, > M,
84 M = +, S, > M, M = - +, S, > M, { }, + : ( 67 ) ( ) 3 - ( - [ ] ) ( = ) ; ( ) + ( [ ] + ) - = (3 ) ( ( + ) ( + ) - ( - ) ( - ) ) ( = + ) ; + (4 ) + ( = ) ; (5 ) - - (6 ) = 3 (7 ) m - m - -, m, = m - ; ( = - ) ; - ( 7) m,, m, = + y,, y, = y = - y m - m - - m - ( + y) m - - ( + y) m C m y + C m y + + y m - C y + C y + + y ( mc - C m ) y + ( mc - C m ) y + = - y ( C m y + C m y + + y m ) ( C y + C y + + y ) ( mc - C ) y + ( ) y 3 + = - y ( C m y + C m y + + y m ) ( C y + C y + + y ) ; = - mc - C m C m C = m - 77
85 m, =, - = y, = - y m - m - - m C m y + C m y + + y m - y C m y + + y m = y ( C m y + C m y + + y m ) y = C m C m = m - m =, m = =, m, m - m - - = m - : ( ) = ; ( ) - ( ) = ; (3 ) + ( ) = ( ) = ( - ) - ( + ) + ( - ), ( - ) =, + =,,, =, =, = -, = ( ) - ( ) + + = ( - ) - ( + ) + ( - ) =
86 ( - ) - ( + ) + - = ,, - =, + =, - +, = -, =, - = = - =, = -, = (3 ) +, ( ) + ( ) ( - ) - ( + ) + - = , - =, + =, +, =, = -,, + = = + =,
87 =, = - 3 f : ( ) f ( ) f ( ) ; ( ) f( ) 4 f, f( ) >, f ( ) =? 5 f( ) = A, ua g( u) = B, g( f ( ) ) = B? U ( )f ( ) A 6 f ( ) = cos ( ) { } ( ), f( ) ( ) ; ( ) { y } y ( ), f ( y ) + ( ) ; (3 ) { z } z ( ), f ( z )- ( ) 7 : { }, { } : ( ) = r > ; + ( ) = s > (, =,, ) ( ) >, r - > N, = r, >, N, r - <, > ( r - ) r - >, ( r - ) = +, { } = +, ( ) >, s - >, + = s, N, > N, k + k > s - ( k = N +,, ) - N + > ( s - ) - N N = + 8 ( ) : 8
88 ( ) + 9 ( = + ) ; ( ) - = +, : ( ) ( ) = + ; ( = ) ( ) > ( =,, ), = + ( ) = +, (, > N, > G > N, - N >, > ), G >, N, ( ) N > - N G + N > G = G, G >, N, > N ( ) > G, ( ) = + = +, e G >, N, > N, > e G, l ( ) l = + > G, (l + l + + l ) = +, l( ) = +, = + : ( ) l(!)!( = + ) ; ( ) ( = + ) 8
89 f U - ( ), : { } U - ( ) ( ), f ( ) = A, f ( - ) = sup U( ) - sup f( ) = A U( ) - ( ) U - ( ), f ( )A f ( ) = A, f( )A U - ( ), =, - >, N, > N, <, f, f ( ) f ( ), f ( ) A,, f ( ) A U - ( ), i, j i j, f, f ( i )f ( ) f ( j )A j, j, f ( )f ( j ) A ( ) >, U - ( ), A - < f( ) f ( ) = A, >, N, > N, A - < f ( ) A, = N +, A - < f( ) sup U - ( ) sup f( ) = A f ( - ) = U( ) - f( ) = A f (, + ) f ( ) = f ( ), f ( ) = A + : f ( )A, (, + ) + (, + ), f ( ) = f ( ) = f ( ) = = f ( ) f ( ) = A, f( ) = A, f ( ) = f ( ) = A, f ( ) A, (, + ) 3 f (, + ) f ( ) = f ( ), + : f ( )f ( ), (, + ) f( ) = f ( ) = f ( ), + >, f ( ) = f ( ) ; <, f( ) = f( ) 4 f (, + ), f (, ), ( f( + ) - f ( ) ) = A : + f( ) = A A =, ( f ( + ) - f ( ) ) =, >, X, X, + f( + ) - f( ) < X, = +, [ X, X + ], N +, 8
90 f ( ) [ X, X + ], M >, [ X, X + ], f ( ) M f ( + k + ) - f ( + k) <, k =,,, f( ) = f( + ) + = f( + ) - f( ) + + f( ) + = f( + ) - f( + - ) + f( + - ) - + f( + ) - f( ) + + f( ) + f( + ) - f( + - ) + + f( + ) - f( ) + + f( ) M + ( > N ), M + <, > X + N +, f( ) + = A, F( ) = f( ) - A f ( ) ( F( + ) - F( ) ) = [ ( f( + ) - f ( ) ) - A] =, + + <, F( ) f( ) + A, F (, ) F( ) + f ( ) + =, = A : ( A) = f ( ) = +, : >, = + 83
