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
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1 1 yousuke.itoh/lecture-notes.html [, 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, sin nx sin mxdx = πδ mn, cos nx sin mxdx = (1.1a) (1.1b) (1.1c) (1.1d) (1.1e) (1.1f)
2 1.1 : [a, b) f(x) b a f p (x) [a + n, b + n) f p (x) f p (x) = f(x n), a + n x < b + n (1.2) 1. [, π) f(x) = x π ( ) f(x) F (x) n nπ x < (n + 1)π F (x) = x nπ F (x) = x nπ for nπ x < (n + 1)π (n Z) (1.3) 2. [, π) f(x) = x 2π ( ) f o (x) f o (x) = x ( x < π) 2π F o (x) n (2nπ 1) x < (2n+1)π F o (x) = x 2nπ 3. [, π) f(x) = x 2π ( ) f e (x) f e (x) = x ( x < π) 2π F e (x) n (2nπ 1) x < (2n+1)π F e (x) = x 2nπ 4. Euler ( ) e i(m+n)x = cos(m + n)x + i sin(m + n)x =e imx e inx = (cos mx + i sin mx)(cos nx + i sin nx) =(cos mx cos nx sin mx sin nx) + i(sin mx cos nx + cos mx sin nx) (1.4) cos(m + n)x = (cos mx cos nx sin mx sin nx), (1.5) sin(m + n)x = (sin mx cos nx + cos mx sin nx) (1.6) 2
3 5. ( ) dx = 2π, (1.7) cos nxdx =, (1.8) sin nxdx =, (1.9) cos mx cos nx = 1 [cos(m n)x + cos(m + n)x], (1.1) 2 cos mx cos nxdx = 1 [cos(m n)x + cos(m + n)x] dx 2 [ ] 2π+ 1 = 2(m n) sin(m n)x + 1 2(m+n) sin(m + n)x = m n π + 1 4m sin 2mx 2π+ = π m = n (1.11) sin mx cos nx = 1 [sin(m + n)x + sin(m n)x], (1.12) 2 sin mx cos nxdx = 1 [sin(m n)x + sin(m + n)x] dx 2 [ ] 2π+ 1 = 2(m n) cos(m n)x 1 cos(m + n)x = 2(m + n) (1.13) sin mx sin nx = 1 [cos(m n)x cos(m + n)x], (1.14) 2 sin mx sin nxdx = 1 [cos(m n)x cos(m + n)x] dx 2 [ ] 2π+ 1 = 2(m n) sin(m n)x 1 2(m+n) sin(m + n)x = m n π 1 4m sin 2mx 2π+ = π m = n (1.15) 3
4 2 x < π f(x) a n = 1 π b n = 1 π f(x) cos nxdx, (2.1) f(x) sin nxdx (2.2) f(x) a 2 + (a n cos nx + b n sin nx) (2.3) 2.1. x < π 1. f(x) = x 2. f(x) = x 4
5 2.1 : x < π 1. f(x) = x ( ) n a n = 1 x cos nxdx π [ ] π 1 = x sin nx 1 sin nxdx nπ nπ [ ] π 1 = n 2 cos nx = (2.4) π b n = 1 x sin nxdx π [ ] π 1 = x cos nx + 1 cos nxdx nπ nπ = 2( 1)n n f(x) (2.5) 2( 1) n sin nx (2.6) n a n cos nx (n N) 2. f(x) = x ( ) a a = 1 π n 1 n x dx = 2 π xdx = π (2.7) a n = 1 x cos nxdx = 2 x cos nxdx π π [ ] π 2 = x sin nx 2 sin nxdx nπ nπ [ ] π 2 = n 2 cos nx = 2(( 1)n 1) π n 2 π b n = 1 π (2.8) x sin nxdx =, (2.9) 5
6 f(x) π 2 + = π 2 k= 2(( 1) n 1) n 2 π cos nx 4 (2k + 1) 2 cos(2k + 1)x (2.1) π b n 6
7 [, π) g(x) 1, x <, g(x) = (3.1) 1, x < π 3.2. g(x) [, π) f(x) = x g(x) = f (x) (3.2) f(x) g(x) [ f(x) ] 7
