Functional integration: Difference between revisions

Revision as of 09:51, 6 March 2017

Revision as of 03:43, 9 April 2017

Line 5:

In an [[Lebesgue integration|ordinary integral]] there is a function to be integrated (the integrand) and a region of space over which to integrate the function (the ___domain of integration). The process of integration consists of adding up the values of the integrand for each point of the ___domain of integration. Making this procedure rigorous requires a limiting procedure, where the ___domain of integration is divided into smaller and smaller regions. For each small region, the value of the integrand cannot vary much, so it may be replaced by a single value. In a functional integral the ___domain of integration is a space of functions. For each function, the integrand returns a value to add up. Making this procedure rigorous poses challenges that continue to be topics of current research.

Functional integration was developed by [[Percy John Daniell]] in aan article of 1919<ref>{{Cite journal

| volume = 20

| issue = 4

Line 22:

==Functional Integration==

{{Confusing|section|date=January 2014}}

{{unsourcedunreferenced section|date=March 2017}}

Whereas standard Riemann integration sums a function ''f''(''x'') over a continuous range of values of ''x'', functional integration sums a [[functional (mathematics)|functional]] ''G''[''f''], which can be thought of as a "function of a function" over a continuous range (or space) of functions ''f''. Most functional integrals cannot be evaluated exactly but must be evaluated using [[perturbation methods]]. The formal definition of a functional integral is

Line 38:

==Examples==

Most functional integrals are actually infinite, but the [[quotient]] of two functional integrals can be finite.{{clarification neededclarify|date=March 2017}} The functional integrals that can be solved exactly usually start with the following [[Gaussian integral]]:

:<math>