arminstraub.com

A trick for playing a multivariate integral

For suitable functions ff the integral ∫0∞∫0∞…∫0∞dx1dx2⋯dxn(f(x1)+f(x2)+…+f(xn))k+1 \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 1k!∫0∞tk(∫0∞e−tf(x)dx)ndt. \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 nn-th power of the inner integral as a multiple integral over variables x1,…,xnx_1, \ldots, x_n. