Validity of integral involving delta function

[SarthakC]

Hi,

Is the following integral well defined? If it is, then what does it evaluate to?

\int_{-1}^{1} \delta(x) \Theta(x) \mathrm{d}x

where \delta(x) is the dirac delta function, and \Theta(x) is the the Heaviside step function.

What about if I choose two functions f_k and g_k, which are such that f_k \rightarrow \delta(x) and g_k \rightarrow \Theta(x)? Will this integral converge? I understand that if i choose f_k and g_k such that f_k = g'_k then this integral will evaluate to 1/2, but what if i choose other independent representation of these functions?

 

[Simon Bridge]

Looks like translation to me.
http://en.wikipedia.org/wiki/Dirac_delta_function#Translation
How did this integral come about?

 

[SarthakC]

It isn't really translation. the Theta function makes a reasonable difference. And I'm evaluatig both funcitons without any delay. Could you elaborate how it is a case of translation?

The integral is coming about in my attempts to solve a differential equation of a particular form (I had asked a question about this earlier: https://www.physicsforums.com/showthread.php?t=760961). In trying to evaluate a some boundary conditions of this equation I'm ending up evaluating integrals of this form.

 

[Simon Bridge]

$$\int_{-1}^1 \Theta(x)\delta(x)\; dx = \int_{-\infty}^\infty \Theta(x)\delta(x)\; dx - \int_{-\infty}^1\Theta(x)\delta(x)\; dx - \int_1^\infty \Theta(x)\delta(x)\; dx$$

 

[SarthakC]

$$\int_{-1}^1 \Theta(x)\delta(x)\; dx = \int_{-\infty}^\infty \Theta(x)\delta(x)\; dx - \int_{-\infty}^1\Theta(x)\delta(x)\; dx - \int_1^\infty \Theta(x)\delta(x)\; dx$$

Um.. Could you elaborate on how this helps?

 

[pwsnafu]

IIRC, we have $\delta \cdot H = \frac{1}{2}H$, so
$\int_{-1}^{1} \delta(x)\, H(x) \, \phi(x)\, dx = \int_{-1}^{1}\frac{1}{2}H(x) \, \phi(x) = \frac{1}{2}\int_{0}^{1}\phi(x)\, dx$
for any test function $\phi \in \mathcal{D}(-1,1)$. Set $\phi(x)=1$ and we get 1/2.

 

[SarthakC]

IIRC, we have $\delta \cdot H = \frac{1}{2}H$

And how do you prove this statement? I would be more inclined to believe it to be $\frac{1}{2}\delta$, instead. But i can't prove that either.

 

[pwsnafu]

And how do you prove this statement? I would be more inclined to believe it to be $\frac{1}{2}\delta$, instead. But i can't prove that either.

Doh! You're right it would be $\frac{1}{2}\delta$ wouldn't it?
supp$(\delta\cdot H)$ = supp$(\delta)\cdot$supp(H) $=\{0\}$, so it can't be Heaviside.

When I go to uni on Monday I'll look it up for you.

 

[Simon Bridge]

The LHS is what you want to solve, on the RHS are a bunch of integrals where you should be able to do at least the first one already.

It's down to properties of the Dirac delta function.
http://tutorial.math.lamar.edu/Classes/DE/DiracDeltaFunction.aspx

 

[HallsofIvy]

The \delta "function" is NOT a true function. It is a "distribution" or "generalized function". Those are defined as operators on functions. Specifically, if f is any function defined on a closed and bounded measurable set that includes "0" then \delta(f)= f(0). Given a "regular function, g(x), we can define a corresponding generalized function using and integral- the generalized function G is given by G(f)= \int_{-\infty}^{\infy}f(t)f(t)dt. But most generalized functions, including the "\delta function" cannot be written that way. The Polish mathematician, Mikuzinski, showed how to "construct" the generalized functions from regular functions in much the way the real numbers can be "constructed" from the rational numbers using equivalence classes of sequences.

 

[pwsnafu]

Okay it is indeed $\frac{1}{2}\delta$. A proof sketch follows:

Let $\rho_n(x)$ be a delta sequence. Then define
$H(x) = \int_{-\infty}^x \rho_n(t) \, dt$.
We see that $\lim_{n\to\infty}H^2_n = H$ and hence
$\lim_{n\to\infty} 2 H_n(x) \rho_n(x) = \delta(x)$.

 

[Incnis Mrsi]

The problem is ill-defined. Considering such well-defined “deformations” of the expression under integral as \delta(x-x_0) \Theta(x),\ x\in{\mathbb R}\setminus\{0\}, we’ll obtain 0 for negative x₀ and \delta(x-x_0) for positive x₀.