Generalized Function

The class of all regular sequences of Particularly Well-Behaved Functions equivalent to a given regular sequence (sometimes also called a Distribution or Functional). A generalized function $p(x)$ has the properties

$$\int_{-\infty}^\infty p'(x)f(x)\,dx=-\int_{-\infty}^\infty p(x)f'(x)\,dx$$

$$\int_{-\infty}^\infty p^{(n)}(x)f(x)\,dx=(-1)^n \int_{-\infty}^\infty p(x)f^{(n)}(x)\,dx.$$

The Delta Function is a generalized function.

See also Delta Function