Feynman parametrization

Feynman parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops. However, it is sometimes useful in integration in areas of pure mathematics as well. It was introduced by Julian Schwinger and Richard Feynman in 1949 to perform calculations in quantum electrodynamics.

Richard Feynman observed that

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.

More generally, using the Dirac delta function ${\displaystyle \delta }$:

This formula is valid for any complex numbers A₁,...,A_n as long as 0 is not contained in their convex hull.

Even more generally, provided that ${\displaystyle {\text{Re}}(\alpha _{j})>0}$ for all ${\displaystyle 1\leq j\leq n}$:

where the Gamma function ${\displaystyle \Gamma }$ was used.