text.dvi
|
|
|
- ありおき しろみず
- 5 years ago
- Views:
Transcription
1 I
2 chap. Fourier Jean-Baptiste-Joseph Fourier ( ) Auxerre Ecole Polytrchnique Napoleon G.Monge Isere Napoleon Academie Francaise [] [ ] [] [] [ ] [ ] []
3 chap. + + Fourier Fourier Abel 3
4 chap. N Z Q C K 3 = K fg p(x) =a + a x + a x + :::+ a n x n dp dx = a +a x + :::+ na n x n p(x)dx = x a x + a + :::+ a n xn+ + C n+ e ax sin(ax) cos(ax) d dx eax e ax dx = ae ax = a eax d sin(ax) = dx a cos(ax) d cos(ax) = a sin(ax) dx sin(ax)dx = cos(ax) a cos(ax)dx = sin(ax) a n cos(mx) cos(nx)dx = n 8 >< cos(mx) sin(nx)dx = n sin(mx) sin(nx)dx = (fg) = f g + fg fgdx = fdx g ( fdx) g dx >: 8 >< >: ; m 6= n ; m = n 6= ; m = n = ; m 6= n ; m = n 6= ; m = n = 4
5 chap. i = p z = x + iy z = x iy jzj = zz = x + y e i = cos()+i sin() z = x + iy = re i (x; y) $ (r;) a n f (n) + a n f (n) + :::+ a f + a f = n a n z n + a n z (n) + :::+ a z + a = n ; ;:::; n c i ( i n) f(x) =c e x + c e x + :::+ c n e nx m (m >) ce x d + e x + d xe x + :::+ d m x (m) e x 5
6 chap. 3 f :! p f p f(x + p) =f(x) f(x) =f(x) f(x) =f(x) f(x) = ((f(x)+f(x))) + ((f(x) f(x))) a sin(x) sin(ax) a cos(x) cos(ax) 6
7 chap. 4 Fourier f :! L a n b n = L = L LL f(x) cos( nx )dx (n =; ; ;:::) L LL f(x) sin( nx )dx (n =; ; 3;:::) L Fourier Fourier f :! L a n b n Fourier Fourier a + X n= a n cos( nx L )+b n sin( nx L ) f f(x) f Fourier 7
8 chap. 4 Fourier Fourier f(x) = ( (5 x<) 3 ( x<5) a n = 55 f(x) cos( nx)dx 5 5 = 3 5 nx 5 cos( ( )dx 5 5 nx [sin( = 3 n 5 )]5 (n 6= ) 5 5 (n =) ( (n 6= ) = 3 (n =) b n = 5 nx 5 5 f(x) sin( )dx 5 = 3 5 nx 5 sin( )dx 5 = 3 5 nx [ cos( 5 n 5 )]5 ( cos(n)+) = 3 n 3 + P 3 nx n= ( cos(n)) sin( ) n 5 = sin( x )+ 3x sin( )+ 5x sin( )+::: Fourier Fourier ( 8 ( x<) f(x) = 8 ( x<4) 8
9 chap. 5 Fourier f(x) Z jf(x)jdx < Fourier D n (x) = X s n (x) = nx n ^f(k)e ikx nx n ^f(k)e ikx e ikx = sin(n + )x sin x Dirichlet D n (x)dx = (u; x) = (f(x + u)+f(x u)) Z s n (x) = (u; x)d n (u)du Dirichlet 9
10 chap. 5 Fourier x c (u; x) = (f(x+u)+f(x u) c) Z (u)d n (u)du! (n!) Dirichlet - Jordan f Fourier (f(x + ) + f(x )) lim n! s n(x) = (f(x +)+f(x )) : (bounded variation f :[a; b]! M [a; b] a = x <x <:::<x n = b nx i= jf(x i ) f(x i )j <M Fourier Fourier
11 chap. 5 Dini - Lipschitz!(f;t) f Z a!(f; t) dt < t Fourier f kf s n k! (n!): f :[a; b]!!(f; t) = sup <<t axb max jf(x + ) f(x)j: jump Gibbs phenomenon f :[; L)! [;c), (c; L) f(c +) f(c ) > kf s n k! ( Z sin x x dx )(f(c +)f(c )) (n!): Z sin x x dx =:8949 :::
12 chap. 6 Parseval f :! L fa n g fb n g Fourier Z L X jf(x)j dx = a L + (a n + b n): Parseval n= Fourier Dirichlet ( x ( x<) f(x) = x ( x<) a = f(x)dx = xdx = a n = nx f(x) cos( )dx = nx x cos( =[x n = 4 nx sin ] [cos nx (n) ] = 4 (cos(n) a) (n) b n = f(x) sin( nx)dx = )dx (n 6= ) nx sin n dx f(x) =+ P 4 n= (cos(n) ) cos( nx) (n) = 8 cos( x )+ cos( 3x)+ cos( 5x)+::: 3 5 Parseval Z jf(x)j dx = Z x dx =[ x3 3 ] = =+64 ( :::) ) :::= Fourier x = Parseval :::=