8 [, π) g(x) 1, x <, g(x) = (3.3) 1, x π g(x) a n = 1 π dxg(x) cos nx =, (3.4) b n = 1 dxg(x) sin nx = 2 dx sin nx π π [ = 2 ] π cos nx = 2(1 ( 1)n ) nπ nπ 2(1 ( 1) n ) nπ 2. x x π 2 sin nx = k= k= (3.5) 4 sin(2k + 1)x (3.6) (2k + 1)π 4 (2k + 1) 2 cos(2k + 1)x (3.7) π d dx ( ) = 4 sin(2k + 1)x (3.8) (2k + 1)π k= g(x) 8
9 4 [, π) f(x) f(x) f(x) a n a n = 1 f(x) cos nxdx = 1 π π x = t a n = 1 π f(x) cos nxdx 1 π f( t) = f(t) a n = 1 π b n f(x) cos nxdx + 1 π b n = 1 π f(x) cos nxdx + 1 π π f(t) cos ntdt = 2 π f(x) sin nxdx f( t) cos( nt)dt f(x) cos nxdx f(x) cos nxdx x = t f( t) = f(t) sin( nt) = sin nt b n = 1 π π f(t)( sin nt)dt = 1 π f(t) sin ntdt = b n f(x) f(x) a n = 2 π b n =, f(x) cos nxdx, (4.1a) (4.1b) 4.1. [, π) f(x) f(x) a n =, (4.2) b n = 2 π f(x) sin nxdx (4.3) 4.2. [, π) (1) x 2 (2) x 3 (4.4) 4.3. [, π) (1) tan x (2) x 3 (3) 3.1 (4.5) 9
10 4.1 : 1. [, π) f(x) a n a n = 1 π dxf(x) cos nx = 1 π dxf(x) cos nx + 1 π dxf(x) cos nx [, ] y = x = 1 π dxf(x) cos nx + 1 π f( y) = f(y) = 1 π b n b n = 1 π dxf(x) cos nx 1 π f(x) sin nxdx = 1 π dxf(x) sin nx + 1 π π ( dy)f( y) cos( ny) dyf(y) cos ny = (4.6) dxf(x) sin nx [, ] y = x = 1 π dxf(x) sin nx + 1 π f( y) = f(y) = 1 π = 2 π dxf(x) cos nx + 1 π π ( dy)f( y) sin( ny) dyf(y) cos ny dxf(x) cos nx (4.7) 2. [, π) 1
11 (a) f(x) = x 2 b n = a n a = 2 π a n = 2 π x 2 dx = 2 3 π2, (4.8) x 2 cos nxdx = 4( 1)n n 2, (n > ) (4.9) (b) f(x) = x 3 a n = b n b n = 2 π x 3 sin nxdx = 2( 1)n n ( ) 6 n 2 π2 (4.1) 3. [, π) (a) tan x x = ±π/2 lim =, x π/2+ lim =, x /2+ lim =, (4.11) x π/2 lim =, (4.12) x /2 (b) x 3 [, π) (c) g(x) = sgn(x) [, π) x = lim = 1, x + lim = 1 (4.13) x 11
12 5 D n (x) 1 n 2 + cos kx (5.1) k= n 1 π D n(x)dx = 1 (5.2) π δ p (x) = δ(x 2πn) (5.3) n= 5.3. f(x) = x ( x < π) 2π 2( 1) n f(x) sin nx (5.4) n f(x) D n (x) f(x) ( y ) 12
13 5.1 : 1. D n (x) D n (x)dx = n ( ) 1 n 2 + cos kx dx = π (5.5) k=1 2. a n = 1 π = 1 π δ p (x) cos nxdx ( + δ(x + 2π) + δ(x) + δ(x 2π) + ) cos nxdx = 1, (5.6) b n = 1 δ p (x) sin nxdx = 1 π π δ(x 2nπ) cos nxdx =, n= (5.7) δ p (x) cos nx = D n (x) (5.8) ( ) 3. f(x) f (x) 2( 1) n cos nx = 2 cos nπ cos nx = {cos(nx + nπ) + cos(nx nπ)} { } 1 = cos(nx + nπ) cos(nx nπ) = 1 D n (x π) D n (x + π) (5.9) 13
