typologio_extasewn_PLH12
description
Transcript of typologio_extasewn_PLH12
-
12 2012-2013
m n
[ ]ijA a= [ ]T jiA a= , (, ). : ( )T TA A= ( )T T TA B A B+ = + ( )T TA A = , R ( )T T TAB B A= m n [ ]ijA a= ij jia a= .
TA A= n n
[ ]ijA a= ( ) 1A 1 1
nAA AA I = = .
: , nn
1 1( )A A = 1 1( ) ( )T TA A =
1 1 1( )AB B A = 1 1( ) ( )k kA A = k Z
Laplace [ ]ijA a= i j : det( )A A= =
11 12 1
21 22 2
1 1
1 2
n
n nn
i k ik k j k jk k
n n nn
a a a
a a aa A a A
a a a
= =
= = =
L
L
M M M
L
( 1)i jij ijA M+= ijM ij-. n n A : det( ) det( )TA A= det( ) det( )nA A = , R det( ) det( )det( )AB A B= [ ]det( ) det( ) kkA A= , \{0}k Z A det( ) 0A
11 ( )
det( )A adj AA =
( )adj A
A . * * * * * * * * * * * *
U V .. V ,k R 1 2, U u u
1 2k U+ u u .
1 2, , , kv v vK
1 1 2 2 1 2 0.0k k k + + + = = = = =L Lv v v 1 2{ , , , }kv v vK .. V V I. 1 2, , , kv v vK
I. .. V 1 2, , , kv v vK V dimV k= . ={ 1 2, , , kKu u u } () V x V ,
1
ki ii
x a u=
= , ia R . [a1 a2 ak ]T
x B [ ]Bx . V ..
,U W .V : dim( ) dim dim dim( )U W U W U W+ = +
,U W V .. V V U W= ( V U W= + { }0U W = ) ( V U W= + dim dim dimV U W= + ).
* * * * * * * * * * * *
nR ( , ) n nx y R R , ox y :
. ( ) ( ) ( )k k + = +o o ox y z x z y z ,
, ,n x y z R , ,k R
. =o ox y y x , , n x y R
. 0ox x 0= = 0ox x x To x = ox x x .
[0, ] , \{ }nx y 0R
: cos=ox y
x y.
, nx y R ( ) 0=ox y .
, nx y R :
. 2 2 20= + = +ox y x y x y
. + +x y x y
I. =x x , R
IV. x y x yo (Cauchy-Schwarz)
p x
y 2 .=x yp y
yo
nE R
{ }: 0,nE E = = y x y xR . , nE E = R , ( )E E = .
1 2, , ,n
k Ku u u R
(. 0i j =ou u i j , 1i =u ) .
1 2, , , nK nR ,
1 1=
1 2 11 2 1
1 1 2 2 1 1
j j j jj j j
j j
= o o o
Lo o o
2,3, ,j n= K ,
11
1
=u
, 2
22
=u
, ,n
n
n
=K u
nR .
n n A T TA A A A I= = ,
1 TA A = , . ,
: I. ( ) ,nR II. det 1,A =
III. ,A =x x IV. A A =o ox y x y V. .
* * * * * * * * * * * *
() :f U V ( ,U V )
( ) ( ) ( )f k k f f + = +x y x y , , U x y ,k R . ( U V= .
U ). ker { : ( ) }f U f U= = 0x x f U . Im { : ( ) , }f V f U V= = y x y x f V . :f U V -- (1-1)
, U x y
( ) ( )f f= =x y x y . :f U V ( )f U V= . :f U V :
. dim dim ker dim ImU f f= +
. f 1-1 ker { }f = 0 .
. 1={ 1 2, , , nKu u u } U 2={ 1 2, , , mKv v v } V ,
1 11 1 21 2 1
2 12 1 22 2 2
1 1 2 2
( )( )
( )
m m
m m
n n n mn m
f a a af a a a
f a a a
= + + +
= + + +
= + + +
L
L
M
L
u v v v
u v v v
u v v v
m n f 11 12 1
21 22 2
1 2
n
n
m m mn
a a a
a a aA
a a a
=
L
L
M M M
L
,
[ ] [ ]1 2
( )B B
A f=x x , Ux . dim dimU V n= = , :f U V .
