What is the product of two Dirac delta functions

In summary, the product of two Dirac delta functions is not well-defined and cannot be simplified. It represents the convolution of two impulses and has infinite area when multiplied by another Dirac delta function. The product has many physical applications, including representing the impulse response of a system and being used in signal processing and engineering.
  • #1

Homework Statement


What is the product of two Dirac delta functions

δ(Real(z-c))δ(Img(z-c))=?
'z' and 'c' are complex numbers.

This is not a problem, But I just need to use this formula in a derivation that I am currently doing. I just want the product of these two Dirac delta functions as a single/double Dirac delta function. I don't need the derivation, just the formula is enough.
 
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  • #2
If your δ are real-valued delta distributions, that product is simply the complex-valued delta distribution, δ(z-c). You can verify this by integrating over the complex plane.
 

1. What is the product of two Dirac delta functions?

The product of two Dirac delta functions is not well-defined. This is because the Dirac delta function is a distribution and not a regular function, so it does not follow the usual laws of multiplication.

2. Can the product of two Dirac delta functions be simplified?

No, the product of two Dirac delta functions cannot be simplified. As mentioned, the Dirac delta function is a distribution and not a regular function, so it does not follow the usual laws of algebra.

3. What does the product of two Dirac delta functions represent?

The product of two Dirac delta functions represents the convolution of two impulses. Convolution is a mathematical operation that combines two functions to produce a third function, which in this case is the product of two Dirac delta functions.

4. Is the product of two Dirac delta functions equal to zero?

No, the product of two Dirac delta functions is not equal to zero. While the Dirac delta function is often represented as a spike at a single point, it has an area of infinity, and when multiplied by another Dirac delta function, the product is also infinite.

5. What is the physical significance of the product of two Dirac delta functions?

The product of two Dirac delta functions has many physical applications, such as representing the impulse response of a system, or the response of a system to an impulsive input. It is also used in signal processing and engineering to represent the response of a system to a delta function input.

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