14 [, π) f(x) f(x) S(x) 1. x = f(x) = sgn(x) + 1 S(x) x = S() 2. x = f(x) = cosh(sin 2 x ) S(x) x = S() [, π) f(x) = x 1/2 2. [, π) f(x) = x 1/ x 2 x 3 x 2 π 3 + 4( 1) n n 2 cos nx, (6.1) x 3 2( 1) n ( ) 6 n n 2 π2 sin nx (6.2) x 3 /3 1 d 2( 1) n 3 dx n = ( ) 6 n 2 π2 sin nx ( 4( 1) n n 2 2( 1)n π 2 3 x 2 ) cos nx (6.3) 14
15 f(+) + f( ) 1. S() = = 2 = ( 1) sgn(+) sgn( ) = 1, (6.4) 2. S() = f() = cosh(sin 2 )) = cosh = 1, (6.5) 2. (a) x 1/2 [, π) (b) x 1/2 sgn(x)/(2 x 1/2 ) x = 3. [, π) f(x) 2π Θ(x) F (x) f(x 2nπ)Θ(x (2n 1)π)Θ((2n + 1)π x) (6.6) n= f(x) = x 3 /3 F { (x) (x 2nπ) 2 Θ(x (2n 1)π)Θ((2n + 1)π x) n= (x 2nπ)3 + δ(x (2n 1)π)Θ((2n + 1)π x) 3 } (x 2nπ)3 Θ(x (2n 1)π)δ((2n + 1)π x) 3 n = a = 1 dx (x 2 + x3 x3 δ(x + π) δ(π x) π 3 3 n > a n = 1 π ) = 2π2 (6.7) 3 π2 3 π2 3 = ) dx (x 2 + x3 x3 δ(x + π) δ(π x) cos nx 3 3 (6.8) = 4( 1)n n 2 2( 1)n π 2, 3 (6.9) b n =, (6.1) F (x) ( 4( 1) n n 2 2( 1)n π 2 ) cos nx (6.11) 3 15
16 x 3 x 2 x 3 2π x = (2n+1)π (n Z) 16
17 7 f(x) a 2 + (a n cos nx + b n sin nx) (7.1) Parseval ( ) 1 π {f(x)} 2 dx = a2 π 2 + (a 2 n + b 2 n) (7.2) 7.1. [, π) x 2 2π x 2 = π ( 1) n n 2 cos nx (7.3) 1 n 2 = π2 6, (7.4a) 1 n 4 = π4 9, (7.4b) g(x) a x < b (b a = > ) a n = 2 b n = 2 b a b a g (ξ) cos 2πnξ dξ, (7.5a) g (ξ) sin 2πnξ dξ (7.5b) g(x) a 2 + ( a n cos 2πnx + b n sin 2πnx ) (7.6) 7.2. [ /2, /2) (1) x 2 (2) x 3 (7.7) 17
18 n 2 = π2 6, (7.8a) 1 n 4 = π4 9, (7.8b) x = π x 2 = π ( 1) n n 2 cos nx (7.9) π 2 = π n 2 (7.1) π 2 6 = 1 n 2 (7.11) x 2 x 2 a = 2π2 3, (7.12) a n = 4( 1)n n 2, (7.13) b n =, (7.14) 1 π {x 2 } 2 dx = 2π4 π 5, (7.15) 2 + (a 2 n + b 2 n) = 1 ( ) 2π 2 2 ( ) 4( 1) n n 2 a 2 = 2π n 4 (7.16) π 4 9 = 1 n 4 (7.17) 18
19 2. (a) [ /2, /2) x 2 a n = 2 b n = 2 /2 /2 /2 /2 x 2 cos 2πnx dx, (7.18a) x 2 sin 2πnx dx (7.18b) ξ = 2πx/ ã n, b n x 2 [, π) a n = 2 4π 2 1 π b n = 2 4π 2 1 π ξ 2 cos nξdξ = 2 4π 2 ãn, ξ 2 sin nξdξ = 2 4π 2 b n (7.19a) (7.19b) [, π) x x 2 = ( 1) n n 2 π 2 cos 2πnx (7.2) (b) [ /2, /2) x 3 x 2 ( ) 3 a n = ã n, (7.21a) 2π ( ) 3 b n = bn (7.21b) 2π [, π) x x 3 3 ( 1) n 4π 3 n ( ) 6 n 2 π2 sin 2πnx (7.22) 19
20 [ /2, /2) f(x) = x 2. [, ) f(x) = x 8.2. [ /2, /2) f(x) = x 8.3. [, π) f(x), g(x) (a m, b m ),(c m, d m ) 1 π π f(x)g(x)dx = 1 2 a c + (a n c n + b n d n ) (8.1) 2
21 8.1 : 1. (a) [ /2, /2) f(x) = x a = 2 a n = 2 b n = 2 = 2 /2 /2 /2 /2 /2 /2 xdx =, (8.2) x cos 2nπx dx =, (8.3) x sin 2nπx dx [ 2nπx x cos 2nπ = ( 1)n nπ ] /2 /2 + 1 /2 nπ /2 cos 2nπx dx (8.4) x ( 1) n nπ sin 2nπx (8.5) (b) [, ) f(x) = x a = 2 a n = 2 = 2 [ = 1 nπ b n = 2 = 2 xdx =, (8.6) x cos 2nπx dx 2nπx x sin 2nπ = nπ ] [ 2nπx cos 2nπ x sin 2nπx dx [ 2nπx x cos 2nπ 1 nπ ] ] sin 2nπx dx = (8.7) + 1 nπ cos 2nπx dx (8.8) x 2 2nπx sin nπ (8.9) 21
