小小的希望 asked in 科學數學 · 1 decade ago

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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  • Eric
    Lv 6
    1 decade ago
    Favorite 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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