Laplace transform of Dirac delta
The Dirac delta (http://planetmath.org/DiracDeltaFunction) δ can be interpreted as a linear functional, i.e. a linear mapping from a function space
, consisting e.g. of certain real functions, to ℝ (or ℂ), having the property
δ[f]=f(0). |
One may think this as the inner product
⟨f,δ⟩=∫∞0f(t)δ(t)𝑑t |
of a function f and another “function” δ, when the well-known
∫∞0f(t)δ(t)𝑑t=f(0) |
is true. Applying this to f(t):=e-st, one gets
∫∞0e-stδ(t)𝑑t=e-0, |
i.e. the Laplace transform
ℒ{δ(t)}= 1. | (1) |
By the delay theorem, this result may be generalised to
ℒ{δ(t-a))}=e-as. |
When introducing some “nascent Dirac delta function”, for example
ηε(t):={1ε for 0≤t≤ε,0 for t>ε, |
as an “approximation” of Dirac delta, we obtain the Laplace transform
ℒ{ηε(t)}=∫∞0e-stηε(t)𝑑t=∫ε0e-stε𝑑t+∫∞εe-st⋅0𝑑t=1ε∫ε0e-st𝑑t=1-e-εsεs. |
As the Taylor expansion shows, we then have
limε→0+ℒ{ηε(t)}= 1, |
being in accordance with (1).
Title | Laplace transform of Dirac delta |
---|---|
Canonical name | LaplaceTransformOfDiracDelta |
Date of creation | 2013-03-22 19:10:56 |
Last modified on | 2013-03-22 19:10:56 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 11 |
Author | pahio (2872) |
Entry type | Result |
Classification | msc 46E20 |
Classification | msc 44A10 |
Classification | msc 34L40 |