22 2. [ /2, /2) f(x) = x /2 c = 1 xdx =, (8.1) /2 c n = 1 /2 ( ) 2nπix x exp dx /2 = 1 [ ( )] /2 2nπix 2nπi x exp + 1 /2 ( ) 2nπix exp dx 2nπi /2 = i( 1)n 2nπ x n=,n /2 i( 1) n 2nπ ( ) 2nπix exp (8.11) (8.12) 3. [, π) f(x), g(x) (a m, b m ),(c m, d m ) 1 π π f(x)g(x)dx = 1 2 a c + 1 π f(x)g(x)dx = 1 π π (a n c n + b n d n ) (8.13) ( ) c dxf(x) 2 + (c n cos nx + d n sin nx) = c dxf(x) 2π + 1 ) (c n dxf(x) cos nx + d n dxf(x) sin nx π = a c + (a n c n + b n d n ) (8.14) 2 22
23 T u(x, t) t = k 2 u(x, t) x 2, (9.1a) u(x, ) = f(x), ( x ), ( ) (9.1b) u(, t) = T, u(, t) =, ( ) (9.1c) ( ) 3 x f(x) = T (9.2) 9.2. x u(x, t) t = k 2 u(x, t) x 2, (9.3a) u(x, ) = f(x), ( x < 2π), ( ) u(, t) = u(2π, t), u(, t) x = (9.3b) u(2π, t), ( ) (9.3c) x f(x + 2π) = f(x) 23
24 T u(x, t) t = k 2 u(x, t) x 2, (9.4a) u(x, ) = f(x), ( x ), ( ) (9.4b) u(, t) = T, u(, t) =, ( ) (9.4c) ( ) 3 x f(x) = T (9.5) ( ) u 1 (x) u 1 (x, t) t = k 2 u 1 (x, t) x 2, (9.6a) u 1 (x, ) = g(x), ( x ), ( ) (9.6b) u 1 (, t) = T, u 1 (, t) =, ( ) (9.6c) u 2 u 2 (x, t) t = k 2 u 2 (x, t) x 2, (9.7a) u 2 (x, ) = f(x) g(x), ( x ), ( ) (9.7b) u 2 (, t) =, u 2 (, t) =, ( ) (9.7c) u = u 1 + u 2 u 2 u 1 (x, t) = T x + T (9.8) g(x) = T x + T (9.9) u(x, t) = T x + T + ( A n exp k π2 n 2 ) t 2 sin πnx (9.1) 24
25 A n = 2 ( f(x) + T ) x T sin πnx dx (9.11) A n = 2T ( ( ) 3 x + x 1 = 2T 1 ( ) 3 x f(x) = T (9.12) ) sin πnx dx ( (1 ζ) 3 + ζ 1 ) sin πnζdζ = 12T n 3 π 3 (9.13) ζ = x/ u(x, t) = T x + T 12T π 3 ( 1 n 3 exp k π2 n 2 ) t 2 sin πnx (9.14) : (9.14) 1 x/ u(x, t)/t 2 /(kπ 2 ) t = f(x) ( ) t/( 2 /(kπ 2 )) =.2,.1 2. x u(x, t) t = k 2 u(x, t) x 2, (9.15a) u(x, ) = f(x), ( x < 2π), ( ) u(, t) = u(2π, t), u(, t) x = (9.15b) u(2π, t), ( ) (9.15c) x 25
26 f(x + 2π) = f(x) ( ) u(x, t) = e kλ2t (Ae iλx + Be iλx ), λ, u(x, t) = Ax + B, λ = (9.16) λ = u(x, t) =( ) λ u(, t) = e kλ2t (A + B) = e kλ2t (Ae i2πλ + Be i2πλ ) = u(2π, t) u(, t) x = e kλ2t iλ(a B) = e kλ2t iλ(ae i2πλ Be i2πλ ) = (9.17a) u(2π, t) x (9.17b) B(1 e 2πiλ ) = A(e 2πiλ 1) (9.18) B(1 e 2πiλ ) = A(1 e 2πiλ ) (9.19) e 2πiλ = 1 (9.2) λ = n ( ) u(x, t) = a 2 + e kλ2t (a n cos nx + b n sin nx) (9.21) a n, b n f(x) 26
