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# Laplace transform 證明題 , 請高手幫忙！

Let f satisfy f(f+T)=f(t) for all t >= 0 & for some fixed positive number T , f is said to be periodic with period T on 0 <= t < infinite .

Show L{f(t)} = (積 0~T e-st f(t) dt)/(1- e-sT)

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### 1 Answer

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- EricLv 61 decade agoFavorite Answer
Denote the Laplace transform of f(t) by F(s). For each n > 0, we can perform a change of variables to obtain

∫nT(n 1)T e-stf(t) dt = ∫0T e-s(t nT)f(t nT) dt

= e-snT∫0T e-stf(t) dt.

Hence,

F(s) = ∫0∞ e-stf(t) dt = ∑n=0∞ ∫nT(n 1)T e-stf(t) dt

= ∑n=0∞ e-snT∫0T e-stf(t) dt

= (1-e-sT)-1∫0T e-stf(t) dt

since the sum of the geometric series ∑n=0∞ e-snT is (1-e-sT)-1.

2007-05-27 06:24:13 補充：

奇怪，加號都不見了...

所有的 "n 1"、 "t nT" 都應該是 "n+1"、"t+nT"

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