13 chap. 6 Fourier Series. f(x) =( x) ( x<) f Fourier Dirichlet X Parseval n 4 n= ( x; ( x<). f(x) = x; ( <x<) f Fourier Dirichlet X Parseval (n ) 4 n= a. (( x) n ) = n( x) (n) ; Z ( x) n = n + ( x)n+. b. Z f(x) sin(ax)dx = Z a f(x) cos(ax)+ a f (x) cos(ax)dx (a 6= ). c. Z f(x) cos(ax)dx = a f(x) sin(ax) a Z f (x) sin(ax)dx: (a 6= ). 3
14 chap. 8 Banach Hilbert X: X: p : X! X ) p(x) ; p(x) =, x = ) p(x + y) p(x) + p(y) 3) p(x) =jj p(x) p(x) x d(x; y) =p(x y) x y Banach X: X Banach : fx n g > N n;m>n ) d(a n ;a m ) < Cauchy Cauchy Cauchy Hilbert X: Banach Hilbert : ) <x;y>=< y;x> ) <x + x ;y >=< x ;y >+ <x ;y > 3) < x; y >= <x;y> 4) <x;x>; <x;x>=, x = kxk =<x;x> j <x;y>jkxkkyk Cauchy-Schwarz c x + c y = (c ;c ) 6= (; ) 4
15 chap. 8 L p (T) f :! L p - 8 < kfk p = : `p p = =p jf(x)j dx p ( p<) sup x< jf(x)j (p = ) <f;g>= Hilbert fa n g `p- kfa n gk p = Z f(x)g(x)dx ( ( P n= ja n j p ) =p ( p<) sup ja n j (p = ) p = < fa n g; fb n g >= X a n bn Hilbert L p () f :! L p - 8 < =p jf(x)j kfk p = p dx ( p<) : sup <x< jf(x)j (p = ) p = <f;g>= Hilbert f(x)g(x)dx 5
16 chap. 8 Fourier f(x) = a + P n= (a n cos(nx)+b n sin(nx)) = P n= c n e inx a n = c n + c n b n = i(c n c n ) c n = Z f(x)e inx dx Parseval T : L (T)! ` T (f) =fc n g kfk = kt (f)k T Hilbert L (T) = ` fe inx g L (T) ; cos(x); sin(x); cos(x); sin(x); cos(3x); sin(3x);::: L (T) : f n g Hilbert X k n k = < n ; m >= (n 6= m) X v v = P c n n Hilbert n ` 6
17 chap. 8 Sturm - Liouville [a; b] d dx p(x)df(x) dx! (r(x)+q(x))f(x) = f(a)+ f (a) =; f(b)+ f (b) = f Sturm - Liouville <f;g>= Z b r(x) a f(x)g(x)r(x)dx [a; b] =[;] sin(nx) f + f =; f() = ; f() = 7
18 chap. 8 Legendre : ( x )y xy + n(n +)y = L ([; ]); P n (x) = d n n n! dx n (x ) n Tchebyche : ( x )y xy + n y = L ([; ]; p dx); T n(x) = ()n p x x dn (n )!! dx ( n x ) n Hermite : y xy + ny = L (;e x = dx); H n (x) =() n e x = dn dx n (ex = ) Laguerre : xy +( x)y + ny = L ([; );e x dx); L n (x) = ex n! d n dx n (ex x n ) Jacobi : x( x)y +[ x]y + n y = L ([; ];x dx); G n (; x) = ()x ( x) ( + n) d n dx n (x+n ( x) n ) 8
19 chap. 7 [; 3] u(x; t) t x ( x 3) u(;t)=u(3;t)= ( u(x; ) = Step : u(x; t) =X(x)T (t) X(x)T (t) =X (x)t (t) (a) T (t) =T(t) T (t) T (t) = X (x) X(x) = T dt = Z dt T (t) =Ce t < = T (t) =Ce t (b) X (x) = X(x) z + = z = 6i X(x) = c e ix + c e ix = d cos(x)+d sin(x) u(x; t) =e t (A cos(x)+b sin(x)) 9
20 chap. 7 Step : A =. =u(;t)=e t A cos(x) =u(3;t)=e t B sin(3) = n 3 (n Z) B = Step 3: u(x; t) = P n= Bn n e( 3 )t sin( nx = P ( n n= B n e 3 )t sin( nx ) 3 ) 3 Step 4: 5 = u(x; ) = X n= B n sin( n 3 x) 5 [3; 3] Fourier a n = b n = 33 f(x) sin( nx)dx 3 3 = sin( nx)dx 3 = 5 3 nx [ cos( 3 n 3 )]3 = 5 n B n = 5 ( cos(n)+) n ( cos(n)+) u(x; t) = P 5 ( n= n cos(n)+)e n t 9 sin( nx = e t 9 sin( x 3 )+ 3 et sin(x)+ 5 e 5 t 9 sin( 5x 3 ) 3 )+:::
