The Laplace Transform of the Delta Function | Massachusetts Institute of Technology - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.03SC | Differential Equations
Academic year:

2021
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293

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1
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DemCedgevexynw

The Laplace Transform of the Delta Function Since the Laplace transform is given by an integral, it should be easy to compute it for the delta function. The answer is 1. L(δ(t)) = 1. 2. L(δ(t − a)) = e−as for a > 0. As expected, proving these formulas is straightforward as long as we use the precise form of the Laplace integral. For (1) we have: L(δ(t)) = � ∞ 0− δ(t)e−st dt = 1. As we saw in a previous session, integrating e−st against δ(t) amounts to evaluating e−st at t = 0, and e0 = 1. Similarly for the shifted version (2), integrating e−st against δ(t − a) amounts to evaluating e−st at t = a: L(δ(t − a)) = � ∞ 0− δ(t − a)e−st dt = e−sa . Notice that the two formulas are consistent: if we set a = 0 in formula (2) then we recover formula (1).

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