I io@hiroshima-u.ac.jp 27 6
A A. /a δx = lim a + a exp π x2 a 2 = lim a + a = lim a + a exp a 2 π 2 x 2 + a 2 2 x a x = lim a + a Sic a x = lim a + a Rect a Gaussia Loretzia Bilateral expoetial Normalized cardial sie Rectagular fuctio Sicx def. = siπx πx Rectx def. = x < 2 x > 2 A
e ax2 dx = e a x b ic2 dx = π a a b A. c x c δcx = lim a + a Rect a = c lim a/ c + x a/ c Rect a/ c δax = a δx a = δx c A.2 d e ix = 2π δx A.3 A.3 A.3 ϵ e ix e ϵ x dx e ix e ϵx2 dx ϵ A.3 = 2πξ A2
dξ e 2πi ξx = δx A.3 D N θ def. = D θ = N e iθ = si N+ 2 θ si 2 θ = N e iθ = 2π = δθ 2π m m= A.4 N e iθ = e inθ e in+θ = e in+ 2 θ e in+ 2 θ = si N+ 2 θ e iθ e iθ/2 e iθ/2 si 2 θ = N A.4 A.4 A.5 D N θ + 2π = D N θ A.6 N A.4 si N+ θ 2π 2 2N + D N θ θ 2π lim D N θ * * 2 θ 2π N A.6 si N+ 2 θ D N = lim θ si 2 θ = 2N + lim D N = + D N θ N A.6 π D N θ dθ = π π dθ π + N = N e iθ = 2π + N = N [ e iθ i ] θ=π = 2π θ= π b * Riema-Lebesgue f x a b lim f x e ix dx = a * 2 distributio A3
θ = 2π N 2π N D θ = 2π δθ 2π m A.5 m= f g x def. = f x gx x dx A.7 f g = g f f g h = f g h f ag + bh = a f g + b f h δ f = f a b f g h δ d [ ] d f x f g x = gx = f x d gx dx dx dx δx δx f x = f x δx x δx x f x = f x x δx a δx a f x = f x a d δx dx d δx dx f x = d f x dx A4
A.2 Fξ = dx e 2πi ξx f x A.8a f x = dξ e 2πi ξx Fξ A.8b A.3 dξ e 2πi ξx = δx a f x + b gx f ax a Fξ + b Gξ ξ a F a f x x Fξ e 2πi ξx f x e 2πi ξ x Fξ ξ f g x f x gx Fξ Gξ F G ξ x d dx f x f X dx 2πi ξ Fξ Fξ 2πi ξ dx 2 f x Fx = dξ Fξ 2 f ξ A5
= 2πξ ξ F = f x = dx e ix f x d 2π e ix F A.a A.b A.3 d 2π e ix = δx a f x + b gx f ax a F + b G A.2 a F a A.3 f x x F e i x f x e i x F A.4a A.4b f g x f x gx F G F G 2π A.5a A.5b x d dx f x f X dx i F A.6 F i A.7 dx 2 f x = Fx d 2π F 2 A.8 2π f A.9 A6
a -4-3 -2-2 3 4-4 -3-2 - 2 3 4 x b -4-3 -2-2 3 4-4 -3-2 - 2 3 4 x c -4-3 -2-2 3 4-4 -3-2 - 2 3 4 x d -8-6 -4-2 2 4 6 8 x -2-8 -6-4 2 4 6 8 e -8-6 -4-2 2 4 6 8 x -8-6 -4-2 2 4 6 8 A. A7
exp π x 2 π 2 x 2 + exp 2 x exp π ξ 2 = exp 2 4π exp 2 ξ = exp π 4 = π 2 ξ 2 + 2 + 4 A.2 A.2a A.2b Rect x Sic x Sic ξ = Sic 2π Rect ξ = Rect 2π A.22a A.22b σ exp π x2 a a 2 exp x2 2πσ 2 2σ 2 a exp π a 2 ξ 2 = a exp a2 2 4π exp σ2 2 2 A.23 A.3 δ x δξ = 2π δ δx a = f x a = b δξ mb = g δ mg m= m= b F ξ δξ mb = g F δ mg m= m= ba = ga = 2π A8
3 a =.25 3.25.5 2 2 2 4 8-2 - 2-2 - 2 x A.2 ξ 2π ξ ξ 2π 2π A9
A.3 f x + L = f x L L F = L dx e ix f x f x = F e ix = 2π L e ix = L δx L L dx e ix = L δ A.24a A.24b A.25a A.25b A.25a A.5 e ix = e i 2πx 2πx L = 2π δ L 2π = L δx L A.25b = L dx e ix = L dx e i 2π L x = L δ = L δ L f g x F G A.26a f x gx F G A.26b A
L dx 2 2 f x = F L A.27 def. F F = F K G K A.7 K A.2 A.4 A.6 A.7 δx L L δ L N δx a = N f x a = a a m= m= L = Na ga = 2π δ,mg F mg L A
A.4 δ 3 r def. = δx δy δz F = f r = dr e i r f r = dx e i xx dy e i yy dz e i zz f r d 2π e i r dx dy F = 3 2π e i xx 2π e i dz yy 2π e izz f f g r f r gr F G F G 2π 3 A.28a A.28b δ 3 r a = 2π δ a 2π m m= δ 3 r R R 2π3 V uc δ 3 G G f r R R 2π3 V uc F δ 3 G G R = a + 2 a 2 + 3 a 3 G = m g + m 2 g 2 + m 3 g 3 g i a j = 2π δ ij V uc = a a 2 a 3 = 2π 3 g g 2 g 3 A2