Re: [分析] 反拉氏轉換
※ 引述《ToMoveJizz ( )》之銘言：
: s/(s^4+a^4)的反拉氏轉換，a>0
令a^4 = 4b^4, 分子改成4b^2s, 導出來好看一些.
s^4 + 4b^4 = (s^4 + 4b^2s^2 + 4b^4) - 4b^2s^2 = (s^2+2b^2)^2 - (2bs)^2
= (s^2 - 2bs + 2b^2)(s^2 + 2bs + 2b^2)
4b^2s/(s^4 + 4b^4) = 4b^2s/((s^2 - 2bs + 2b^2)(s^2 + 2bs + 2b^2))
= b/(s^2 - 2bs + 2b^2) - b/(s^2 + 2bs + 2b^2)
= b/((s-b)^2 + b^2) - b/((s+b)^2 + b^2)
L^{-1}[b/((s-b)^2 + b^2)](t) = e^{bt} sin(bt)
L^{-1}[b/((s+b)^2 + b^2)](t) = e^{-bt} sin(bt).
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◆ From: 163.22.18.20
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