91 3 4 : cos cos 5 f( ) = A : >, >, < - <, f( ) - A < 6 : f ( ) = si U ( ), 7, A [, ], m A A m =, f( ) = [, ], f ( ) =, A ( =,, ),, k A k : (B) = f( ) = -, 3 : 4 : 5 : 84 ( ) - ( ) f f + [ + l = - + ( + )( + ) - ] + si - cos = - f ( ) ; = f ( )
92 6 f ( t)tt, P( )Q( ) : tt, P( ) = ,, Q( ) = m m + m - m , m, ( f( t) ), > m, P( f ( t) ) + Q( f ( t) ) ( + ) ( f ( t) ), = m, 7 f ( ) ( ), > = ( ) ( ) = f i = i = i - ; i -, = m ( f ( t) ) m, < m 85
93 f U ( ), f f( ) = f( ) y = f( ) = f f U + ( ), >, >, - <, f( ) - f( ) < f f ( ) = f ( ), f ( + ) = f ( ) + f U - ( ), f f ( ) = f ( ), f ( - ) = f ( ) - f f f U ( ), f, f, f : f( - ) = f( + ) = A, f, f( )A : f( - )f( + ) f ( + ), f ( - ),,,, f ( + ) - f( ), 86
94 f ( ) :,,, f ( ) - f ( ), :,, <, f ( ) - f ( ) - f U ( ), = ( =,, ), >, - <, f ( ) - f ( )? f ( ) <,, f( ), : >, N, N, < ;, N N >, - < N, f ( ) - f ( ) < ; N >, ( = N ) >, - <, f( ) - f( ) < N, N, N,, >, = f ( ) U( ), f ( )? f ( ), f ( ), f ( ) f ( ) : f( ) >, >,, U ( ; ), f( ) - f( ) ; ( ) f( )f( ) >, >, U ( ; ), f ( ) - f ( ) ( ) D( ) =,,, 87
95 R, D( ),, D( ) R ( ) = q, = p q ( p, qn +, p, q ),, =, (, ) 8 R( ) =,, (, ), ; (, ), R ( ) R ( ), (, ) ( ) : = p q, = q, >, U ( ; ), f ( ) - f ( ) = - q =, ( ),R ( ) = p q - : ( ) y = ; ( ) y = 3 ( ) - < = mi =, >, =, <, <,, - = , 3, - <, - < <, ( ) =, >, = 3, <, 3 <, - < ,, 3 >, 3 = <, = mi, 3, - 3
96 - <, 3-3 < f ( ) = 3,,, =,, =, f( ) 3, f ( ) =, f ( ) = >, = 3, >, U + ( ; ), f( ) - f( ) = 3 > 3 = f ( ), f ( ) < 3 f ( ) = l =, = l =, = f ( ) f ( )U ( ), l = =, 3 4 f ( ) = sg si si U ( ),,, f( ) -, f ( ), = 89
97 sg u u =, si, sg si, k = k ( k =,, ) k = k ( k =,, ), k k f ( k + ) = + k f( k - ) = - k f( k + ) = + k f ( k - ) = - k sg si = -, sg si = ; sg si =, sg si = - ; k = k ( k =,, ) = =, >,, + <, + 3 <, = +, = + 3, f ( ) - f ( ) = sg si + - sg si + 3 = >,, f ( ), = f ( ) 5 f ( ) (, ), e f ( ) e -, f( )(, ) f( ) (, ) (, ), f ( ), f ( + ) = f ( - ) = f ( ) e - f ( ) (, ) f ( )(,, f ( + ), e - f( ) (, ), f ( )(, ) (, ), f ( + ) = f( ) >, f ( )f ( ), 9
98 f( ), + f ( + ) = + f ( )f ( ), e f ( )(, ), >, +, f ( )f ( e f ( )e f ( ), e f ( )e f( + ), + ), f ( ) = f ( + ) f ( - ) = f ( ), (, ), f ( )(, ) e ( 73 ) 7 f, g g( ) = y g( ) = y f( y), f( y), >, >, < y - <, f ( y ) - g ( ) <, < - <, y, f, y f ( y), g( ) - g( ) >, >, - <, g ( ) - g ( ), g ( ) =,, g( ) ( ) f, f U ( ) ( ) f, f ( ) > (< ), r < f ( ) (- r > f( ) ), U( ), U ( ), f ( ) > r (f( ) < - r) 9
99 ( 3) f g, fg, fg, f g ( g( ) ) (4) f, g u, u = f ( ), g( f ( ) ) = g( f ( ) ) = g( f ( ) ) ( ) (5 ) ( ) f, f( ) =, g u =, g( f( ) ) = g( f ( ) ) ( +, -, +, - ),, ( ) : f [, ], f [, ], ( ) : f [, ], f ( ) f( ), (, ), f ( ) =, f ( ) =, f ( ) = (3 ) : f [, ], f - [ f( ), f ( ) ] [ f ( ), f ( ) ],, (4 ) : f [, ], f [, ], >, = () >,, [, ], - <, f ( ) - f ( ) < [, ], [, ],,, ; 9