21 chap. 7 L [;L] u(x; t) t x ( x L) u(;t)=u(l; t) = ( u(x; ) = f(x) u t (x; ) = Step : u(x; t) =X(x)T (t) u(x; t) =(A cos(at)+b sin(at))(a cos(x)+b sin(x)) Step : =u(;t)=u(l; t) A = B = = n L (n Z) Step 3: u(x; t) = P n= Cn sin( nx nat ) cos( ) L L = P n= C n sin( nx nat ) cos( ) L L Step 4: u(x; ) = f(x) f(x) [L; L] Fourier u(x; t) C n = L Z L L f(x) sin( nx L )dx = P n= C n sin( nx nat ) cos( ) L L = P C n n= sin( n(x+at) ) + sin( n(xat) L L (f(x + at)+f(x at)) =
22 chap. 9 Fourier Fourier L f :! fe i n L x g (n Z) L L! f(x) = P n= c n e i n L x = P LL n= f(y)e = L i n L LL P f(y) n= L y dy e i n L x L ein L (xy) dy L! f(x) = f Fourier ^f() = f(x) = f(y)e iy dy e ix d f(y)e iy dy ^f()e ix d f f L () k ^fk kfk ^f j ^f()j! (jj!) iemann - Lebesgue Fourier Sin Cos F s () = f(x) sin(x)dx; F c () = Fourier Sin Cos ^f() =(F s () if s ()) f(x) cos(x)dx
23 chap. 9 f(x) = ( (jxj a) (jxj >a) Fourier ^f() = f(x)e ix dx = a a e ix dx =[ i eix ] a a = i (eix e ix ) = sin(a) f(x) = ( x (jxj < ) (jxj ) Fourier e ix dx x e ix dx =[ = sin() i eix x ] + i = i (ei e i )+ i = sin() + (e i + e i ) xe ix dx [ eix x] + sin() e ix dx ^f() = ( x )e ix dx = 4 cos sin Fourier Cos f(x) =e mx (m>) e ax cos(bx)dx = eax (a cos(bx)+bsin(bx)) a +b e ax sin(bx)dx = eax (b cos(bx)+asin(bx)) a +b 3
24 chap. Fourier Dirichlet f L () f f ^f()e ix d = (f(x +)+f(x ) Parseval f; g L () \ L () f(x)g(x)dx = ^f()^g()d jf(x)j dx = j ^f()j d Plancherel Fourier L () L () f()e ix d f; g L () f 3 g(x) = f(t)g(x t)dt (f 3 g)^() = ^f()^g() 4
25 chap. f(x) = ( (jxj a) (jxj >a) Fourier sin(a) sin(a) e ix d = 8 >< >: Dirichlet jxj <a jxj = a jxj >a sin(a) cos(x) d = 8 >< >: jxj <a jxj = a jxj >a a = x = sin d = e x (x ) Fourier Sin x(mx) x dx = + em (m>) m = x x +) dx = 4 ( ( x<) 3 f(x) = Fourier (x ) cos x dx = ; sin x dx = x x 5
26 chap. [; ) u(x; t) t x ( x<) u(;t)= ( u(x; ) = Step : u(x; t) =X(x)T (t) u(x; t) =Be at (A cos(x)+bsin(x)) Step : =u(;t)=e at A A =. Step 3: Step 4: u(x; t) = f(x) =u(x; ) = B() = B()e at sin(x)d B() sin(x)d f(x) sin(x)dx u(x; t) == f(u) sin(u)du e at sin(x)d 6
27 chap. (; x) =i^u(; ^ (; x) =^u(; (; t) = d ^u(; t) dt Fourier ^u(; t) =a ^(; t) ^u(; t) =Ce a t u(x; ) = f(x) ^u(; ) = ^f() C = ^u(; t) = ^f()e a t 4at x e 4at A ^ = e a t E t (x) u(x; t) = f 3 E t (x) = f(y) q 4at (xy) e 4at dy = f(x z p at)e z dz 7
28 chap. [; ) u(x; t) t x ( x L) u(;t)= ( u(x; ) = f(x) u t (x; ) = Step : u(x; t) =X(x)T (t) u(x; t) =(A cos(at)+b sin(at))(a cos(x)+b sin(x)) Step : =u t (x; t) B = Step 3: u(x; t) = Step 4: u(x; ) = f(x) f(x) = (A() cos(x) +B() sin(x)) cos(at)d (A() cos(x) +B() sin(x))d A() = f(x) cos(x)dx; B() = f(x) sin(x)dx; u(x; t) = = = = f(v)(cos(v) cos(x) + sin(v) sin(x)) cos(at)ddv f(v) cos((x v)) cos(at)dvd f(v)(cos((x v + at)) + cos((x v at)))dvd (f(x + at)+f(x at)) 8
29 chap. Fourier - f L () x jf(x)j dx j ^f()j d 4 kfk4 f(x) =ce ax (a>) f Cc () f L () j ^f()j = j ^f()j = ji ^f()j = j(f )^()j x jf(x)j dx j ^f()j d = x jf(x)j dx j(f )^()j = x jf(x)j dx jf (x)j dx j x f(x)f (x)dxj <(x f(x)f (x))dx = x(f f + f f ) dx = x(jf(x)j ) dx = [xjf(x)j ] jf(x)j dx = 4 kfk4 f(x) x =, x jf(x)j dx ^f() =, j ^f()j d f(x) x = ^f() = 9
