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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.

Last change: 2009/05/13 | 2622 reads |