100 f (, ), f ( + ), f ( - )? f (, ) f (,, >, ( ) >,, (, ), - <, f( ) - f( ) <, U + ( ; ), - <, f ( ) - f ( ) < f +, f ( + ) f ( - ) + f (, ) f( + ), f ( - ) [, ] F( ) : F( ) = + F( ) = f( + ), =, f ( ), (, ), f ( - ), =, f ( ) = f ( + ) = F( ), - F( ) = - f ( ) = f ( - ) = F( ), F( ) [, ], (, ) f [, ] F F ( ) [, ], (, ), (, )F ( ) = f ( ), (, ) (, ), I, : ( ) ; ( ) f, f ( 4 ) ; (3 ) ; (4 ) I ci, I ci, f I, I, f I = I I (5 ) (, ) f ( ) f ( + ), f( - ), f ( ) = si (, ), 93
101 si si = + - = si 3 I f( ) f I : >, >,, I, - <, f( ) - f( ) y = si, [, + ) >, >,, - <, f ( ) - f ( ) si [, ] ( > ),, U( + ),, = +, =, < - = + - = < + +,, - <, f ( ) - f ( ), =, >, = +, =, >, 4 - <, si - si =, si [, + ) f ( ) (, ), + f ( ) (, ) + f ( ) = - f ( ) = - f ( ) =, f ( ) =, f ( ) [, ], f ( ) [, ] F( ), F( ) = f ( ), (, ),, =, F( )[, ], F( ) = f ( ) = = F( ), + F( ) = - f ( ) = = F( )
102 [, ] F ( ),, [, ], F( ), =, =, f ( ) (, ), f ( )(, ) ;, (, ), f( )(, ) 3 : (, ) f ( ), (, ) f ( )(, ),, f( + ), f( - ) f ( ) [, ] F( ), F( )[, ] + f ( )(, ),, f ( ) - F( ), f ( ),, f ( ) [, ] + f( ), =, F( ) = - f ( ), (, ), f( ), =, F( )[, ], f ( ) (, ) 4 f I, f I f, < < 3, f ( ) > f ( ), f ( 3 ) > f ( ), ( f ( ) < f ( ), f ( 3 ) < f ( ) ), 4 -,, y =, y = f ( ) (, f ) ), (, f ( ) ), f 4-95
103 , f I,,, 3, < < 3, ( f ( ) - f ( ) ) ( f ( ) - f ( 3 ) ) < f ( ) > f ( ), f( ) < f( 3 ) : f ( ) < < mi{ f( ), f ( 3 )} [, ], [, 3 ],, < <, < < 3, f( ) = f ( ) =, f,, 5 f ( ) I, f ( )I : { }, { } I, ( - ) =, ( f ( ) - f( ) ) = [ ] f ( )I, >, ( ) >,, I, - <, f( ) - f( ) < I { }, { }, ( - ) =, >, N >, > N, - <,, f ( ) - f( ) <, ( f ( ) - f( ) ) = [ ] I { }{ }, ( f ( ) - f( ) ) = f( )I f( )I, ( - ) =, >, >,,, - <, f( ) - f( ) =,, I, - <, f ( ) - f ( ), =,, I, - <, f ( ) - f ( ), =,, I, - <, f ( ) - f ( ), ( - ) =, ( f ( ) - f ( ) ), 96
104 f ( ) I f ( ) I,, { }, { }, ( 8 ) 6 f [, + ), f( ) : f [, + ) + f [, + )? f [, + ) [, + ), f ( ) = B > A = f ( ), + X, > X, f ( ) < B + A, [, X] f f [, + ) [, + ), f ( ) < A, f [, + ) : f ( ) = =, k ( k =,,, + ) f ( ) = +, f ( ) = + - -, X >, f ( - X) <, f ( X) > 6 f 6 f [, + ) f ( ) = A,, >, X,, + > X, f ( ) - f ( ) < [, X + ], >, ( ) >,, [, X + ], - <, f( ) - f( ) < ( ) = mi{, },, [, + ), - <, : f( ) - f( ) < ( ), [, X], - <, ( ) f ( ) - f ( ) < ( ) [, X ], ( X, + ), - <, [, X + ], ( ), f ( ) - f ( ) < (3 ), ( X, + ), - <, f( ) - f( ) <, >, ( ) >,, [, + ), - <, f [, + ) f( ) - f( ) <, 97
105 3, : = sup { r < r r } ( > ), = if { r < r r } ( < < ) : = + ; ( ) = ( 3 ) ( ) ( >, )R, - - =, + = + ( > ) ; = +, + = ( < < ) ( ) log (, + ), log + = -, log = + ( > ) ; + log + = +, log = - ( < < ) + ( ) u ( ) = >, v( ) =, u( ) v( ) = ( 3 ) ( ) u = >, v =, 3 u v = ( 3 3),,?, f ( ) = - + -, =,, 98
