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单词 LaplaceTransformOfIntegral
释义

Laplace transform of integral


On can show that if a real function  tf(t)  isLaplace-transformable (http://planetmath.org/LaplaceTransform), as wellis 0tf(τ)𝑑τ.  The latter is alsocontinuousMathworldPlanetmath for  t>0  and by theNewton–Leibniz formula (http://planetmath.org/FundamentalTheoremOfCalculus),has the derivative equal f(t).  Hence we may apply theformula for Laplace transform of derivative, obtaining

F(s)={f(t)}=s{0tf(τ)𝑑τ}-00f(t)𝑑t=s{0tf(τ)𝑑τ},

i.e.

{0tf(τ)𝑑τ}=F(s)s.(1)

Application.  We start from the easily derivable rule

1s 1,

where the curved from the Laplace-transformed functionMathworldPlanetmath to the original function.  The formula (1) thus yields successively

1s20t1𝑑τ=t,
1s30tτ𝑑τ=t22!,
1s40tτ22!𝑑τ=t33!,

etc.  Generally, one has

1sntn-1(n-1)!n+.(2)
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更新时间:2025/5/4 15:39:59