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. 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 formula_3:  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 formula_5 for all formula_6:  where the Gamma function formula_8 was used. Now just linearly transform the integral using the substitution, and 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 formula_14, 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 formula_20 denotes integration over the region formula_21 with formula_22. The next step is to perform the formula_23 integration.  where we have defined formula_25  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, :formula_28 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. An alternative form of the parametrization that is sometimes useful is  This form can be derived using the change of variables formula_31. We can use the product rule to show that formula_32, then  More generally we have where formula_35 is the gamma function. This form can be useful when combining a linear denominator formula_36 with a quadratic denominator formula_37, such as in heavy quark effective theory (HQET).