106 f ( ), - ( ) ( + ) ( ) ( + ) = e = p, p, q, q >, q : ( + ) = + p q = + p q p q p q = = e p q q + p q q p + p q, (3 ) ( ),, ( p) q p ( + ) = [ ( + ) ], u( ) = ( + ), v( ) =, u( ) = ( + ), (3 ) = e, v( ) = ( + ) = [ ( + ) ] = [ ( + ) ] = e =, ( + ) = = e, R, ( + ) - ( > ) y = -, - = = e log ( + y) y, 99
107 - = y, = log ( + y), y - = y = y = y = l y log ( + y) y l( + y) l l l( + y) y e - = e, 3 y l( + y) y = l y ( + y) y = >, 3( ), = = > = = >,, l = l l + l + + l, 4 >, = =, = l e l l + l + + l = e = e l = = =,, l = - 9( ) l + l + + l, = -
108 = e l l + l + + l = e = 3 4 : >, =, 5 + = ( >, > ) + =, , + - = ( + ) + = e = = + (3 3 ) u = + ( - ) + ( - ) , v = ( - ) + ( - ) ( + ) = e, u = = e ; = l, v = ( - ) + ( - )
109 = l + l = l + v = u = e l = ( 84 ) f (, ), f ( + )f( - ) : ( ) f (, ) ; ( ) (, ), f ( )m{ f ( + ), f ( - ) }, f (, ) ( ), f [, ] F( ) = f( + ), =, f( ), (, ), f( - ), =, F( )[, ] F (, ), F( ) = f ( + ) = + F( ) = f( - ) = - f( ) = + F( ), f ( ) = - F( ) [, ], M >, F( ) M, [, ] (, ) f ( ) = F( ), f( ) M, (, ) () F( )[, ],, [, ], F()F( ) [, ] =, F() = f( + ) f ( - ), f ()F () ; f ( ) m{ f ( + ), f ( - ) }, f () F() f ( ) = F(), f (, ) (, ), f (, ) f : >, f (, + ) ( -, ), [ +, - ] f
110 f (, ), f ( + ) = f( - ) = +, f (, ) f ( + ) = f( - ) = +, (, )M > f ( ), > < -, (, + ) ( -, ), f ( ) > M f ( ) [ +, - ],, [ +, - ], f()f ( ) [ +, - ] f ( ) f ( ) (, ), [ +, - ] ; (, + ) ( -, ), f ( )f ( ) < M < f ( ) f ( ) f (, ) 3 f I, : ( ) ri f ( r) =, I f ( ) ; ( ) r, r, r < r, f ( r ) < f ( r ), f I ( ) I, R, { r }, r = f r I, f ( r ) =, f ( ) = f ( r ) =, f ( ) ( ), I, <, R, r, r, < r < r < { r ( ) }, { r ( ) }, < r ( ) < r < r < r ( ) ) ), r( =, r( = f ( r ( ) ) < f ( r ) < f ( r ) < f ( r ( ) ) ;, f f ( ) r( ) = f ( ), f( r ( ) ) = f ( ) ;, f I f ( )f ( r ) < f ( r ) f ( ), 4,, 3, < < 3 : (, )(, 3 ) 3 = - 3 < f ( ) = ( - ) ( - 3 ) + ( - ) ( - 3 ) + 3 ( - ) ( - ) 5 f [, ], [, ], y[, ], 3
111 : [, ], f ( ) = f ( y) f( ) f ( ) [, ] m = f ( ), m =, ; m >, 6 f [, ],,,, [, ],,,, = : [, ], + f( ) = f( ) + f ( ) + + f ( ) 9, = = = = 7 f [, + ), f( ), [, + ), = f ( ), =,, : ( ) { } ; ( ) = t, f ( t) = t ; (3 ) f ( ) <, (, + ), t = ( ) +, { } = f ( ), { }, ( ) + = f( ), t = + = f ( ) = f( ) = f( t) = t, f : (3 ) f ( ) <, (, + ), t >, f( t) < t, ( )f( t) = t, t = 8 f [, ], f ( ) = f ( ) :, [, ], f + = f( ) =, = > F ( ) = f + - f ( ), F( ) + F + + F - 9 f =,, yr, =, f( + y) = f ( ) + f ( y) : ( ) f R ; ( ) f ( ) = f ( ) ( ) f( + y) = f ( ) + f ( y) y = -, f ( ) = f ( ) + f ( - ), f =, 4
