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, sin nx sin mxdx = πδ mn, cos nx sin mxdx = (1.1a) (1.1b) (1.1c) (1.1d) (1.1e) (1.1f) 1.3. 1
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
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
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
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
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
3 3.1. [, 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
3.1 1. [, 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
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
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
(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
5 D n (x) 1 n 2 + cos kx (5.1) k=1 5.1. n 1 π D n(x)dx = 1 (5.2) 5.2. 2π 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
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) 1 2 + 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 = 1 2 + cos(nx + nπ) + 1 2 + cos(nx nπ) = 1 D n (x π) D n (x + π) (5.9) 13
6 6.1. [, 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() 6.2. 1. [, π) f(x) = x 1/2 2. [, π) f(x) = x 1/2 6.3. 4.3 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
6.1 1. f(+) + f( ) 1. S() = = 2 = 1 + 1 + ( 1) + 1 2 sgn(+) + 1 + sgn( ) + 1 2 = 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
x 3 x 2 x 3 2π x = (2n+1)π (n Z) 16
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 = π2 3 + 4( 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
7.1 1. 1 n 2 = π2 6, (7.8a) 1 n 4 = π4 9, (7.8b) x = π x 2 = π2 3 + 4( 1) n n 2 cos nx (7.9) π 2 = π2 3 + 4 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 2 + 2 3 n 2 a 2 = 2π4 9 + 16 1 n 4 (7.16) π 4 9 = 1 n 4 (7.17) 18
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 2 4.2 x 2 = 2 12 + 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 3 4.2 x 3 3 ( 1) n 4π 3 n ( ) 6 n 2 π2 sin 2πnx (7.22) 19
8 8.1. 1. [ /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
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
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
9 9.1. 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
9.1 1. 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
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) 1..8.6.4.2.2.4.6.8 1. 1: (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
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
1 1.1. x, y 1 2 u(x, y, t) c 2 s t 2 + 2 u(x, y, t) x 2 + 2 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
11 3 11.1. 24 ( 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] 2 11.2. D = {(x, y) x x, y y } 2 u(x, y) x 2 + 2 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 2 864 = 24 6 6 28
12 Gibbs 1..5-3 -2-1 1 2 3 -.5-1. 2: f(x) = sgn(x) ( x < π) S f n(x) : n = 5 : n = 15 : n = 5 1.2 1.1 1..9.8.1.2.3.4.5 3: f(x) = sgn(x) ( x < π) S f n(x) x 29