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
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1 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 = u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1
2 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 (x) = y 1 (x) y 2 (x) dx y + ay + a2 4 y = 0 1 y = e ρx ρ ρ 2 + aρ + a2 4 = 0 1 (ρ + a 2 )2 = 0 ρ = a 2 y = e a 2 x R y 2 (x) = e a e 2 x P (x)dx R y 2 (x) dx = e a e 2 x adx e ax 2 dx = e a e 2 x ax dx = xe e ax a 2 x
3 1.3 y + P (x)y + Q(x)y = R(x) y + P (x)y + Q(x)y = R(x) y(x) y + P (x)y + Q(x)y = 0...(1) Y 1 (x) = c 1 y 1 (x) + c 2 y 2 (x) y + P (x)y + Q(x)y = R(x)...(2) 1 ()y 0 (x) y(x) = c 1 y 1 (x) + c 2 y 2 (x) + y 0 (x) (2) 4 (2)y 0 (x) (1)y 1 (x), y 2 (x), y 0 (x) = y 1 (x) R(x)y2 (x) W (y 1, y 2 )(x) dx + y 2(x) R(x)y 1 (x) W (y 1, y 2 )(x) dx W (y 1, y 2 )(x) y 1 (x), y 2 (x) x 2 y 3xy + 3y = 2x 3 x 2...(1) (1) x 2 y 3xy + 3y = 0...(2) y = x k y = kx k 1, y = k(k 1)x k 2 (2) k(k 1)x k 3kx k + 3x k = 0 k(k 1) 3k + 3 = 0 3
4 k 2 4k + 3 = 0 (k 3)(k 1) = 0 k = 1, 3 (2) y 1 = x, y 2 = x 3 W (x, x 3 x x 3 ) = 1 3x 2 = 2x3 y 1 R(x) = 2x 1 R(x)y2 (x) y 0 (x) = y 1 (x) W (y 1, y 2 )(x) dx + y 2(x) R(x)y 1 (x) W (y 1, y 2 )(x) dx (2x 1)x 3 y 0 (x) = x dx + x 3 2x 3 = x( 1 2 x x) + x3 ( 1 + log x) 2x = 1 2 x2 ( x (log x) x) (1) (2x 1)x dx 2x 3 y = C 1 x + C 2 x x2 ( x (log x) x) = C 1 x + C 2 x 3 + x 2 (1 + (log x) x) 4
5 (1) 1 y 1 (x) (1) y 2 (x) (2) (1) 1 y 1 (x) 5 y 1 (x) (1) (2) y 0 (x) y 0 (x) = y 1 (x) φ(x)dx R P (x)dx φ(x) = e y 2 (x) y 1 (x)r(x)e R P (x)dx dx y 1 (x) (1) e R P (x)dx y 2 (x) = y 1 (x) y 2 (x) dx y 1 (x) (1) (2) y(x) e R P (x)dx y(x) = c 1 y 1 (x) + c 2 y 1 (x) y 2 (x) dx + y 1 (x) R P (x)dx φ(x) = e y 2 (x) y 1 (x)r(x)e R P (x)dx dx φ(x)dx, (x + 1)y + xy y = 1 y 1 y 1 = e kx y 1 y = y 1 u(x) y 1 = e kx y 1 = ke kx, y 1 = k 2 e kx k 2 (x + 1)e kx + kxe kx e kx = 0 k 2 (x + 1) + kx 1 = 0 (k 2 + k)x + (k 2 1) = 0 5
6 k k 2 + k = 0 k 2 1 = 0 k = 1 y = e x u(x) y 1 = e x y = e x u + e x u y = e x u 2e x u + e x u (x + 1)y + xy y = 1 (e x u 2e x u + e x u )(x + 1) + ( e x u + e x u )x e x u = 1 e x u (x + 1) e x (x + 2)u = 1 u (x + 1) (x + 2)u = e x u = U U (x + 1) (x + 2)U = e x du dx (x + 2)U x + 1 = ex x + 1 U (x) = e x + C 1 e x (x + 1) u(x) = U (x) dx = ( e x + C 1 e x (x + 1))dx = e x + C 1 e x x + C 2 6
7 y = e x u(x) = e x ( e x + C 1 e x x + C 2 ) = 1 + C 1 x + C 2 e x (1)x 2 y 3xy + 4y = 2x 3 + x 2 (2)x 2 y 2xy + 2y = 2x + 2 (3)(1 + x 2 )y 2xy + 2y = 1 x2, (hint; (1) (3)y = x xk ) (4)xy (2x + 1)y + (x + 1)y = (x 2 + x 1)e 2x (5)(x + 1)y (3x + 4)y + 3y = (3x + 2)e 3x (6)xy (2x 1)y + (x 1)y = xe x, (hint; (4) (6)y = e kx ) (7)x 2 y (x + 2)xy + (x + 2)y = x 4 e x, (hint; y = x k ) (8) x 2 y + xy + y = x, (hint; y = x k ) 1.4 y = y 2,x = 0 y = c y = a n x n = a 0 + a 1 x + a 2 x a k x k +... n=0 y = na n x n 1 = a 1 + 2a 2 x ka k x k n=1 y 2 = a 2 0+(a 0 a 1 +a 1 a 0 )x+...+(a 0 a k +a 1 a k a j a k j +...+a k a 0 )x k (k + 1)a k+1 = a 0 a k + a 1 a k a j a k j a k a 0, k = 0, 1,... 7
