A trick for playing a multivariate integral

Date: May 13, 2009

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

$$ \int_0^{\infty} t^k e^{- t (f(x_1) + \ldots + f(x_n))} dt = \frac{\Gamma(k+1)}{(f(x_1) + \ldots + f(x_n))^{k+1}} $$

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.

in

Comments

Post new comment

The content of this field is kept private and will not be shown publicly.