. f ( 1f ) II. f 1-1 III. ker { }f = 0
IV. f * * * * * * * * * * * *
n n A i n
11 12 1
21 22 2
1 2
11 1 0
( ) detn
n
n n nn
n n
n
a a a
a a ap
a a a
a a a
=
= + + + +
L
L
M M M
L
L
A , .
i , 1, 2, ,i n= K , - -
[ ]1 2T
nx x x= Kx
11 1 12 2 1
21 1 22 2 2
1 1 2 2
( ) 0( ) 0
( ) 0
i n n
i n n
n n nn i n
a x a x a x
a x a x a x
a x a x a x
+ + + =
+ + + =
+ + + =
L
L
M
L
A :
1 2 0det ( 1)nnA a = = L 1 2 1n ntrA a = + + + = L ,
0 1, na a
( )p . i ix
A , ,ki i x kA .
,
.
n n A , D , . P , 1A PDP= . D A P -
nR .
: k k , , ,
( ) n . . T Q ,
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12 2012-2013
1 2( , , , ) TnA Q diag Q= K .
( )f , ( )1 11 2( ) ( ) ( ), ( ), , ( )nf A P f D P Pdiag f f f P = = K
A 1
1 1 0( ) n nnp A A a A a A a I= + + + + =L O . ( ) ( )f ( )p , ( ) ( )f A A= .
* * * * * * * * * * * *
1 2, , , nx x xK ( ) TF A=x x x , [ ]1 2
Tnx x x= Kx A n n
, .
1 2( , , , ) TnA Qdiag Q = K , ( )F x
2 2 21 1 2 2( ) n nF y y y = + + +Ly ,
[ ]1 2T T
ny y y Q= =Ky x . ( )1 2, , , 0 0n > , 1 2,x x A 1 2x x< .
:f AR s ( f ) : ( )f x s , x A . ( ).
.
1-1 :f AR 1 2,x x A 1 2x x ,
1 2( ) ( )f x f x , : 1 2( ) ( )f x f x= , 1 2x x= . :f AR :g B R , ( )( ) ( ( ))g f x g f x=o , x A
( )f x B . 1-1 f 1 : ( )f f A A , ( )y f A x , ( )y f x= , . 1( ) ( )f y x f x y = = .
0x -
0lim ( )x x
f x
= l 0 0
lim ( ) lim ( )x x x x
f x f x+
= = l
:
( ) ( ) ( )g x f x h x 0x
0 0
lim ( ) lim ( )x x x x
h x g x
= = l , 0
lim ( )x x
f x
= l .
x + , x .
0
sinlim 1x
x
x= ,
0
cos 1lim 0x
x
x
=
:f AR 0x A
00lim ( ) ( )
x xf x f x
= .
* * * * * * * * * * * *
( ( , )A a b= R) :f AR 0x A
0
00
0
( ) ( )lim ( )x x
f x f x f xx x
=
R
fC
0 0( , ( ))x f x 0 0 0( ) ( )( )y f x f x x x = f f f f .
: f, g ( ) ( )( ) ( ) ,cf x c f x c = R
( ) ( ) ( )( ) ( ) ( ) ( )f x g x f x g x =
( )( ) ( ) ( ) ( ) ( ) ( )f x g x f x g x f x g x = +
2
( ) ( ) ( ) ( ) ( ), ( ) 0( ) ( )
f x f x g x f x g x g xg x g x
=
A 0f f 1f
( )1 1 ,f f =
( ( ))f g x
( ) ( ( )) ( ) ( )( ( )) df g x df g dg xf g xdx dg dx
= =
( ) ' 0c = , cR ( )1
'k kx k x = ,
k R
( )sin ' cos( )x x= ( )cos ' sin( )x x=
( ) 21
tan 'cos
xx
= ( ) 'x xe e=
( ) 1ln 'xx
= ( ) ' ln( )x xa a a= ,
1 0a >
( )2
1arcsin '
1x
x=
( ) 21
arctan '1
xx
=+
l Hospital : ( ) ( ) 0f a g a= =
( ), ( )f a g a ( ) 0g a , ( ) ( ) ( )lim lim( ) ( ) ( )x a x a
f x f x f ag x g x g a
= =
: 0 0( ) ( ) 0f x g x= = , ( ), ( )f x g x ( , )a b ,
( ) 0g x , 0 ( , )x a b ,
0 0
( ) ( )lim lim( ) ( )x x x xf x f xg x g x
=
( ), ( )f x g x .
, , 0 m
1 // :1 /
f gg f =
( )0 :1/
ffgg
=
1 / 1 /:
1 /g ff g fg
=
0 00 , , 1+
lim ( ( ) ln ( ))( )lim ( ) x a g x f xg x
x af x e >
= ,
( ) ( ) ln ( )( )g x g x f xf x e=
fC :f AR .
( ) 0,f x > x I A , f .