30 chap. 4 Fourier w(x); xw(x) L () w b; (x) =w(x b)e ix ^w b; () ^f(; b) = w(x b)e ix e ix dx = w(x)e i(x+b) e i(x+b) dx = e ib e ib w(x)e ix(+) dx = e ib e ib ^w( ) = V b; () = f(x)w(x b)e ix dx =< f;w b; > =< ^f; ^w b; > =< ^f;v b; > w b; f x = b V b; ^f = (x; ) =(b; ) W V W V V (); V () L () ^f(; b) Fourier STFT w(x) = p x e ^f(; b) = p f(x)e (xb) e ix dx Gabor 946 3
31 chap. 5 Wavelet L () \ L () a > b a;b(x) = p a x b a! ^a;b () = p xb a = e ib p a a = p ae ib ^(a) e ix d x a e ix d f L () f Wavelet W f(b; a) =<f; a;b >= p a f(x) x b a! dx W f(b; a) =< f; a;b > =< ^f; ^a;b > = p a ^f() ^(a)e ib d admissible condition d j ^()j jj = C < ^ ^() = (x)dx = L () H () Hardy H () =ff L () ; ^f() = ( <)g 3
32 chap. 5 f H () W f(b; a) = p a ^f() ^(a)e ib d ^W f(; a) = p a ^f() ^(a) a;b (x) b Fourier ^9(a; x; ) ^9(a; x; ) = p xb a = e ix a p a b = p ae ix ^(a) W f(b; a) e ib db e ib db a p a xb a db da a = <W f(;a); ;a(x) > da a = < ^W f(;a); ^9(a; x; ) > da = = = = C a aj ^(a)j e ix da ^f()d a aj ^(a)j e ix da ^f()d a j ^(a )j da e ix ^f()d a ^f()e ix d = C ^f()e ix d = C f(x) L ()\L () f H () f(x) = W f(b; a) b;a (x)db da C a Z d j ^()j jj = d j ^()j jj = C psi < f L () 3
33 chap. 5 Wavelet (x) = p x e a e ix Gabor W f(b; a) = p a f(x) p e (xb) a e xb i( a ) dx (x) =( x )e x (x) = 8 >< >: ( x<) (3 x ; x 3) (jxj > 3) (x) =f(x) f(x); f(x) = sin(x) x Shannon C () Meyer C c (n) () Daubechies 33
34 chap., f(x) (x) j (x)j j (x)j dx = (x; x +x) j (x)j x j ^()j ^() = p (x)eix dx j ^()j d = (h; h( +)) j ^()j (x; t) =! + V (x) (x; t) V, m V, f(x) = (x; ) = (x; t) = h ix ^f()e = a 4 e x a a +im x m t) d 4 e a (m ) h ix e m t) d! j (x; t)j = (a +( h ma ) t ) e a +( ma h (x hm ) t m t) 34
35 chap. t hm m t, a +( h ma ) t e i!()t!()!() =!() =!() =q c + c tanh c 3 3!() =( c ) ( ) +c 3!() = p c + c 35
body.dvi
..1 f(x) n = 1 b n = 1 f f(x) cos nx dx, n =, 1,,... f(x) sin nx dx, n =1,, 3,... f(x) = + ( n cos nx + b n sin nx) n=1 1 1 5 1.1........................... 5 1.......................... 14 1.3...........................
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
main.dvi
Chpter 5 5. [; ] f(x) 3 f(x) = X + ( n cos nx + b n nx ) (5:) n= (5.) ; ;...;b ;b ;... n; m > 3 Z Z Z mx cos mx mx nx nx cos ( : n 6= m dx = : n = m ( : n 6= m dx = : n = m nx cos dx = (5.) (5.) (5.) 3
(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)
1. 1.1...,. 1.1.1 V, V x, y, x y x + y x + y V,, V x α, αx αx V,, (i) (viii) : x, y, z V, α, β C, (i) x + y = y + x. (ii) (x + y) + z = x + (y + z). 1 (iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y
30 I .............................................2........................................3................................................4.......................................... 2.5..........................................