112 f ( ) = R, f ( ) = f ( ) + f ( - ) = f ( ) + f( ), f R f( ) = [ f ( - ) + f ( ) ] = f ( - ) + f( ) = f( ) + f ( ) = f( ), ( ) p, f( p ) = f ( p ) = p f ( ) p, - p f( p - p) = f ( p) + f ( - p) = f ( ) =, p, f ( p) = pf ( ) : q, f f( p ) = - f( - p) = - ( - p) f ( ) = pf ( ), qf q = f q = q f ( ), q + q + + q r = p, ( p, q q ), f ( r) = f p q = pf q q = p q = f( ) f( ) = r f( ) : R, f ( ) = f ( ), { r }, f ( ) = f ( r ) = r f ( ) = f( ) r =, R f,, R, f ( ) = f ( ) f f ; R, f ( ) = f ( ) f ( ) ( ), : f( ) = f( ) ; f ( ) = f ( ) = f ( ) ( A) y = si ( ) f ( ), g ( ) ; ( ) 5
113 f ( ), g( ) f( ) + g( )f ( )g( )? 3 : 4 : h - + ( >, > ) log ( + h ) + log ( - h) - log h ( > ) 5ABC, y, + 6 : ( ) f ( ) = cos (, ) ; ( ) g( ) = cos (, ) 7 f ( ) [, + ), ( ) [, + ), [ f( ) - ( ) ] =, ( )[, + ) y = : 3 : (B) + t c c f ( )I, : ( ), I, f + ( ), I, ; f ( ) + f ( ) f( + ( - ) )f( ) + ( - ) f ( ) 5 f ( -, + ),, f ( ) >, f( ) = + f ( ) =, f ( ) - 6 f [, ], ( ) f (, ), [, ] 6 ;
114 (, ) (, f () ) (, ) ( z, f( z ) ) ( ) R [, ] 4-7 f [, ), : (, ) [, ), f( ) (, ), f [ f ( ), + ) 7
115 ( ) ( ) y f ( ) = = (3 ) : f ( ), f ( +) - f ( ) y = f ( ) + o(),, (4 ) : f ( ),, y = =, (5 ) : y f + ( ) = + = f ( +) - f ( ) + y f - ( ) = - = f ( +) - f ( ) - f ( ) f + ( ) = f - ( ),, ( 3) ( ) : f ( ), f ( ) y = f ( ) (, f( ) ) 8
116 y - y = f ( ) ( - ), y - y = - f ( ) ( - ), ( f ( ) ) ( ) : f ( ),, f, f ( ) = : f( ) =, f ( ) = ( ) f ( ),, 3 f [, ], f + ( ) f - ( ), k f + ( ), f - ( ), (, ), f ( ) = k f ( ), f ( ),,, 638 :,, - A = ( - ) + E, A = ( + E) ( - - E), A, E, = A - A E = = ( + E) ( - - E) - ( - ) E E( - ) - E E = - - E =, E =, - =, = ) 9
117 E, A ( ) =, = 67,,, y y y =, +, y + y y,, y + y = ( + ) = + C - + C - ( ) + + C ( ) y =,,, y = - ( )= -, f ( ), f ( ) ( f( ) )? f ( ) ( f ( ) ) f ( ) f ( ) ; ( f ( ) ) f ( ), f ( ) =, f (3 ) = = 3 = 6, ( f (3 ) )= (9 )= f( ) =,, +, < f + ( )f - ( ) : f ( ) =, >,, < ; =, f + ( ) = =, f - ( ) =?,? f ( ) = : ( +) - f + ( ) = + = + ( + ) =, (3 +) - f - ( ) = -
118 = - + = -, f + ( ) =, f - ( ) = 3 f ( )? ( ) ( ) ( ) f ( ) = si, >,,, f - ( ) =, f + ( ) (3 ) f( ) =, f + ( ) =, f - ( ) = - : f( ) = g( ) =,, -, ;,, -, f ( ) =, f ( +) - f ( ) f( + ) - f ( ) =,, -,,, f ( ), f ( ), f ( ) g( ) =, g( + ) - g( ) =, g( + ) - g( ), -,, g ( ) =, g( ), g( ) f ( ) [ -, ] ( > ), f ( ), f ( ) = =,
119 f ( ), f( ) = f ( ) - f( ) ( ) f() - f ( ) =, f ( ) = 3 f ( ), g( ) [, ], (, ), f ( ) = g( ), f - ( ) = g + ( ), h( ) h( ) = f ( ) = g( ), f - ( ) = g + ( ) =, f( ),, g( ), > h( + ) - h( ) h + ( ) = + g( + ) - f ( ) = + g( + ) - g ( ) = + = g + ( ) ; h - ( ) = f - ( ) h + ( ) = h - ( ), h( ) 4 f ( ), P(, f ( ) ) N M, P T ( 5 - ) : P T N T PN = N T = f ( ) f ( ) f ( ) f ( ), TM = f ( )f ( ), + f ( ), PM = f ( ) + f ( ), P, t = f ( ) = t, PT = f ( ), N T = f ( ) f ( )
120 5 - T PM, TM P T = t, T M = f( )f ( ) PN = N T + P T = f ( ) f ( ) = f( ) f ( ) + f ( ) + f ( ) PM = T M + P T = f ( ) f ( ) + f ( ) = f( ) + f ( ) 5 y = y, y, 4 P(, ), y Q P y R, ( ) =, PR R y y - = ( - ), y = -, 5 -, 3