( ) 0,f x < x I A , f .
0( ) 0f x = , 0x A >0 : ( ) 0f x > , 0 0x x x < x I A , f .
( ) 0,f x < x I A , f .
>0 ( ) 0f x > 0 0x x x < , 0x .
) 0( ) 0f x =
0( ) 0f x < , 0x .
x a= R ,
lim ( )x a
f x
= lim ( )x a
f x +
=
y b= , bR , lim ( )
xf x b
= lim ( )
xf x b
=
fC
y ax b= + , ( )lim ( ) 0x
f x ax b
=
( )lim lim( ( ) )x x
f xa f x ax b
x = = R R
* * * * * * * * * * * *
: [ , ]f a b R . Bolzano: f [ , ]a b
( ) ( ) 0f a f b < , 0 ( , )x a b 0( ) 0f x = .
: f [ , ]a b ( ) ( )f a f b , , ( )f a ( )f b 0 ( , )x a b 0( )f x = . - : f [ , ]a b , f [ , ]a b . 1 2, [ , ]x x a b 1 2( ) ( ) ( )f x f x f x , [ , ]x a b . (): f [ , ]a b ( , )a b , ( , )a b
: ( ) ( ) ( ).f b f a f
b a
=
Rolle: f [ , ]a b , ( , )a b , ( ) ( )f a f b= , ( , )a b :
( ) 0f = . f ( , )a b ,
( ) 0,f x = ( , )x a b , ( ) .f x c= Cauchy: ( ), ( )f x g x [ , ]a b , ( , )a b
( ) 0g x , ( , )x a b ,
( , )c a b : ( ) ( ) ( )( ) ( ) ( )f b f a f cg b g a g c
=
Darboux: f [ , ]a b ( ) ( )f a f b > cR ( ) ( )f b c f a < < , ( , )a b ( ) .f c = (, ( ) ( )f a f b < ).
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12 2012-2013
( )x f x=
a , f [ ], ,a h a h +
'( ) 1f x m< < , [ ],x a h a h + , [ ]0 ,x a h a h +
1( ),n nx f x = 1, 2,...,n = a . * * * * * * * * * * * *
f
( ) ( )b a
a b
f x dx f x dx= ( ) ( ) ( )
b c b
a a c
f x dx f x dx f x dx= + ( ) ( )b b
a acf x dx c f x dx=
( )( ) ( ) ( ) ( )b b b
a a a
f x g x dx f x dx g x dx+ = +
( ) ( ) ( ) ( )b b
a a
f x g x f x dx g x dx : f , [ , ]a b
( ) ( )( )b
a
f x dx f b a=
() ( )( ) ( ) ( ) ( )F x c f x dx F x c f x+ = + =
( ) ( )df x f x c= +
( )1 2 1 2( ) ( ) ( ) ( )c f x c h x dx c f x dx c h x dx+ = +
( )x g t= , ( ( )) '( ) ( )f g t g t dt f x dx= o
( ) ( ) ( ) ( ) ( ) ( )f x g x dx f x g x f x g x dx =
kdx kx c= +
1
, { 1}1
a
a xx dx c aa
+
= + + R
1 lndx x cx
= +
cos sinx dx x c= + sin cosxdx x c= +
1
2 2tan ( ) tan( )adx x xc arc c
x a a a
= + = ++
1
2 2sin ( ) sin( )dx x xc arc c
a aa x
= + = +
x xe dx e c= +
( ) ln | ( ) |( )
f x dx f x cf x
= +
. f [ , ]a b F
f , ( ) ( ) ( )b
a
f x dx F b F a= . f [ , ]a b ,
( ) ( )x
a
dF d f t dt f xdx dx
= = * * * * * * * * * * * *
( ) ( ) ( )limb
ba a
f x dx f x dx+
+=
( ) ( )limb b
aa
f x dx f x dx
=
( ) ( ) ( )0
limb b
a af x dx f x dx
+
=
( b )
( ) ( )0
limb b
ea ea
f x dx f x dx+
+
+
= ( a )
( ) = ,
( ) ( ) ( )0 0
lim limb c b e
e ea e ca
f x dx f x dx f x dx
+ ++
+
= +
a c b< < ( ,a b )
( ) ( ) ( )lim limc b
a ba c
f x dx f x dx f x dx+
+
= +
( ) ( ) ( )0
lim lima c a e
b eb c
f x dx f x dx f x dx
+
= + ( a )
( ) ( ) ( )0
lim limc b
bea e ca
f x dx f x dx f x dx+
+
+
++
= +
( a ) ( ),c a b
( ) ( ) ( )0 0
lim limb c e b
e ea a c e
f x dx f x dx f x dx+ +
+
= +
Cauchy
( ) ( ) ( )0
limb c e b
ea a c e
f x dx f x dx f x dx+
+
= +
(c ) Laplace : [0, )f + R
0
{ ( )}( ) ( )xtL f t x e f t dt+
= ,
x
.