8 a 1 = a 2 0, 2a 2 = a 0 a 1 + a 0 a 1 = 2a 3 0, a 2 = a 3 0 3a 3 = a 0 a 2 + a 1 a 1 + a 2 a 0 = 3a 4 0, a 3 = a a k = a k+1 0, k = 1,... a 0 = c a k = c k+1 y = c n+1 x n = c(1 + cx + c 2 x c k x k +..) = n=0 cx < 1 x < 1 c ( cx < 1) 1 cx c c 1 cx y = y 2 y = 1 x + C = y = y 2 dy y 2 = dx 1 y = x + C, 1 C(1 ( x C )) = c 1 cx (c = 1 C ) x 2 y = y x(x = 0, y = 0) x(x 1)y + (3x 1)y + y = 0((x = 0, y = a) 8
9 1.5 n d n y a 0 dx + a d n 1 y n 1 dx a dy n 1 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 (n 1) a n 1 y + a n y = 0...(3) (3) 6 (2) y(x) (3) y 1 (x) (2) 1 y 2 (x) y(x) = y 1 (x) + y 2 (x) (i)(3) y 1 (x) (ii)(2) 1 y 2 (x) 2 (3) (3) y 1 (x) n Y 1 (x),..., Y n (x) C 1,..., C n y 1 (x) = C 1 Y 1 (x) C n Y n (x) 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)...(4) (x)... Y n (n 1) (x) Y (n 1) 9
10 n (3) y = e ρx ρ a 0 ρ n + a 1 ρ n a n 1 ρ + a n = 0...(5) (5) (3) ρ 1,..., ρ n (i)ρ i y = e ρ ix (ii) ρ j = α j ± β j i y ± = e α jx (cos β j x ± i sin β j x) y = e α jx cos β j x, e α jx sin β j x d dx = D (2) (3) (a 0 D n + a 1 D n a n 1 D + a n )y = f(x)...(2) (a 0 D n + a 1 D n a n 1 D + a n )y = 0...(3) (3) (5) (D a)(e ax y) = e ax Dy (D a) n y = 0. (D a) n y = (D a) n (e ax e ax y) = (D a) n 1 (D a)(e ax e ax y) 10
11 = (D a) n 1 e ax D(e ax y) = (D a) n 2 e ax D 2 (e ax y) =... = e ax D n (e ax y) e ax D n (e ax y) = 0 D n (e ax y) = 0 n e ax y = (n 1) n (i) a n y = x n 1 e ax, x n 2 e ax,..., xe ax, e ax n. {x n 1 e ax, x n 2 e ax,..., xe ax, e ax }, (ii) a = α ± iβ n {x n 1 e αx cos βx, x n 2 e αx cos βx,..., xe αx cos βx, e αx cos βx}, {x n 1 e αx sin βx, x n 2 e αx sin βx,..., xe αx sin βx, e αx sin βx} 2n 11
12 1.5.1 f(x)=e kx 7 (a 0 D n + a 1 D n a n 1 D + a n )y = e kx F (D) = a 0 D n + a 1 D n a n 1 D + a n F (D) = (D k) m G(D), G(k) 0 y = xm G(k)m! ekx 0! = 1 1.y + y 2y = e 2x 2.y y 2y = e 2x 3.y 4y + 4y = e 2x 1 F (D) = D 2 + D 2, F (2) = = 4 0 y = 1 4 e2x 2.F (D) = D 2 D 2 = (D 2)(D + 1) y = x (2+1) e2x = x 3 e2x 3.F (D) = D 2 4D + 4 = (D 2) 2 y = x 2! e2x = x 2 e2x (a 0 D n + a 1 D n a n 1 D + a n )y = e kx F (ρ) = a 0 ρ n + a 1 ρ n a n 1 ρ + a n F (k) 0 y = Ae kx F (ρ) = (ρ k) m G(ρ), G(k) 0 y = Ax m e kx y = Ae kx y = Ax m e kx A 12
13 1.5.2 f(x) = sin ωx( sin ωx e iθ = cos θ + i sin θ (a 0 D n + a 1 D n a n 1 D + a n )y = cos ωx( sin ωx)...(1) (a 0 D n + a 1 D n a n 1 D + a n )Y = e iωx...(2) Y y 1, y 2 (a 0 D n + a 1 D n a n 1 D + a n )(y 1 + iy 2 ) = cos ωx + i sin ωx...(3) y 1, y 2 2 (a 0 D n + a 1 D n a n 1 D + a n )y 1 = cos ωx...(4) (a 0 D n + a 1 D n a n 1 D + a n )y 2 = sin ωx...(5) (2) (a 0 D n + a 1 D n a n 1 D + a n )y = e kx ; k 8 (a 0 D n + a 1 D n a n 1 D + a n )Y = e iωx F (D) = a 0 D n + a 1 D n a n 1 D + a n 13
14 F (D) = (D iω) m G(D), G(iω) 0 Y = x m G(iω)m! eiωx (4), (5) Y = e iωx = cos ωx + i sin ωx xm G(iω)m! eiωx x m (cos ωx + i sin ωx) G(iω)m! y 3y + 2y = cos x( sin x) y + 4 = cos 2x sin 2x F (D) = D 2 3D + 2 y 3y + 2y = e ix F (i) = i 2 3i + 2 = 1 3i 0 Y = 1 1 3i eix = ( i)(cos x+i sin x) = 1 10 cos x 3 10 sin x+i( 1 10 sin x+ 3 cos x) 10 y 1 = 1 10 cos x 3 sin x, 10 y 2 = 1 10 sin x cos x 14