121 5 -, = PQR,, PQR Q y q, PQ T PQ = QR, PQ = Q T + P T = + ( - q) + ( - q) = ( q + ), q = 4 6 : f( ), f ( ), f ( ), f ( ), f( ), f ( ) = =, f ( ) =, f ( ), f ( ) f ( ) =,, -, f( ) =, f ( ) =,, f( ), f ( ) f ( ) f ( ),, U ( ), f ( ), f f ( ) f( ) =, f, f, 4
122 f ( ) =, f ( ) = f ( ),, U ( ), f( ) U ( ), f ( ), f( ) f ( ) =, f ( ), f ( ), f ( ) =, f ( ) =, f ( + ) - f ( ) f ( + ) - f( ) = f ( ) = ( 94 ) 8 f ( ) = m si,,, = : ( ) m, f = ; ( ) m, f = ; (3 ) m, f = =, ( m ) ( ) m, f ( ) = m si m, f ( ) = f ( ), f = ( ) m, m si - f ( ) = f ( ) = m m m - si - m - cos, (3 ) m3, f = = m - si =,, =, f ( ) =, 5
123 g( ) = g ( ) =, f ( ) f( ) = g() = f ( ) = g( ) si,, f() - f ( ), = = g() si, f R,, R, f ( + ) = f ( )f ( ) f ( ) =, R, f ( f ( ) = f( ) + ) = f ( )f ( ) = =, f( ) = f ( ), f ( ) =, f ( ) ; f ( ) =, f ( + ) = f ( )f ( ) =, =, f ( + ) = f( )f ( ), f ( ) = f( ) ( ) ( uv) = u v ( ) ( uv) = u v + uv, ( cu)= cu ( c ) (3 ) u v u v - uv =, v v = - v v ( ) ( c) = ( c ) ( ) ( ) = - ( ) (3 ) ( ) = l, (e ) = e (4 ) ( log ) = l, (l )= (5 ) ( si ) = cos, ( cos )= - si, 6 ( t ) = sec, (cot )= - csc, ( sec ) = sec t, (csc )= - csc cot
124 (6 ) ( rcsi ) = -, (rccos )= - -, ( rct ) =, (rccot )= - + +, 3 d y d =, d d y d y d = d y d u d u d,,, 4 ( ) f ( ) = u ( ) v ( ) y = f( ), (l f ( ) )= f ( ) f( ), f ( g( ) )( f ( g( ) ) )? f ( g( ) ) y = f ( u) u = g( ) f ( g( ) ) f ( u) u, u g( ), f ( g( ) ) = f ( u ) u = g( ) ( f ( g( ) ) ) f( g( ) ), ( f( g( ) ) )= f ( g( ) )g ( ), f ( ) = ( ) + ( ), g( ) = ( ) (, f( )g( ), ( ), ( )? ( ), ( ),, f ( ) = ( ) + ( )g( ) = ( ) (, ( ) =, ( ) = - =, f ( ) = ( ) + ( ) =, =, g( ) = ( ) ( = - = - = 3 f ( ) U ( ), f ( + ), f ( - ) 7
125 f + ( ) = f ( + ), f - ( ) = f ( - )? f + ( )f ( + ) f + ( ) f, f ( + ) f ( ), f ( + ) - f( ) f + ( ) =, + f ( + ) = + f( ) =, f ( ), +, <, U ( ) f ( ) =, >,, <, f ( + ) = = + f ( - ) = = - = f + ( ), f - ( ), f - ( ) = f ( - ), f ( + ), f ( - ) f ( ) = si,,, =, + f ( ) - f ( ) = si - cos,,, =, f ( ) = ( ) si f ( ) = : f ( + ), f ( - ), f ( ), f ( ) =,,, =, f ( + ) = f ( - ) = ; f( ) =,, 8
126 - f + ( ) = + - f - ( ) = - =, = f + ( ) = f ( + ), f - ( ) = f ( - ) f ( ) f( ) = + + l( + + ),, : + = = ; u = + +, f ( ) l + + = f ( ) = f ( ) = = = + - rct - + t, y= ( > ), sec + t 9
127 = = ( - ) t - + sec = ( + )cos + ( - ) si + cos 3 y =, y, y =, l y = l y y = l l +, y = l l + 4 : y = ( - ) ( + ) 3, y : y = ( - ) ( + ) 3, -, - ( - ) ( + ) 3, < -, - : y = ( - ) ( + ) ( 5 - ), > -, - ( - ) ( + ) (5 - ), < - = -, ( - ) ( ) 3 - y + ( - ) = + =, y - ( - ) =, y ( - ) = y = 5 ( - ) ( + ) ( 5 - ), -, - ( - ) ( + ) (5 - ), < - f( ) = + e,,, =
