The Dirac Delta Function

The Dirac Delta Function Swapnil Sunil Jain December 27, 2006

The Dirac Delta Function


The Dirac delta function can be defined as


where Π(t) is the pulse function. The delta function can also be defined as

δ(t)={0for t0for t=0,  s.t -+δ(t)𝑑t=1

or it can also be defined as an operator s.t.



1. -f(t)δ(t-a)𝑑t=f(a)


-f(t)δ(t-a)𝑑t =limϵ0(-a-ϵf(t)δ(t-a)𝑑t+1-ϵa+ϵf(t)δ(t-a)𝑑t+a+ϵf(t)δ(t-a)𝑑t)
2. -f(t)δ(t)𝑑t=f(0)

Proof: Readily seen if we set a=0 in Property #1.

3. -f(t)δ(at)𝑑t=1af(0)

Proof: Set u=at, then

-f(t)δ(at)𝑑t =-f(ua)δ(u)dua=1af(0)
4. -f(t)δ(at-t0)𝑑t=1af(t0a)

Proof: Set u=at-t0, then

-f(t)δ(at-t0)𝑑t =-f(u-t0a)δ(u)dua=1af(t0a)
5. -f(t)δ(g(t))𝑑t={xi|g(xi)= 0}1|g(xi)|f(xi)
6. -f(t)δ(t)𝑑t=-f(0)

Proof: If we let u=f(t) and dv=δ(t)dt, then, using integration by parts,

-f(t)δ(t)𝑑t =f(t)δ(t)|---f(t)δ(t)𝑑t=-f(0)
Title The Dirac Delta Function
Canonical name TheDiracDeltaFunction1
Date of creation 2013-03-11 19:30:28
Last modified on 2013-03-11 19:30:28
Owner swapnizzle (13346)
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Numerical id 1
Author swapnizzle (0)
Entry type Definition