5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (
![5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (](/thumbs/92/107866626.jpg "5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (")
5 5.1 [ ] ) d f(t) + a d f(t) + bf(t) : f(t) 1 dt dt ) u(x, t) c u(x, t) : u(x, t) t x : ( ) ) 1 : y + ay, : y + ay + by : ( ) 1 ) : y + ay, : yy + ay 3 ( ): ( ) ) : y + ay, : y + ay b [],,, [ ] au xx
n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)
D d dx 1 1.1 n d n y a 0 dx n + a d n 1 y 1 dx n 1 +... + a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) +... + a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y
1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2
1 Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier analog digital Fourier Fourier Fourier Fourier Fourier Fourier Green Fourier
Fourier Fourier Fourier etc * 1 Fourier Fourier Fourier (DFT Fourier (FFT Heat Equation, Fourier Series, Fourier Transform, Discrete Fourier Transform, etc Yoshifumi TAKEDA 1 Abstract Suppose that u is
y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x
![, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x](/thumbs/93/113428934.jpg ", 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x")
1 1.1 4n 2 x, x 1 2n f n (x) = 4n 2 ( 1 x), 1 x 1 n 2n n, 1 x n n 1 1 f n (x)dx = 1, n = 1, 2,.. 1 lim 1 lim 1 f n (x)dx = 1 lim f n(x) = ( lim f n (x))dx = f n (x)dx 1 ( lim f n (x))dx d dx ( lim f d
simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a =
II 6 ishimori@phys.titech.ac.jp 6.. 5.4.. f Rx = f Lx = fx fx + lim = lim x x + x x f c = f x + x < c < x x x + lim x x fx fx x x = lim x x f c = f x x < c < x cosmx cosxdx = {cosm x + cosm + x} dx = [
1 yousuke.itoh/lecture-notes.html [0, π) f(x) = x π 2. [0, π) f(x) = x 2π 3. [0, π) f(x) = x 2π 1.2. Euler α
1 http://sasuke.hep.osaka-cu.ac.jp/ yousuke.itoh/lecture-notes.html 1.1. 1. [, π) f(x) = x π 2. [, π) f(x) = x 2π 3. [, π) f(x) = x 2π 1.2. Euler dx = 2π, cos mxdx =, sin mxdx =, cos nx cos mxdx = πδ mn,
No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y
No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ
z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy
f f x, y, u, v, r, θ r > = x + iy, f = u + iv C γ D f f D f f, Rm,. = x + iy = re iθ = r cos θ + i sin θ = x iy = re iθ = r cos θ i sin θ x = + = Re, y = = Im i r = = = x + y θ = arg = arctan y x e i =
i 6 3 ii 3 7 8 9 3 6 iii 5 8 5 3 7 8 v...................................................... 5.3....................... 7 3........................ 3.................3.......................... 8 3 35
untitled
40 4 4.3 I (1) I f (x) C [x 0 x 1 ] f (x) f 1 (x) (3.18) f (x) ; f 1 (x) = 1! f 00 ((x))(x ; x 0 )(x ; x 1 ) (x 0 (x) x 1 ) (4.8) (3.18) x (x) x 0 x 1 Z x1 f (x) dx ; Z x1 f 1 (x) dx = Z x1 = 1 1 Z x1
x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f
,,,,.,,,. R f : R R R a R, f(a + ) f(a) lim 0 (), df dx (a) f (a), f(x) x a, f (a), f(x) x a ( ). y f(a + ) y f(x) f(a+) f(a) f(a + ) f(a) f(a) x a 0 a a + x 0 a a + x y y f(x) 0 : 0, f(a+) f(a)., f(x)
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
main.dvi
Chapter 6 6. (5.8)(5.9) f(x) c n = a n= a a c n exp(i nx a )= [f(x +)+f(x )]; x [a; a] f (x) exp(i nx )dx (6.) a k n = n a ; k n = k n+ k n = a (6:) (6.) c n c n = k n (6.) a a df()e ikn : a [f (x +)+f(x
Microsoft Word - 信号処理3.doc
Junji OHTSUBO 2012 FFT FFT SN sin cos x v ψ(x,t) = f (x vt) (1.1) t=0 (1.1) ψ(x,t) = A 0 cos{k(x vt) + φ} = A 0 cos(kx ωt + φ) (1.2) A 0 v=ω/k φ ω k 1.3 (1.2) (1.2) (1.2) (1.1) 1.1 c c = a + ib, a = Re[c],
leb224w.dvi
2 4 i Lebesgue Fourier 7 5 Lebesgue Walter. F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Publ. Inc., New York (99) ( 49 ) 2. ( 8 ) 3. A.2 Fourier Laplace (957 ) 4. (98 ) 5. G. G. Walter, Wavelets
W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
2S III IV K A4 12:00-13:30 Cafe David 1 2 TA 1 appointment Cafe David K2-2S04-00 : C
2S III IV K200 : April 16, 2004 Version : 1.1 TA M2 TA 1 10 2 n 1 ɛ-δ 5 15 20 20 45 K2-2S04-00 : C 2S III IV K200 60 60 74 75 89 90 1 email 3 4 30 A4 12:00-13:30 Cafe David 1 2 TA 1 email appointment Cafe
() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.
![() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.](/thumbs/89/98384840.jpg "() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.")
() 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >
Lebesgue Fubini L p Banach, Hilbert Höld
II (Analysis II) Lebesgue (Applications of Lebesgue Integral Theory) 1 (Seiji HIABA) 1 ( ),,, ( ) 1 1 1.1 1 Lebesgue........................ 1 1.2 2 Fubini...................... 2 2 L p 5 2.1 Banach, Hilbert..............................
