What is the integral of delta function

1. Why is the integral of delta 1?
2. What is the derivative of the delta function?
3. What is the Laplace transform of delta function?

Why is the integral of delta 1?

The definition of the delta function is that integrating a “test function” against it is evaluating that test function at 0. If that “test function” is the constant function 1, evaluation of that constant function at 0, or anywhere, is 1.

What is the derivative of the delta function?

In the theory of electromagnetism, the first derivative of the delta function represents a point magnetic dipole situated at the origin. Accordingly, it is referred to as a dipole or the doublet function.

What is the Laplace transform of delta function?

The Laplace transform of the Dirac delta function is easily found by integration using the definition of the delta function: Lδ(t−c)=∫∞0e−stδ(t−c)dt=e−cs.