Integral after alpha parametrization

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SUMMARY

The discussion centers on evaluating the double integral ∫(0 to inf) dα1 ∫(0 to inf) dα2 a1^(n1) * exp(-i (α1+α2) m^2) * (α1+α2)^(n2) using alpha parametrization techniques. The user was advised to apply the substitutions α1 = 1/2 (t+u) and α2 = 1/2 (t-u) to simplify the integral. However, the user encountered difficulties in seeing the benefits of these substitutions and requested further guidance on the algebra involved in this process.

PREREQUISITES
  • Understanding of double integrals in calculus
  • Familiarity with Feynman integrals and their parametrization
  • Knowledge of substitution methods in integral calculus
  • Basic proficiency in complex exponentials and their properties
NEXT STEPS
  • Review techniques for evaluating double integrals using substitution methods
  • Study the application of alpha parametrization in Feynman integrals
  • Learn about the properties of complex exponentials in integrals
  • Practice algebraic manipulation of integrals post-substitution
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Mathematicians, physicists, and students working on advanced calculus problems, particularly those involving Feynman integrals and complex analysis.

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I have been stuck on the following double integral for some time:

∫(0 to inf) dα1 ∫(0 to inf) dα2 a1^(n1) * exp(-i (α1+α2) m^2) * (α1+α2)^(n2)

which arose after using alpha paremetrization on a Feynman integral. I was advised by my supervisor to use the substitution α1 = 1/2 (t+u) and α2 = 1/2 (t-u) but after doing this I fail to see how it helps.

Any advice on how to do this?

Thanks.
 
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Show your work using the recommended substitutions. Perhaps you made an error in your algebra.
 

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