Schwinger parameterization ($A^{-v}=\Gamma (v{)}^{-1}{\int }_0^{\infty }{t}^{v-1}{e}^{-tA}dt$)

For $\mathrm{Re}A>0$ and $\mathrm{Re}v>0$, rescaling the defining gamma function integral proves this representation, with compatible complex-power branches. It turns denominator powers into exponential factors, making Gaussian integrals possible. Combining two parameters by their sum and ratio gives the usual Feynman parameter identity. Outside its convergence domain a further continuation or pole prescription is needed.