Distribution (mathematics)

Distribution (mathematics)

In mathematical analysis, distributions (or generalized functions) are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used to formulate generalized solutions of partial differential equations. Where a classical solution may not exist or be very difficult to establish, a distribution solution to a differential equation is often much easier. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function (which is historically called a "function" even though it is not considered a proper function mathematically).

Generalized functions were introduced by Sergei Sobolev in 1935. They were re-introduced in the late 1940s by Laurent Schwartz, who developed a comprehensive theory of distributions.