15 F (D) = D y 3y + 2y = cos x, y 3y + 2y = sin x F (2i) = 0 F (D) = D = D + 2i)(D 2i) y + 4y = e i2x Y = x 4i ei2x = 1 4 ix(cos 2x + i sin 2x) = 1 4 x sin 2x 1 ix cos 2x 4 y 1 = 1 x sin 2x, 4 y 2 = 1 x cos 2x 4 y + 4 = cos 2x, y + 4 = sin 2x iω F (D) = 0 F (D) D 2 + ω 2 F (D) = D 2 + ω 2 k G(D) 15
16 G(iω) 0 (a 0 D n + a 1 D n a n 1 D + a n )y = cos ωx( sin ωx) F (ρ) = a 0 ρ n + a 1 ρ n a n 1 ρ + a n F (iω) 0 y = A cos ωx + B sin ωx F (ρ) = (ρ iω) m G(ρ), G(iω) 0 y = x m (A cos ωx + B sin ωx) y = A cos ωx + B sin ωx y = x m (A cos ωx + B sin ωx) A, B f(x)=e kx cos ωx( e kx sin ωx) 2 (a 0 D n + a 1 D n a n 1 D + a n )y = e kx cos ωx( e kx sin ωx) (a 0 D n + a 1 D n a n 1 D + a n )Y = e (k+iω)x Y Y = y 1 + iy 2 y 1, y 2 16
17 (a 0 D n + a 1 D n a n 1 D + a n )y 1 = e kx cos ωx (a 0 D n + a 1 D n a n 1 D + a n )y 2 = e kx sin ωx 9 (a 0 D n + a 1 D n a n 1 D + a n )Y = e (k+iω)x F (D) = a 0 D n + a 1 D n a n 1 D + a n F (D) = (D (k + iω)) m G(D), G(k + iω) 0 Y = x m G(k + iω)m! e(k+iω)x y 1, y 2 e iωx = cos ωx + i sin ωx xm G(k+iω)m! e(k+iω)x xm G(k+iω)m! ekx (cos ωx + i sin ωx) y 3y + 2y = e 3x cos 2x( sin 2x) F (D) = D 2 3D + 2 F (3 + 2i) = (3 + 2i) 2 3(3 + 2i) + 2 = 2 + 6i 0 y 3Y + 2Y = e (3+2i)x Y = i e(3+2i)x 17
18 Y = ( i)e3x (cos 2x + i sin 2x) = 1 20 e3x cos 2x + 3 ( 20 e3x sin 2x + i 3 20 e3x cos 2x 1 ) 20 e3x sin 2x y 1 = 1 20 e3x cos 2x e3x sin 2x = 1 20 e3x (cos 2x 3 sin 2x) y 2 = 3 20 e3x cos 2x 1 20 e3x sin 2x = 1 20 e3x (3 cos 2x + sin 2x) (a 0 D n + a 1 D n a n 1 D + a n )y = e kx cos ωx(e kx sin ωx) F (ρ) = a 0 ρ n + a 1 ρ n a n 1 ρ + a n (i)f (k + iω) 0 y = e kx (A cos ωx + B sin ωx) (ii)f (ρ) = (ρ (k + iω)) m G(ρ), G(k + iω) 0 y = x m e kx (A cos ωx + B sin ωx) y = e kx (A cos ωx + B sin ωx) y = x m e kx (A cos ωx + B sin ωx) A, B 18
19 1.5.4 f(x) 10 (a 0 D n + a 1 D n a n 1 D + a n )y = x n ) F (ρ) = a 0 ρ n + a 1 ρ n a n 1 ρ + a n (i)f (0) 0 y = x n (ii)f (0) = 0 0 m y = x (n + m) 19
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
() 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 ()
(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)
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
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
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
,. 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,
ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
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(,,
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 ),
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 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
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 = [
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 (
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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
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
(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 )
() 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 >
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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
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
(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
, 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
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...........................
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 =
n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
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
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(
06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,
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....
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..............
1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
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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 =
dy + P (x)y = Q(x) (1) dx dy dx = P (x)y + Q(x) P (x), Q(x) dy y dx Q(x) 0 homogeneous dy dx = P (x)y 1 y dy = P (x) dx log y = P (x) dx + C y = C exp
+ P (x)y = Q(x) (1) = P (x)y + Q(x) P (x), Q(x) y Q(x) 0 homogeneous = P (x)y 1 y = P (x) log y = P (x) + C y = C exp{ P (x) } = C e R P (x) 5.1 + P (x)y = 0 (2) y = C exp{ P (x) } = Ce R P (x) (3) αy
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
mugensho.dvi
1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
(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 ]
II 2 II
II 2 II 2005 yugami@cc.utsunomiya-u.ac.jp 2005 4 1 1 2 5 2.1.................................... 5 2.2................................. 6 2.3............................. 6 2.4.................................
) ] [ 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
![) ] [ 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
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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,
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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)
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
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1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
微分積分 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.
微分積分 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. ttp://www.morikita.co.jp/books/mid/00571 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. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (
II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )
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1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
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r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
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