Re: [理工] fourier transform
※ 引述《jody0113 (Jody)》之銘言：
: 求g(x)=x exp(-x^2)的fourier transform 是多少?
: 算不出來 ><" 麻煩各位高手解題
∞ -x^2 -iwx
F(w) = ∫ xe e dx
-∞
∞ -(x+0.5iw)^2 - 0.25w^2
= ∫ xe dx
-∞
-0.25w^2 ∞ -(x+0.5iw)^2
= e ∫ (x+0.5iw)e dx
-∞
-0.25w^2 ∞ -(x+0.5iw)^2
- 0.5iw * e ∫ e dx
-∞
令 x + 0.5iw = v
-0.25w^2 ∞ -v^2 -0.25w^2 ∞ -v^2
F(w) = e ∫ ve dv - 0.5iw * e ∫ e dv
-∞ -∞
∞ -x^2
考慮 I = ∫ e dx
-∞
2 ∞ -x^2 ∞ -y^2
I = ∫ e dx ∫ e dy
-∞ -∞
∞ ∞ -(x^2 + y^2)
= ∫ ∫ e dxdy
-∞ -∞
2π ∞ -r^2
= ∫ ∫ e rdrdθ
0 0
∞ -r^2 2
= π∫ e d(r )
0
-r^2｜∞
= -πe ｜ = π
｜0
ˍ
=> I = √π
1 -0.25w^2 ∞ -v^2 2 -0.25w^2 ˍ
故 F(w) = ---e ∫ e d(v ) - 0.5iw * e * √π
2 ∞
-0.25w^2 ˍ
= 0 - 0.5iw * e * √π
ˍ 2
√π -0.25w
= - ---- iw * e
2
--
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推
12/23 18:35, , 1F
12/23 18:35, 1F
推
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12/24 07:36, 2F
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12/24 14:17, , 11F
12/24 14:17, 11F
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12/24 14:47, , 12F
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09/11 14:41, , 13F
09/11 14:41, 13F
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