Laplace transform of Dirac's delta distribution

A Dirac δ symbol can be interpreted as a linear functional, that is, a linear mapping from a suitable function space to ℝ (or ℂ), having the property

δ[f] = f(0).

One may write this formally as the pairing

∫ ∞ ⟨f,δ⟩ = f(t)δ (t) dt, 0

when the standard relation

∫ ∞ f (t)δ(t)dt = f(0) 0

is being used.

Applying this to

f(t) = e−st,

one obtains

∫ ∞ −st 0 e δ(t)dt = e = 1. 0

Hence, in the convention used here, the Laplace transform is

ℒ {δ(t)} = 1.

(1)

By the delay theorem, this generalizes to

−as ℒ{δ(t − a)} = e , a > 0.

When introducing the approximation

( 1 { -, 0 ≤ t ≤ 𝜀, η𝜀(t) := ( 𝜀 0, t > 𝜀,

we obtain

∫ ∞ ℒ {η (t)} = e−stη (t)dt 𝜀 0 𝜀 ∫ 𝜀 e−st = ----dt 0 𝜀 1 − e−𝜀s = --------. 𝜀s

Using the Taylor expansion of e^−𝜀s,

e−𝜀s = 1 − 𝜀s + O (𝜀2),

we find

l𝜀→i0m+ ℒ {η𝜀(t)} = 1.

The Dirac delta, δ, is rigorously defined as a linear functional on a suitable space of test functions. Its defining action is

δ[f] = f(0).

Equivalently, one may write

∫ ∞ ⟨f, δ⟩ = f(t)δ(t)dt = f(0), 0

using the half-line convention adopted in this article.

Therefore,

ℒ {δ(t)} = 1,

and, for a > 0,

ℒ {δ(t − a )} = e−as.

[1] L. Schwartz, Théorie des distributions, vols. 1–2, Hermann, Paris, 1950–1951.

[2] W. Rudin, Functional Analysis, McGraw–Hill Book Company, 1973.

[3] L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer–Verlag, 1990.

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