* * * * * * * * * * * *
( ) , ( ) 0b
a
E f x dx f x=
( )2
1b
a
S f x dx= +
( ) ( )2
2 1b
ox
a
E f x f x dx = +
( )2
b
ox
a
V f x dx=
( ) ( )2 1b
a
E f x f x dx=
( ) ( )2 22 1b
ox
a
V f x f x dx = * * * * * * * * * * * *
-
.
: ( )na a n= .
: 1n na a+ = + , 1 ( 1) n a n = +
n ..: 1[2 ( 1) ]
2nn a nS + =
: 1n na a+ = 1
1n
na a =
n ..: 111
n
nS a
=
,
1 . : , ,a b c 3
.. 2b a c= .
xR n ( x )
1lim 0n n
= lim 1n
nn
= lim 0, 1n
nx x
=
lim 1n
x
n
xe
n + =
, lim 0!
n
n
x
n= .
: MR: ,na M n N.
: mR: ,nm a n N .
: ,
. ,m M R :nm a M , n N .
.
.
na , nN
, 1n na a + , n N .
, 1n na a + , n N .
, .
-
.
R .
.
lim 0nn
= ,n na n N
lim 0nn
a
= .
1lim /n nn
a a+= : 1p :
) :1
n
n
a
= , 1n n na b b +=
lim nn
b
.
: 1 lim nn
b b
.
) : ( )0
1 n nn
a
=
, 0na >
0na < 0 1 2n , , ,...= ) Taylor: f (1) (2) ( ), ,..., nf f f [ , ]a b ( )nf ( , )a b , ( ),a x
( ) ( )
( )
(1) (2)2
( )
( ) ( )( ) ( )1! 2!
( ) ( )!
nn
n
f a f af x f a x a x af a
x a R xn
= + + +
+ +
L
L
( )( 1)
1( )( ) ( 1)!n
n
n
fR x x an
+ +=
+
n-.
0a = , Maclaurin.
Taylor ( 0a = ) xR
2
12! !
nx x xe x
n= + + + + +L L
3 5 2 1
sin ( 1)3! 5! (2 1)!
nnx x xx x
n
+
= + + ++
L L
2 4 2
cos 1 ( 1)2! 4! (2 )!
nnx x xx
n= + + +L L
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12 2012-2013 -1< x < 1
2 3 1
ln(1 ) ( 1)2 3 1
nnx x xx x
n
+
+ = + + ++
L L
3 5 2 1
arctan 1 ( 1)3 5 2 1
nnx x xx
n
+
= + + ++
L L
) Fourier: :[ , ]f L L R 2L . Fourier f
0( ) ~ ( cos sin )n n
n
n x n xf x a bL L
=
+
01 ( )
2
L
L
a f x dxL
=
1 ( )cosL
n
L
n xa f x dx
L L
= , 1, 2,n = K
1 ( )sinL
n
L
n xb f x dxL L
= , 1, 2,n = K
. lim 0nn
a
, 0
n
n
a
= .
. ) 0
n
n
a
= ,
0n
n
b
= ,
,k R
0 0 0( )n n n n
n n n
ka b k a b
= = =
+ = + .
) 0
n
n
a
=
0n
n
b
= ,
0( )n n
n
a b
=
+ .
I. 0
n
n
| a |
= ,
0n
n
a
=
. .
V. ( ) 0 n na b .
0
n
n
b
= ,
0n
n
a
=
0
n
n
a
= ,
0n
n
b
= .
V. ( ) 0 na , 0 nb< , lim 0n
nn
ac
b= > .
0n
n
a
=
0n
n
b
=
.
VI. ( - d Alembert) 0na
0n n 1lim n
nn
a
a+
= . :
1 < , 0
n
n
a
=
1 > , 0
n
n
a
=
1 = , . VI. ( - Cauchy). 0na > lim n nn
a
=
1 < , 0
n
n
a
=
1 > , 0
n
n
a
=
1 = , .
VII. ( Leibnitz) ( )0
1 n nn
a
=
.
( )na , lim 0,
nn
a
= .
IX. ( ) - :[1, )f + R
( )1
I f x dx+
= ( )1n
S f n
=
=
: (1)I S I f< < + .
