Dirac delta function how did they prove this?

In summary, the conversation discusses the Dirac Delta function and its properties. The function is not a usual function, but rather a distribution or generalized function that is applied to other functions. It is mentioned that \delta applied to an odd function is 0. There is a question about proving the relationship \delta(x)=\delta(-x) on the interval -a to b when integral signs are present, with a suggestion to use a change of variables and changing limits of integration. However, it is stated that the proof is not clear.
  • #1
galactic
30
1
Hi all,

I'm familiar with the fact that the dirac delta function (when defined within an integral is even)

Meaning delta(x)= delta(-x) on the interval -a to b when integral signs are present

I want to prove this this relationship but I don't know how to do it other than with a limit maybe

Book said they proved it using a change of variables and changing limits of integration but I can't see how they proved it? Does anyone know how?
 
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  • #2
What you have written is pretty much meaningless. The "Dirac Delta function" is not a function at all. It is a "distribution" or "generalized function" which means it is applied to functions, not numbers as are ordinary functions. In particular, that means that we do not define either [itex]\delta(x)[/itex] or [itex]\delta(-x)[/itex] for specific values of x. What is true is that [itex]\delta[/itex] applied to an odd function is 0.
 

Related to Dirac delta function how did they prove this?

1. What is the Dirac delta function?

The Dirac delta function is a mathematical function that is defined as zero for all values of its argument except for when the argument is equal to zero. At the point where the argument is zero, the Dirac delta function is defined as infinity.

2. How did they prove the existence of the Dirac delta function?

The existence of the Dirac delta function was first postulated by physicist Paul Dirac in the 1920s. He used the concept of generalized functions, also known as distributions, to define the Dirac delta function. This approach was later mathematically formalized by mathematician Laurent Schwartz in the 1950s.

3. What is the physical significance of the Dirac delta function?

The Dirac delta function is often used in physics to model point-like objects or events, such as a particle at a specific location or an impulsive force acting on a system. It is also used in signal processing to represent a sudden change or spike in a signal.

4. How is the Dirac delta function used in mathematical calculations?

The Dirac delta function is a useful tool in mathematical calculations, particularly in integration and Fourier analysis. It allows for the simplification of certain integrals and the representation of functions as infinite sums of Dirac delta functions.

5. Are there any limitations or criticisms of the Dirac delta function?

While the Dirac delta function is a powerful mathematical tool, it has its limitations. It is not a conventional function and cannot be evaluated at any point other than zero. It also does not satisfy some of the usual properties of functions, leading to criticisms and debates about its use in mathematical and physical contexts.

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