Ramp function, Dirac delta function and distributions

In summary, the conversation discusses the function r(x) and its second derivative, as well as the integral involving r(x) and the function phi(x). The first point is proven using integration by parts, but the second point raises a question about why the second derivative of r(x) cannot simply be written as 0. The reason for this is that r(x) only has a second derivative in the distribution sense, which is why it cannot be written as 0.
  • #1
Amok
256
2
[tex]r(x) = x[/tex] if [tex]x \geq 0[/tex] and [tex]r(x) = 0 [/tex] if [tex] x<0[/tex]

I have to show that:

1-[tex]\[ \int_{- \infty}^{+ \infty} r(x) \varphi ''(x) dx = \varphi(0) \][/tex]

And 2- that the second derivative of r is the Dirac delta.

And I managed to do this by integrating by parts. Howver, I don't get why I can't just write:

[tex]\[ \int_{- \infty}^{+ \infty} r''(x) \varphi (x) dx = \varphi(0) \][/tex]

Wouldn't that be correct considering distributions (I actually used this to show the second point)? I guess my question is, why can't I write the second derivatives of the ramp function (the derivative of the Heaviside function) simply as

[tex]r(x) = 0[/tex] if [tex]x \geq 0[/tex] and [tex]r(x) = 0 [/tex] if [tex] x<0[/tex]

i.e. 0

Which would make the integral = 0

Does it only have a second derivative in the distribution sense? Why?

EDIT: I don't get why my message is being displayed like this...
 
Last edited:
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  • #2
Change the backslash in your closing tex tags to a forward slash, i.e. /tex instead of \tex.
 

1. What is the definition of a ramp function?

A ramp function is a mathematical function that increases at a constant rate from a specific starting point. It is typically represented by the equation f(x) = ax, where a is the slope of the ramp.

2. How is a Dirac delta function defined?

A Dirac delta function is a mathematical function that is defined as zero everywhere except at a single point, where it is infinitely large. Its integral over any interval is equal to 1, making it a useful tool in solving differential equations and representing impulses in physics and engineering.

3. What is the relationship between a ramp function and a Dirac delta function?

The ramp function can be seen as the integral of a Dirac delta function. This means that the ramp function can be thought of as the accumulation of infinitely small impulses represented by the Dirac delta function.

4. How are distributions used in mathematics?

Distributions are used in mathematics to generalize the concept of a function and to extend the notion of a derivative to functions that are not necessarily continuous. They are also useful in solving differential equations and representing physical systems that involve impulses or discontinuities.

5. Can distributions be visualized like regular functions?

No, distributions cannot be visualized in the same way as regular functions because they are not continuous. They are defined as generalized functions and do not have an explicit graph like traditional functions. Instead, they are represented by their properties and relationships with other functions.

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