第一章.doc

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炭擂汲
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1 = c < < + + = S = c( ) = k =, k =,,, Λ < < + = 4 = = = = 4

2 k = k =,,, Λ X R X X = f () X X = f ( ) k = + k =,,, Λ = f () X X f ( ) = = = = n n = an + an a + a a n =a +a +a = a + a + a a n f ( ) = = =

3 Q( ) = P() Q() P() Q() P( ) = ( ) = α < < + α = α α π π π =

4 α > = α [, + ) α > <α < α < = α [, + ) = a a > a = = = / = - =a (a<) =a (a>) > a (, + ) a < (, + ) e =.788 = e = log = a ) (, = log a = a 8 o a = log a = a = log = lg log = ln e e 4

5 = log ( a > ) a = log ( a < ) a = sin < < + = cos < < + = tg k + π = ctg π k = sin π = cos π - = tg π = ctg π = f () (, + ) l > (, + ) 5

6 f ( + l) = f () l f () l kl e e = sh( ) = e + e = ch( ) = sh( ) e e = th( ) = = ch( ) e + e ch ( ) sh ( ) = cos + sin = ch ( ) + sh ( ) = ch() cos sin = cos() sh ( ) = sh( ) ch( ) sin( ) = sin cos X = ch() Y = sh() X Y = Poincar = sh = ch = th 6

7 = 4 q q = q 7q = 4 q q = 4 q q = 4 q q z = 4 q q = z X X RX X R f X X R f X R f : X R f = f () X f f f () ( X, f ) f ( X ) = { : X } R 7

8 R E = {(, ) : X } f () = = sgn( ) sgn = > = < = Gauss = [] ] [ [ ] < [ ] + [ ] < = [ ] 5 ( 8 ) = ( ) ( 8 ) Dirichlet 8 8 < < 8 8 < 58 8

9 = D( ) = Q P Q = f (P) P P = P(Q) C = C( ) = a + b b a C() C( ) C( ) = R = p = p( ) p = p() L( ) = R( ) C( ) f : X R M > X f ( ) M f () X = sin = cos = D() = X = [,] (, + ) f : X R M > f : X R n =,,Λ, f ( ) = X f ( ) M n X f ( n ) > n X [ a, b] ( a, b) ( a, b] [ a, b) (, a] [ a, + ) (, + ) f : X R X 9

10 < f ) f ( ) f () ( f () < > f () = = [] X X X f : X R M m X m M f : X R M X M M M m = M M = M M m X m M M = ma( M +, ) M M > A X f : X R: A A f A X m M f : X R ϕ A ϕ R ( ) = f ϕ X ϕ f + g( ) g( ) g( ) g ( ) = g ( )

11 = f () u = g() g () = f ( g ()) f g g f ο f ο g ( ) = f ( g( )) ο f ο g = g ο f f ( g ο h) = ( f ο g ο h f ο f ολ ο f n ο ) = f () z = z = ( ) f : X R X f ( ) f ( ) = f ( ) = f X f : X Y Y = f (X ) f f f : X : X Y f R ) = f ( f : X Y Y = f () f : X Y Y = f () X = f () = f ( ) f : X Y f : Y X X f ο f = I : X Y f ο f = I : Y Y ( f ) = f : X

12 = f ( ) = f () = =f() =f(), < =, f : X Y X Y Y = = f ( ) = f ( ) e e = sh( ) = R R e e = e = z z = z z z = z = ± + + =f - ()

13 ln( = + + ) = ln( + + ) = sh ( ) g( ) ) = f ( g ( ) = g( ) = f ( ) = f () g( ) = f ( ) X Y Y = f () f () X Y X f ( ) = f ( ) g ( f ( )) = g ( f ( )) = = f : X Y g( ) = f ( ) Y X = f () = f ( ) g ( ) = g( ) = g( ) = f ( ) : f R R f ( ) f () * * * * * = f [ f ( )] f ( ) = f ο f [ f ( )] f ( ) f ο f f ( * ) = * ) f ( * = f () = = f ( ) f ο f * A, B, C, Λ,, z,λ A A N } = {,,, Λ Z = { Λ,,,,,, Λ }

14 p Q = : p, q Z, q q R C } = { + i :, R, i = { R = } { R : = } A B A B A B A B A = B : A B B A A B A B b B b A A Υ B = { : A B} A Ι B = { : A B} A B = { A, B} A Ω A c = Ω A X Y f X Y = f () f : X Y X Y f ( X ) Y, X f ( ) f ( ) f f X, Y f f ( X ) = Y f f : X Y e sh = e ch e + e = ch sh = ch ± ) = chch ± shsh ( 4

15 sh ± ) = shch ± chsh ( ( ) = + 4 f f () f ( f ()) f ( ) f ( a + b) f ( a) + f ( b) + = + f () f ( f ()) f ( f ( f ())) ch sh = ch + [ f ( )] f ( ) f ( ) f > ( ) ( ) < f () f ( ) = 6 = = = + + = log f ) log ( = 5 = log f ) = 6 = [log( )] ( = log + log( ) = log ( ) ) = cos = log cos arccos( = f ( = [, 5] = + b = a + ab > = ) = + (, + ) a + = = = log( + e) = cos 5

16 π 4 4 = sin + sin( + ) = sin + cos = = [] = ln( + ) π = ln a ( a =, ) = sin ( + ) = sin( + ) 8 : 45 ο ABCD S h l ( < < + ) π 8 D C S h 45 A B o ( ( ( = sin = [] = tg = sec = cos + sin ln( + + ) () = (, + ) f [, a) ( a > ) () f ( a, a) = ( ) 6

17 () f (, + ) a () () f T ( T > ) f g () ( a, b) f ( ) ma(, g( )) min(, g( )) ( a, b) (, + ) f () f () f () f () () () () () f ( a, b) f ( a, b) f ( a, b) f ( a, b) T = ( < < + ) ( ) ( e e ) = ( < < + ), =,, > ;, g ) = (,, > ; f [ g ( )] g[ ] = ( f ο f ολ ο f )( ) n 7

18 = ( f ο f ολ ο f )( ) n sin + cos R() = R ( ) = R ( ) R ( ) = R( ) R ( ) = R ( ) R( ) = R( ) f () R R = ( + ) 4 = sin( + ) = = < = = e = a + b + c > b a f () (, ] a b a < f () (, ] a = + = = log( ) = (,) [, + ) b a b [, + a ), ) = (,) (, + ) = (,) = + sin (, + ) (, π ), ) sin( + a) sin ( + ( + = 8

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