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# A trick for playing a multivariate integral

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 $$x_1, \ldots, x_n$$. Then change order of integration and evaluate $$\int_0^{\infty} t^k e^{- t (f(x_1) + \ldots + f(x_n))} dt = \frac{\Gamma(k+1)}{(f(x_1) + \ldots + f(x_n))^{k+1}}$$ which follows from the integral representation $$\Gamma(z) = \int_0^{\infty} t^{z-1} e^{- t} dt$$ of the gamma function. Note that we don't need to assume that $$k$$ is an integer.

I came across this trick while browsing the Journal of Experimental Mathematics and reading the article A Proof of a Recurrence for Bessel Moments. In this article Jonathan M. Borwein and Bruno Salvy are interested in the case where $$f$$ is the hyperbolic cosine and the integral $$K_0(t) = \int_0^{\infty} e^{- t \cosh(x)} dx$$ is a modified Bessel function.