128 , =, f ( ) = + e + e + e = f + ( ) = e + e = + + e =, f - ( ) = - + e =, + e - f = 6 : ( ) [, + ), { }, N + ( ) f R, ff R ( ) f ( ) = ( N + f ( ) =, =, N +,, N +, ), f ( ) = ; N+, f, f ( ) D( ) R, u = D( ) D( u), D( u)d( ), D( ) =, ( DD ) ( ) = D( ) = ;, D( ) =, ( DD) ( ) = D( ) = ; ( DD) ( ), DD R, ( DD) ( ), R ( ) 9 sh -, ch -, th -, coth - : ( ) y = sh - ; ( ) y = ch - ;
129 (3 ) y = th - ; ( 4) y = coth - ; (5 ) y = th - - coth - ; ( 6) y = sh - ( t ), : ch y - sh y =, ch y = - th y, sh y = coth y - ( ) ( ) ( 3), y = - ( ), ( y ) ( ) = sh y,, d y d = d d y = ( sh y) = ch y,, ( )ch y = + sh y, ( ) = ch y, d y d = d d y = (3 ) = th y, ( ), ch y = (4 ) = coth y, ( 3 ), sh y = d y d = ch y = + sh y = + (ch y) = sh y = ch y - = - ( > ) d y d = d d y = ( th y) = ch y, - th y =, ( < ), - d y d = d d y ( th - ) =, ( < ) - = (coth y) = = - sh y, - sh y coth y - = -, (coth - )= - ( > )
130 (5 ) y = th - - coth -, (6 ) y = sh - ( t ), y = y = = + t sec = sec 3 ( 3 4) = ( t), y = ( t), t, ( t) ( t), ( t) = ( ), d y d = ( t) ( t) ( 3 ) d y d = ( ) t + ( ) ( ) - ( ) t ( 3 ) = ( t), C: y = ( t), t,, [, ], ( t) + ( t), C, ( t)t C (( t), ( t) ) ( Y - ( t) ) ( t) - ( X - ( t) ) ( t) =, ( 3 3) ( Y - ( t) ) ( t) + ( X - ( t) ) ( t) = ( 3 4) 3 f ( ) f ( ), f ( ) f( ), f ( ), f ( ) = f ( ) f ( ) - f ( ), - 3
131 , f ( ), f ( - ) ( ) f ( - ), f f ( ) ( ) = ( f ( - ) 4 ( ) ( ) ( ) = (l ) ( > ), (e ) ( ) = e ; ( ) ( si ) ( ) = si + ; (3 ) ( cos ) ( ) = cos + ; ( ) ) (4 ) ( m ) ( ) = m( m - ) ( m - + ) m - ( m ) ; (5 ) ( l ) ( ) = ( - ) - ( - )! ; (6 ) + ( ) = ( - )! ( + ) + 5 u( ), v ( ), ( 3 5) ( uv) ( ) = C u ( k ) v ( - k) k = ( 3 6) 6 ( t), ( t) [, ], = ( t), y = ( t), t[,] y = ( t) ( - ( t) ( ( 3 7) d [ ( t) ] 3 d = ( t), C:, t y = ( t) d y d = ( t) ( t) y d d = d d t?? d d t y = d d d d y d 4 ( t) = ( t) ( - ( t) ( ( t) [ ( t) ] d d y d y, d, d
132 d y = d d d t d y d d d t = d ( t) d t ( t) = ( t) ( - ( t) ( ( t) [ ( t) ] 3 : ( ) ( 3 5) ( ) ( 3 6) ( 5 (4 ) ) (3 ) ( 5 (6 ) ) (4 ), ( 5 (3 ), (5 ) ) (5 ), ( 9 ) (6 ) ( 7 ) = e t cos t, y = e t si t, d y d d y d t t = e si t + e t si tcos t, d d t = e t cos t - e t si tcos t, d y d = d y d t d d t = si t + si tcos t cos t - si tcos t r =, d y d (3 ), d y d = r ( ) t + r ( ) r ( ) - r ( ) t = = t + - t t + - t = f ( t), 3 f ( t), f ( t), C: y = t f ( t) - f ( t), 5
133 d y d, d y d d y d t = f ( t) + t f ( t) - f ( t) = tf ( t), d d t = f ( t), y = d d d y d = d d t d y d t d d t d y d d d t = tf ( t) f ( t) = = t 4 : d d t ( t) f ( t) = f ( t) ( ) [ e si( + c) ] ( ) = e ( + ) si ( + c + ), ( ) [ e cos( + c) ] ( ) = e ( + ) cos( + c + ), si = ( ) =, + + [e cos ( + c) ] = e cos( + c) - e si( + c) = e + + cos( + c) - si( + c) + = e + cos cos( + c) - si si( + c) = e + cos( + c + ), cos = = + +, = k -, [e cos ( + c) ] ( k - ) = e ( + ) k - cos[ + c + ( k - )], = k, 6 [e t cos( + c) ] ( k ) = ( [e t cos( + c) ] ( k - ) ) = [e ( + ) k - cos( + c + ( k - )) ] = ( + ) k - [ e cos( + c + ( k - )) ]
