Two denominators
Richard Feynman observed that [2]

which is valid for any complex numbers A and B as long as 0 is not contained in the line segment connecting A and B. The formula helps to evaluate integrals like:

If A(p) and B(p) are linear functions of p, then the last integral can be evaluated using substitution.
Multiple denominators
More generally, using the Dirac delta function
: [5]

This formula is valid for any complex numbers A1,...,An as long as 0 is not contained in their convex hull.
Even more generally, provided that
for all
:

where the Gamma function
was used. [6]
Derivation

By using the substitution
, we have
, and
, from which we get the desired result

In more general cases, derivations can be done very efficiently using the Schwinger parametrization. For example, in order to derive the Feynman parametrized form of
, we first reexpress all the factors in the denominator in their Schwinger parametrized form:

and rewrite,

Then we perform the following change of integration variables,


to obtain,

where
denotes integration over the region
with
.
The next step is to perform the
integration.

where we have defined 
Substituting this result, we get to the penultimate form,

and, after introducing an extra integral, we arrive at the final form of the Feynman parametrization, namely,

Similarly, in order to derive the Feynman parametrization form of the most general case,
one could begin with the suitable different Schwinger parametrization form of factors in the denominator, namely,

and then proceed exactly along the lines of previous case.
This page is based on this
Wikipedia article Text is available under the
CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.