Editing Functional integration (section)

==Functional integration==
{{Confusing|section|date=January 2014}}
{{unreferenced section|date=March 2017}}
Whereas standard [[Riemann integral|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
<math display="block">
\int G[f]\; \mathcal{D}[f] \equiv \int_{\mathbb{R}}\cdots \int_{\mathbb{R}} G[f] \prod_x df(x)\;.
</math>

However, in most cases the functions ''f''(''x'') can be written in terms of an infinite series of [[orthogonal functions]] such as <math>f(x) = f_n H_n(x)</math>, and then the definition becomes
<math display="block">
\int G[f] \; \mathcal{D}[f] \equiv \int_{\mathbb{R}} \cdots \int_{\mathbb{R}}  G(f_1; f_2; \ldots) \prod_n df_n\;,

</math>

which is slightly more understandable. The integral is shown to be a functional integral with a capital <math>\mathcal{D}</math>. Sometimes the argument is written in square brackets <math>\mathcal{D}[f]</math>, to indicate the functional dependence of the function in the functional integration measure.