Properties of Dirac delta function

In summary, the student is trying to prove \delta'(y)=-\delta'(-y) by using a test function f(y) and integrating by parts. They have found the LHS to be -f'(0) and the RHS to be f'(0), but are unsure of where the minus sign comes from. Another student suggests using the chain rule to prove the statement.
  • #1
signalcarries
4
0

Homework Statement



I'm trying to prove that [tex]\delta'(y)=-\delta'(-y)[/tex].

Homework Equations





The Attempt at a Solution



I'm having trouble getting the LHS and the RHS to agree. I've used a test function [tex]f(y)[/tex] and I am integrating by parts.

For the LHS, I have
[tex]\int_{-\infty}^{\infty} f(y)\delta'(y)dy = \int_{-\infty}^{\infty} \frac{d}{dy}[f(y)\delta(y)]dy - \int_{-\infty}^{\infty} \delta(y)\frac{df(y)}{dy}dy = 0 - f'(0) = -f'(0)[/tex]

For the RHS, I have
[tex]-\int_{-\infty}^{\infty} f(y)\delta'(-y)dy = \int_{\infty}^{-\infty} f(-t)\delta'(t)dt = -\int_{-\infty}^{\infty} f(-t)\delta'(t)dt = -\int_{-\infty}^{\infty} \frac{d}{dt} [f(-t)\delta(t)]dt + \int_{-\infty}^{\infty} \frac{df(-t)}{dt} \delta(t)dt = 0 + \int_{-\infty}^{\infty} \frac{df(-t)}{dt} \delta(t)dt = f'(0)[/tex].

I seem to be off by a minus sign, but I can't figure out where. Any help would be appreciated.
 
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  • #2
df(-t)/dt=-df(t)/dt at t=0.
 
  • #3
Yes, I suppose it does. Thanks!
 
  • #4
Dick said:
df(-t)/dt=-df(t)/dt at t=0.

A proof of the above statement would be more helpful.
 
  • #5
the_amateur said:
A proof of the above statement would be more helpful.

Chain rule.
 

Related to Properties of Dirac delta function

What is the Dirac delta function?

The Dirac delta function, also known as the Dirac delta distribution, is a mathematical concept used to represent a point mass or impulse at a specific point. It is often described as an infinitely narrow and tall function that is zero everywhere except at the origin, where it is infinity.

What are the properties of the Dirac delta function?

Some of the key properties of the Dirac delta function include:
- It is an even function, meaning that it is symmetric about the y-axis.
- It has a total area of 1.
- It has a value of 0 everywhere except at the origin, where it is infinity.
- It follows the sifting property, which states that the function is equal to 1 when the argument is 0 and 0 otherwise.
- It can be shifted, stretched, and scaled without changing its properties.

How is the Dirac delta function used in physics?

The Dirac delta function is commonly used in physics to represent point sources or idealized impulses. It is used in the study of electromagnetism, quantum mechanics, and signal processing, among other fields. In physics, the Dirac delta function is often used to simplify equations and make them more manageable.

What is the relationship between the Dirac delta function and the Kronecker delta?

The Dirac delta function and the Kronecker delta are two different mathematical concepts, but they are related. The Kronecker delta is a discrete analog of the Dirac delta function, used to represent impulses in discrete systems. The Kronecker delta has a value of 1 when the arguments are equal and 0 otherwise, whereas the Dirac delta function has a value of 0 everywhere except at the origin, where it is infinity.

What are some real-life applications of the Dirac delta function?

The Dirac delta function has many real-life applications, including:
- In electrical engineering, it is used to model idealized impulses in electronic circuits.
- In signal processing, it is used to represent impulses in digital signals.
- In quantum mechanics, it is used to represent point particles in certain calculations.
- In fluid dynamics, it is used to model point sources of mass or energy.
- In economics, it is used to represent sudden changes in demand or supply.
- In image processing, it is used to sharpen edges and enhance details in images.

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