:( )
!! !
n
r
n nCr r n r
= =
( ) ( )( )
PP /
PA B
A BB
=
: ( ) ( ) ( )P P P .A B A B = i jA A = , i j 1 2 ... ,nA A A = :
( )P B = 1 1P( )P( / ) P( )P( / )n nA B A A B A+ +L
Bayes: P( )P( / )P( / )
P( )k k
kA B AA B
B=
(..) ( )X
.
.. ( )E X X : ( ) = ( )
x
E X xf x
.., : ( ) = ( ) E X xf x dx .
.., ( )f x (..) ( ..) (...) ( ..). H .. :
( ) ( )2 2var( ) = = ( )x xx
X E X x f x .. :
( ) ( )2 2var( ) = = ( ) x xX E X x f x dx
: [ ]( )22var( ) = X E X E X . ..
X () , : = var( )X X . X .. ( ). Y aX b= + :
( ) ( ) ( )E Y E aX b aE X b= + = + 2( ) ( ) ( )Var Y Var aX b a Var X= + =
: ( , ) : ( ) (1 )k n knB n p f k p pk
=
0,1,...,k n= ( ) ,E X np= ( ) (1 ).Var X np p=
Poisson ( )P : ( ) ,!
k
f k ek
= 0,1,...k =
( )E X = , ( )Var X = :
1(1 ) 1,2,...( ) : ( )0
kp p kG p f k ==
( ) 1 / ,E X p=
2( ) (1 ) /Var X p p= :
1( ) (1 ) ,1
kkf k p p
= , 1,...k = +
( ) /E X p= , 2( ) (1 ) /Var X p p=
:
1 2
( )
N Nk n kf k
Nn
=
10,..., min( , )k n N= , 1 2N N N+ = 1( ) NE X n
N= , 1 2( )
1N N N nVar X nN N N
=
: ( , )U a b 1
( )0
a x bf x b a
=
( ) ( ) / 2E X a b= + , 2( ) ( ) /12Var X b a=
( )2,N : 21
21( )2
x
f x e
=
x < <
( )E X = , 2( )Var X =
0( ) : ( )
0
a xa e xE a f x
=
( ) 1/E X a= , 2( ) 1/Var X a=
1 2, ,..., nX X X ( )iE X = . 2( )iVar X = ,
( )~ (0,1)n X
2
1~ ( , )
n
ii
X N n n = 30n .
* * * * * * * * * * * *
:
( ) 1 ... ...1
n n n n r r nn n
a b a a b a b br
+ = + + + + +
2 2 2( ) 2a b a ab b = + 3 3 2 2 3( ) 3 3a b a a b ab b = +
2 2 ( )( ) a b a b a b = + 3 3 2 2( )( ) a b a b a ab b = +m
1 2 3 2
2 3 2 1( )( ...
), 1, 2, 3, ...n n n n n
n n n
a b a b a a b a ba b ab b n
= + + +
+ + + =
(1 ) 1 , 0, 1, 2,3,...na na a n+ + > = * * * * * * * * * * * *
( xR ) sin( ) sin( ), cos( ) cos( )x x x x= =
2 2sin cos 1x x+ = , sintancos
xx
x=
sin( ) sin cos sin cosx y x y y x = cos( ) cos cos sin sinx y x y x y = m
( ) tan tantan1 tan tan
x yx y
x y
=m
22 tan
sin 2 2sin cos1 tan
xx x x
x= =
+
22 2 2
21 tan
cos 2 cos sin 2cos 11 tan
xx x x x
x
= = =
+
2
2 tantan 2
1 tanx
xx
=
, 2
211 tan
cos
+ =
sin sin 2sin cos2 2
x y x yx y = m
cos cos 2cos cos2 2
x y x yx y + + =
cos cos 2sin sin2 2
x y x yx y + =
sin(0) cos( / 2) 0, cos(0) sin( / 2) 1 = = = = sin( / 6) cos( / 3) 1/ 2 = =
2 3sin( ) cos( ) , sin( ) cos( )
4 4 2 3 6 2= = = =
* * * * * * * * * * * *
{ }| ,z x i y x y= = + C R : z x iy=
: 1
2
1 zz
z z
= =
:
2 2r z x y= = + 2 2| |r z z z= =
(cos sin )z r i = + , . De Moivre
( ) ( )( )cos sinn n in nz r e r n i n = = + , n n
nx z= , nN , ( n - z ),
2 2cos sin , 0,1, 1nk
k kz r i k n
n n
+ + = + =
K .