作者Honor1984 (奈何上天造化弄人？)
看板Grad-ProbAsk
标题Re: [理工] 傅立叶转换
时间Thu Jul 19 01:10:14 2018
※ 引述《shirley10631 (leilei_0822)》之铭言：
: https://i.imgur.com/8r26BbI.jpg
: 想问大家，我把f(x)取转换後，
: F(w)=pi/2*(1/1+jw)後面就卡住了，
: 麻烦各位指点！
题目不清楚，x < 0时f(x)是什麽？
假设f(x) = (π/ 2)exp(-x)u(x)
F(k) = F[f(x)] = (π/ 2)∫exp(-x)u(x)exp(-ikx)dx
= (π/ 2)/[1 + ik]
= (π/ 2)[1 - ik]/[1 + k^2]
=> F(k) + F(-k) = π/[1 + k^2]
其中F(-k) = (π/ 2)∫exp(-x)u(x)exp(ikx)dx
∞
= (π/ 2)∫exp(v)u(-v)exp(-ikv)dv
-∞
= F[g(x)]， g(x) = exp(-|x|)u(-x)
h(x) = f(x) + g(x) = exp(-|x|)
∞
= [1/(2π)]∫[π/[1 + k^2]] * exp(ikx) dk
-∞
∞
= ∫ cos(kx)/[1 + k^2] dk
0
= exp(-|x|)
所以答案是exp(-|w|)
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