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In using Feynman parametrisation, I have noticed different expressions given in the literature that seem to imply

$$ \int_0^1dx\int_0^1dy\int_0^1dz\delta(1-x-y-z)f(x,y,z)=\int_0^1dx\int_0^{1-x}dyf(x,y,z)|_{z=1-x-y}. $$

However I have been unable to prove this. Is this statement true and if so why?

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I have figured out the answer to this. The result is not totally general but may be found in each case by writing the integrals as integrals over an infinite range and putting Heaviside step functions in the integrand to restore the finite range of integration.

The Dirac-deltas change the arguments of the step functions and hence the limits of integration.

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