Proving an identity of Dirac's delta function

Thread starter Chen
Start date Nov 22, 2005
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Delta Delta function Function Identity

In summary, Dirac's delta function is a mathematical function used to model point-like sources or impulses. It is defined as a function that is zero everywhere except at the origin, where it has an infinite value, and has an area of 1 under its curve. The Dirac delta function is related to the identity function as it can be thought of as the identity function with an infinitely narrow peak at the origin. Proving identities of Dirac's delta function is significant as it allows for simplification of mathematical expressions and understanding its behavior and relationships to other functions. There are various methods for proving identities, including using properties of the delta function, manipulating integrals, and using Fourier transforms. Some common applications of Dirac's delta function include

Nov 22, 2005

Chen

Hello,

I need to prove (7) here:
http://mathworld.wolfram.com/DeltaFunction.html

http://mathworld.wolfram.com/images/equations/DeltaFunction/equation5.gif

The instructions were to start with the definition of the delta function by integral, and then chagne variables u -> g(x). But I couldn't get anywhere really.

Thanks,
Chen

Last edited by a moderator: May 2, 2017

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Nov 22, 2005

Chen

Never mind.

1. What is Dirac's delta function?

Dirac's delta function is a mathematical function used in physics and engineering to model point-like sources or impulses. It is defined as a function that is zero everywhere except at the origin, where it has an infinite value, and has an area of 1 under its curve.

2. How is Dirac's delta function related to the identity function?

The Dirac delta function can be thought of as the identity function, but with an infinitely narrow peak at the origin. This means that the integral of the Dirac delta function with any other function is equal to the value of that function at the origin.

3. What is the significance of proving an identity of Dirac's delta function?

Proving an identity of Dirac's delta function is important because it allows us to use the properties of this function to simplify mathematical expressions and solve problems in physics and engineering. It also helps us understand the behavior of this function and its relationship to other functions.

4. How can we prove an identity of Dirac's delta function?

There are several methods for proving an identity of Dirac's delta function, including using properties of the delta function, manipulating integrals, and using Fourier transforms. It is important to carefully follow the rules and properties of the delta function to ensure a valid proof.

5. What are some common applications of Dirac's delta function?

Dirac's delta function has many applications in physics and engineering, including modeling point sources in electrical circuits, analyzing impulse responses in signal processing, and solving differential equations in quantum mechanics. It is also used in probability and statistics to model random variables and distributions.