Home

Trick

A trick for playing a multivariate integral

Wed, 05/13/2009 - 14:35 — arminstraub

For suitable functions f the integral

$$ \int_0^{\infty} \int_0^{\infty} \ldots \int_0^{\infty} \frac{dx_1 dx_2 \cdots dx_n}{(f(x_1) + f(x_2) + \ldots + f (x_n))^{k+1}} $$

is equal to

$$ \frac{1}{k!} \int_0^{\infty} t^k \left( \int_0^{\infty} e^{- t f(x)} dx \right)^n dt . $$

To see why this is true, start with the last integral and write the n-th power of the inner integral as a multiple integral over variables x1, …, xn. Then change order of integration and evaluate read more »

Syndicate content