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29
![[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29](/thumbs/93/114156928.jpg "[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29")
A p./29 [ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29 [ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx [ ] F(x) f(x) C F(x) + C f(x) A p.2/29 [ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx [ ] F(x) f(x) C F(x)
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
17 3 31 1 1 3 2 5 3 9 4 10 5 15 6 21 7 29 8 31 9 35 10 38 11 41 12 43 13 46 14 48 2 15 Radon CT 49 16 50 17 53 A 55 1 (oscillation phenomena) e iθ = cos θ + i sin θ, cos θ = eiθ + e iθ 2, sin θ = eiθ e
211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3)
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3) 6 y = g(x) x = 1 g( 1) = 2 ( 1) 3 = 2 ; g 0 ( 1) =
( z = x 3 y + y ( z = cos(x y ( 8 ( s8.7 y = xe x ( 8 ( s83.8 ( ( + xdx ( cos 3 xdx t = sin x ( 8 ( s84 ( 8 ( s85. C : y = x + 4, l : y = x + a,
[ ] 8 IC. y d y dx = ( dy dx ( p = dy p y dx ( ( ( 8 ( s8. 3 A A = ( A ( A (3 A P A P AP.3 π y(x = { ( 8 ( s8 x ( π < x x ( < x π y(x π π O π x ( 8 ( s83.4 f (x, y, z grad(f ( ( ( f f f grad(f = i + j
(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t
![(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t](/thumbs/67/56612994.jpg "(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t")
6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]
,
2014 10 3, 2018 2 28 F ( ) A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 24 ( ) A α a alpha
ft. ft τfτdτ = e t.5.. fx = x [ π, π] n sinnx n n=. π a π a, x [ π, π] x = a n cosnx cosna + 4 n=. 3, x [ π, π] x 3 π x = n sinnx. n=.6 f, t gt n 3 n
![ft. ft τfτdτ = e t.5.. fx = x [ π, π] n sinnx n n=. π a π a, x [ π, π] x = a n cosnx cosna + 4 n=. 3, x [ π, π] x 3 π x = n sinnx. n=.6 f, t gt n 3 n](/thumbs/92/108735706.jpg "ft. ft τfτdτ = e t.5.. fx = x [ π, π] n sinnx n n=. π a π a, x [ π, π] x = a n cosnx cosna + 4 n=. 3, x [ π, π] x 3 π x = n sinnx. n=.6 f, t gt n 3 n")
[ ]. A = IC X n 3 expx = E + expta t : n! n=. fx π x π. { π x < fx = x π fx F k F k = π 9 s9 fxe ikx dx, i =. F k. { x x fx = x >.3 ft = cosωt F s = s4 e st ftdt., e, s. s = c + iφ., i, c, φ., Gφ = lim
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
2 2 L 5 2. L L L L k.....
L 528 206 2 9 2 2 L 5 2. L........................... 5 2.2 L................................... 7 2............................... 9. L..................2 L k........................ 2 4 I 5 4. I...................................
J1-a.dvi
4 [ ] 4. ( ) (x ) 3 (x +). f(x) (x ) 3 x 3 3x +3x, g(x) x + f(x) g(x) f(x) (x 3)(x +)+x+ (x 3)g(x)+x+ g(x) r(x) x+ g(x) x (x+)+ x r(x)+ g(x) x x 4x+5 r(x) g(x) x (f(x) (x 3)g(x)) g(x) x f(x) f(x)g(x) 4
29
9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n
main.dvi
9 5.4.3 9 49 5 9 9. 9.. z (z) = e t t z dt (9.) z z = x> (x +)= e t t x dt = e t t x e t t x dt = x(x) (9.) t= +x x n () = (n +) =!= e t dt = (9.3) z
B2 ( 19 ) Lebesgue ( ) ( ) 0 This note is c 2007 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercia
B2 ( 19) Lebesgue ( ) ( 19 7 12 ) 0 This note is c 2007 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercial purposes. i Riemann f n : [0, 1] R 1, x = k (1 m
6. Euler x
...............................................................................3......................................... 4.4................................... 5.5......................................
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt
S I. x yx y y, y,. F x, y, y, y,, y n http://ayapin.film.s.dendai.ac.jp/~matuda n /TeX/lecture.html PDF PS yx.................................... 3.3.................... 9.4................5..............
z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy
z fz fz x, y, u, v, r, θ r > z = x + iy, f = u + iv γ D fz fz D fz fz z, Rm z, z. z = x + iy = re iθ = r cos θ + i sin θ z = x iy = re iθ = r cos θ i sin θ x = z + z = Re z, y = z z = Im z i r = z = z
1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2
filename=mathformula58.tex ax + bx + c =, x = b ± b 4ac, (.) a x + x = b a, x x = c a, (.) ax + b x + c =, x = b ± b ac. a (.3). sin(a ± B) = sin A cos B ± cos A sin B, (.) cos(a ± B) = cos A cos B sin
ルベーグ積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
ルベーグ積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/005431 このサンプルページの内容は, 初版 1 刷発行時のものです. Lebesgue 1 2 4 4 1 2 5 6 λ a
4................................. 4................................. 4 6................................. 6................................. 9.................................................... 3..3..........................
2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( (
(. x y y x f y = f(x y x y = y(x y x y dx = d dx y(x = y (x = f (x y = y(x x ( (differential equation ( + y 2 dx + xy = 0 dx = xy + y 2 2 2 x y 2 F (x, y = xy + y 2 y = y(x x x xy(x = F (x, y(x + y(x 2
Fourier (a) C, (b) C, (c) f 2 (a), (b) (c) (L 2 ) (a) C x : f(x) = a 0 2 + (a n cos nx + b n sin nx). ( N ) a 0 f(x) = lim N 2 + (a n cos nx + b n sin
( ) 205 6 Fourier f : R C () (2) f(x) = a 0 2 + (a n cos nx + b n sin nx), n= a n = f(x) cos nx dx, b n = π π f(x) sin nx dx a n, b n f Fourier, (3) f Fourier or No. ) 5, Fourier (3) (4) f(x) = c n = n=
phs.dvi
483F 3 6.........3... 6.4... 7 7.... 7.... 9.5 N (... 3.6 N (... 5.7... 5 3 6 3.... 6 3.... 7 3.3... 9 3.4... 3 4 7 4.... 7 4.... 9 4.3... 3 4.4... 34 4.4.... 34 4.4.... 35 4.5... 38 4.6... 39 5 4 5....