134 = e ( + ) k cos( + c + k ), [e cos( + c) ] ( ) = e ( + ) cos( + c + ) ( ) 5 f ( ) = ( + + 3)e -, f ( ) ( ) u( ) = + + 3, v ( ) = e -, k3 u ( k ) ( ) =,, f ( ) ( ) = ( + + 3) (e - ) ( ) + C ( + + 3) (e - ) ( - ) + C ( + + 3) (e - ) ( - ) = ( - ) ( + + 3)e - + ( - ) - ( + )e - + ( - ) - ( - )e - = ( - ) e - [ - ( - ) ] 6 f( ), l, m, F( ) = f ( ),, l( - ) + m( - ) +, > f ( ) ( -, ], F( ), >, < F ( ), F ( ), l, m, F( ), ; F ( ) m, F l, f - ( ), f - ( ) F( ), = f ( ) + [ l( - ) + m( - ) + ] = f( ) F ( ), F - ( ) = F + ( ), F - ( ) = f - ( ), F + ( ) = + = + F( ) - F( ) - l( - ) + m( - ) + f ( ) - f ( ) - = m, 7
135 m = f - ( ) F( ) F ( ) = f ( ), <, f - ( ), =, l( - ) + f - ( ), > F =, F - ( ) = F + ( ), F - ( ) = f - ( ), F + ( ) = + = + = l, F ( ) - F ( ) - l = f - ( ), l( - ) + f - ( ) - f - ( ) - l = f - ( ), m = f - ( ), = f ( ) 7 + ( ) = ( - )! ( + ) + si[ ( + ) rccot ], (rct ) ( ) = ( - ) - ( - )! si( rccot ) ( + ) + = i - i - + i + ( ) = i = ( - )! i - i - + i ( ) ( - i) + -, ( + i) + = ( - )! [ ( + i) + i ( + ) +, cos + isi = e i, - ( - i) + ], 8
136 + ( ) + i = + = ( - )! [ ( + i) + i ( + ) + - ( - i) + ] + + i +, + = cos, = + si, cot =, + i = + (cos + isi ) = + e i, - i = + ( cos - isi ) = + e - i, ( + i) + = ( + ) + e i( + ) = ( + ) + [cos( + )+ isi( + )], ( - i) + = ( + ) + e - i( + ) = ( + ) + [cos( + )- isi( + )],, + ( ) = ( - )! [ isi ( + )] i ( + ) + = ( - )! ( + ) + (rct ) ( ) = 3 ( 5 ) 4 : si[ ( + ) rccot ], + ( - ) = ( - ) - ( - )! si( rccot ) ( + ) = (cos t + tsi t), y = ( si t - tcos t), ( t) ( t) = tcos t, y ( t) = tsi t, (3 4 ), [ Y - ( si t - tcos t) ] y ( t) + [ X - ( cos t + tsi t) ] ( t) =, ( t), y ( t), si ty + cos tx =,, 4 ( 9 ) 9 y = rct ( ) ( + ) y+ y= ; 9
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年 月总第 8 期 数学方法与数学思想 编辑点评 数学与哲学都是研究最普遍的事物的 但是研究的角度 目的 方法 过 程和成果并不一样 所以两者之间有联系也有区别 该文通过对像 先有鸡 还是先有蛋 这样一些通俗又典型的例子 说明数学家与哲学家对于同一 个问题思维和处理的方式如何不同 便于读者形象地理解文中的论点 文 章的论述比较恰当 准确 深刻 写作也通顺流利 是一篇可读性较强的 文章 值得读者体会和学习
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Chapter 01 1.1 1.2 1.3 1.4 2 1.1 truth Tfalse F (proposition) (1) (T) (2) 2+3=6(F) (3) 1+2=3 (atom proposition)(primary proposition) (compound proposition)(1)(negation)(2) (and)(3)(or) 1+1=2 1+1=2 52=6
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名人养生.doc
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常见病防治（二）.doc
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警界风采录（六）.DOC
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ISBN 7-80173-166-2/ S 01 698.00 I..................................................................... ...... 177...................................................... II 177
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