27 x, y 1 2 u(x, y, t) c 2 s t u(x, y, t) x u(x, y, t) y 2 =, (1.1a) u(x, y, ) u(x, y, ) = f(x, y), =, (1.1b) t u(, y, t) = u( x, y, t) = u(x,, t) = u(x, y, t) =, (1.1c) f(x, y) = x( x x)y( y y) (1.1) 27
28 ( x < ) u(x, t) = k 2 u(x, t) t x 2, (11.1a) u(, t) = T sin ωt, lim u(x, t) =, ( ) (11.1b) x 1 t ω = 2π/864[Hz] D = {(x, y) x x, y y } 2 u(x, y) x u(x, y) y 2 =, (11.2) u(, y) =, u( x, y) =, (11.3) u(x, ) y =, u(x, y) y = sin πx x (11.4) (x, y) 1 T T T =
29 12 Gibbs : f(x) = sgn(x) ( x < π) S f n(x) : n = 5 : n = 15 : n = : f(x) = sgn(x) ( x < π) S f n(x) x 29
(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
1 1 u m (t) u m () exp [ (cπm + (πm κ)t (5). u m (), U(x, ) f(x) m,, (4) U(x, t) Re u k () u m () [ u k () exp(πkx), u k () exp(πkx). f(x) exp[ πmxdx
1 1 1 1 1. U(x, t) U(x, t) + c t x c, κ. (1). κ U(x, t) x. (1) 1, f(x).. U(x, t) U(x, t) + c κ U(x, t), t x x : U(, t) U(1, t) ( x 1), () : U(x, ) f(x). (3) U(x, t). [ U(x, t) Re u k (t) exp(πkx). (4)
() 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 >
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)
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 ),
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..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...........................
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.
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chap1.dvi
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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)
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1 1.1 [] f(x) f(x + T ) = f(x) (1.1), f(x), T f(x) x T 1 ) f(x) = sin x, T = 2 sin (x + 2) = sin x, sin x 2 [] n f(x + nt ) = f(x) (1.2) T [] 2 f(x) g(x) T, h 1 (x) = af(x)+ bg(x) 2 h 2 (x) = f(x)g(x)
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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
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
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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
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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
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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],
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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
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