2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p
2012 IA 8 I 1 10 10 29 1. [0, 1] n x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 2. 1 x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 1 0 f(x)dx 3. < b < c [, c] b [, c] 4. [, b] f(x) 1 f(x) 1 f(x) [, b] 5.
1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D
1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00
f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a
3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (
f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f
22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
A 2008 10 (2010 4 ) 1 1 1.1................................. 1 1.2..................................... 1 1.3............................ 3 1.3.1............................. 3 1.3.2..................................
4 R f(x)dx = f(z) f(z) R f(z) = lim R f(x) p(x) q(x) f(x) = p(x) q(x) = [ q(x) [ p(x) + p(x) [ q(x) dx =πi Res(z ) + Res(z )+ + Res(z n ) Res(z k ) k
f(x) f(z) z = x + i f(z). x f(x) + R f(x)dx = lim f(x)dx. R + f(x)dx = = lim R f(x)dx + f(x)dx f(x)dx + lim R R f(x)dx Im z R Re z.: +R. R f(z) = R f(x)dx + f(z) 3 4 R f(x)dx = f(z) f(z) R f(z) = lim R
ii
ii iii 1 1 1.1..................................... 1 1.2................................... 3 1.3........................... 4 2 9 2.1.................................. 9 2.2...............................
2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta
009 IA 5 I, 3, 4, 5, 6, 7 6 3. () Arcsin ( (4) Arccos ) 3 () Arcsin( ) (3) Arccos (5) Arctan (6) Arctan ( 3 ) 3. n () tan x (nπ π/, nπ + π/) f n (x) f n (x) fn (x) Arctan x () sin x [nπ π/, nπ +π/] g n
(u(x)v(x)) = u (x)v(x) + u(x)v (x) ( ) u(x) = u (x)v(x) u(x)v (x) v(x) v(x) 2 y = g(t), t = f(x) y = g(f(x)) dy dx dy dx = dy dt dt dx., y, f, g y = f (g(x))g (x). ( (f(g(x)). ). [ ] y = e ax+b (a, b )
.1 1,... ( )
1 δ( ε )δ 2 f(b) f(a) slope f (c) = f(b) f(a) b a a c b 1 213 3 21. 2 [e-mail] nobuo@math.kyoto-u.ac.jp, [URL] http://www.math.kyoto-u.ac.jp/ nobuo 1 .1 1,... ( ) 2.1....................................
<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>
電気電子数学入門 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/073471 このサンプルページの内容は, 初版 1 刷発行当時のものです. i 14 (tool) [ ] IT ( ) PC (EXCEL) HP() 1 1 4 15 3 010 9 ii 1... 1 1.1 1 1.
A
A 2563 15 4 21 1 3 1.1................................................ 3 1.2............................................. 3 2 3 2.1......................................... 3 2.2............................................
0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9
1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ
,
2014 10 3, 2017 1 28 ( ) A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 24 ( ) A α a alpha ǽlf@
II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K
II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d
S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....
1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1
sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V
2014 S hara/lectures/lectures-j.html r 1 S phone: ,
14 S1-1+13 http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 1 S1-1+13 14.4.11. 19 phone: 9-8-4441, e-mail: hara@math.kyushu-u.ac.jp Office hours: 1 4/11 web download. I. 1. ϵ-δ 1. 3.1, 3..
- II
- II- - -.................................................................................................... 3.3.............................................. 4 6...........................................
i I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................
入試の軌跡
4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf
Chap11.dvi
. () x 3 + dx () (x )(x ) dx + sin x sin x( + cos x) dx () x 3 3 x + + 3 x + 3 x x + x 3 + dx 3 x + dx 6 x x x + dx + 3 log x + 6 log x x + + 3 rctn ( ) dx x + 3 4 ( x 3 ) + C x () t x t tn x dx x. t x
DVIOUT
A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3
2 2 1 5 5 Schrödinger i u t + u = λ u 2 u. u = u(t, x 1,..., x d ) : R R d C λ i = 1 := 2 + + 2 x 2 1 x 2 d d Euclid Laplace Schrödinger 3 1 1.1 N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3,... } Q
振動と波動
Report JS0.5 J Simplicity February 4, 2012 1 J Simplicity HOME http://www.jsimplicity.com/ Preface 2 Report 2 Contents I 5 1 6 1.1..................................... 6 1.2 1 1:................ 7 1.3
Morse ( ) 2014
Morse ( ) 2014 1 1 Morse 1 1.1 Morse................................ 1 1.2 Morse.............................. 7 2 12 2.1....................... 12 2.2.................. 13 2.3 Smale..............................
