Dirac delta function and Heaviside step function

In summary, the Dirac delta function and Heaviside step function are closely related and used in the evaluation of integrals. The value of H(x-a) at the point a can be ignored in the integral due to its vanishing nature, making the contribution of a single point negligible. This is similar to how Riemann sums work, where the contribution of the point can be represented by a small rectangle.
  • #1
pedroobv
9
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[SOLVED] Dirac delta function and Heaviside step function

In Levine's Quantum Chemistry textbook the Heaviside step function is defined as:

[tex]H(x-a)=1,x>a[/tex]
[tex]H(x-a)=0,x<a[/tex]
[tex]H(x-a)=\frac{1}{2},x=a[/tex]​

Dirac delta function is:

[tex]\delta (x-a)=dH(x-a) / dx[/tex]​

Now, the integral:

[tex]\int ^{\infty}_{-\infty}f(x)\delta (x-a)dx[/tex]​

Is evaluated using integration by parts considering

[tex]u=f(x), du=f'(x)[/tex]
[tex]dv=\delta (x-a)dx, v=H(x-a)[/tex]​

We have then:
[tex]\int ^{\infty}_{-\infty}f(x)\delta (x-a)dx=f(x)H(x-a)|^{\infty}_{-\infty}-\int ^{\infty}_{-\infty}H(x-a)f'(x)dx[/tex]

[tex]\int ^{\infty}_{-\infty}f(x)\delta (x-a)dx=f(\infty)-\int ^{\infty}_{-\infty}H(x-a)f'(x)dx[/tex]​

Since [tex]H(x-a)[/tex] vanishes for [tex]x<a[/tex], the integral becomes:

[tex]\int ^{\infty}_{-\infty}f(x)\delta (x-a)dx=f(\infty)-\int ^{\infty}_{a}H(x-a)f'(x)dx=f(\infty)-\int ^{\infty}_{a}f'(x)dx[/tex]​

This is the point where my question arrives. [tex]H(x-a)[/tex] is considered to have a value of unity for all the integral and that's why it is pulled out of the integral as a constant, however the lower bound of the integral is [tex]a[/tex] and in this point [tex]H(x-a)=1/2[/tex]. Could you please tell me if the following explanation is correct?

I think that because in all the integral, except in [tex]a[/tex], [tex]H(x-a)=1[/tex] and since the upper bound is infinity the value of the integral at the point [tex]a[/tex] can be ignored.

If I'm wrong, any suggestion for correcting my explanation will be appreciated.
 
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  • #2
You can always ignore the value of a function at a single point when you are integrating. If you remember how Riemann sums work, the contribution of the single point can be put into a rectangle of height 1/2 and width 0. So it makes no contribution.
 
  • #3
Ok, thank you very much. That answers my question.
 

1. What is the Dirac delta function and what is its significance in mathematics?

The Dirac delta function, denoted as δ(x), is a mathematical function that is defined to be zero everywhere except at the origin, where it is infinite. Its integral over the entire real line is equal to one. It is often used as a model for an idealized point source or impulse in physics and engineering, and has many applications in signal processing, calculus, and quantum mechanics.

2. How is the Dirac delta function related to the Heaviside step function?

The Heaviside step function, denoted as H(x), is a mathematical function that is defined to be zero for negative input and one for positive input. It is often used to model a switch or a step change in a system. The Dirac delta function can be represented as the derivative of the Heaviside step function, making them closely related and often used together in mathematical models.

3. Can the Dirac delta function be graphed or visualized?

No, the Dirac delta function cannot be graphed or visualized because it is an idealized mathematical concept. It has a value of infinity at a single point and is zero everywhere else, making it impossible to represent on a graph. However, its properties and effects can be observed through its applications.

4. What is the Laplace transform of the Dirac delta function?

The Laplace transform of the Dirac delta function is equal to one, denoted as 1. This means that when the Laplace transform is applied to a function containing the Dirac delta function, it simplifies to just the constant value of one. This property is useful in solving differential equations and analyzing systems with impulse inputs.

5. Are the Dirac delta function and Heaviside step function interchangeable?

No, the Dirac delta function and Heaviside step function are not interchangeable. While they are related, they serve different purposes and have different mathematical properties. The Dirac delta function represents an idealized point source, while the Heaviside step function represents a step change in a system. They can be used together in mathematical models, but they cannot be interchanged for one another.

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