i
i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,
I y = f(x) a I a x I x = a + x 1 f(x) f(a) x a = f(a + x) f(a) x (11.1) x a x 0 f(x) f(a) f(a + x) f(a) lim = lim x a x a x 0 x (11.2) f(x) x
11 11.1 I y = a I a x I x = a + 1 f(a) x a = f(a +) f(a) (11.1) x a 0 f(a) f(a +) f(a) = x a x a 0 (11.) x = a a f (a) d df f(a) (a) I dx dx I I I f (x) d df dx dx (x) [a, b] x a ( 0) x a (a, b) () [a,
i 3 Mathematica Fourier Fourier Dirac Dirac Fourier Haar 7.4 (MRA) 8 R L 2 {V j } j Z j Z V j V j+1 j Z V j = L 2 (R) f(2x) V j+1 f(
Fourier 解析とウェーブレットの基礎 水谷正大著 大東文化大学経営研究所 i 3 Mathematica Fourier 1 5.4 Fourier Dirac Dirac 2.3 4 Fourier 8 6 7 Haar 7.4 (MRA) 8 R L 2 {V j } j Z j Z V j V j+1 j Z V j = L 2 (R) f(2x) V j+1 f(x) V j V 0
App. of Leb. Integral Theory (S. Hiraba) Lebesgue (X, F, µ) (measure space)., X, 2 X, F 2 X σ (σ-field), i.e., (1) F, (2) A F = A c F, (3)
Lebesgue (Applications of Lebesgue Integral Theory) (Seiji HIABA) 1 1 1.1 1 Lebesgue........................ 1 1.2 2 Fubini...................... 2 2 L p 5 2.1 Banach, Hilbert..............................
000 001
all-round catalogue vol.2 000 001 002 003 AA0102 AA0201 AA0701 AA0801 artistic brushes AA0602 AB2701 AB2702 AB2703 AB2704 AA0301 AH3001 AH3011 AH3101 AH3201 AH3111 AB3201 AB3202 AB2601 AB2602 AB0701 artistic
I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google
I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59
..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A
![..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A](/thumbs/93/112175219.jpg "..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A")
.. Laplace ). A... i),. ω i i ). {ω,..., ω } Ω,. ii) Ω. Ω. A ) r, A P A) P A) r... ).. Ω {,, 3, 4, 5, 6}. i i 6). A {, 4, 6} P A) P A) 3 6. ).. i, j i, j) ) Ω {i, j) i 6, j 6}., 36. A. A {i, j) i j }.
1/1 lim f(x, y) (x,y) (a,b) ( ) ( ) lim limf(x, y) lim lim f(x, y) x a y b y b x a ( ) ( ) xy x lim lim lim lim x y x y x + y y x x + y x x lim x x 1
1/5 ( ) Taylor ( 7.1) (x, y) f(x, y) f(x, y) x + y, xy, e x y,... 1 R {(x, y) x, y R} f(x, y) x y,xy e y log x,... R {(x, y, z) (x, y),z f(x, y)} R 3 z 1 (x + y ) z ax + by + c x 1 z ax + by + c y x +
X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
Z: Q: R: C: 3. Green Cauchy
7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................
(2000 )
(000) < > = = = (BC 67» BC 1) 3.14 10 (= ) 18 ( 00 ) ( ¼"½ '"½ &) ¼ 18 ¼ 0 ¼ =3:141596535897933846 ¼ 1 5cm ` ¼ = ` 5 = ` 10 () ` =10¼ (cm) (1) 3cm () r () () (1) r () r 1 4 (3) r, 60 ± 1 < > µ AB ` µ ±
実解析的方法とはどのようなものか
(1) ENCOUNTER with MATHEMATICS 2001 10 26 (2) 1807 J. B. J. Fourier 1 2π f(x) f(x) = n= c n (f) = 1 2π c n (f)e inx (1) π π f(t)e int dt Fourier 2 R f(x) f(x) = F[f](ξ)= 1 2π F [f](ξ)e ixξ dξ f(t)e iξx
応力とひずみ.ppt
in yukawa@numse.nagoya-u.ac.jp 2 3 4 5 x 2 6 Continuum) 7 8 9 F F 10 F L L F L 1 L F L F L F 11 F L F F L F L L L 1 L 2 12 F L F! A A! S! = F S 13 F L L F F n = F " cos# F t = F " sin# S $ = S cos# S S
( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n
04.dvi
22 I 4-4 ( ) 4, [,b] 4 [,b] R, x =, x n = b, x i < x i+ n + = {x,,x n } [,b], = mx{ x i+ x i } 2 [,b] = {x,,x n }, ξ = {ξ,,ξ n }, x i ξ i x i, [,b] f: S,ξ (f) S,ξ (f) = n i= f(ξ i )(x i x i ) 3 [,b] f:,
2009 IA I 22, 23, 24, 25, 26, a h f(x) x x a h
009 IA I, 3, 4, 5, 6, 7 7 7 4 5 h fx) x x h 4 5 4 5 1 3 1.1........................... 3 1........................... 4 1.3..................................... 6 1.4.............................. 8 1.4.1..............................
II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1
II 2014 1 1 I 1.1 72 r 2 72 8 72/8 = 9 9 2 a 0 1 a 1 a 1 = a 0 (1+r/100) 2 a 2 a 2 = a 1 (1 + r/100) = a 0 (1 + r/100) 2 n a n = a 0 (1 + r/100) n a n a 0 2 n a 0 (1 + r/100) n = 2a 0 (1 + r/100) n = 2
IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................
18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C
8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,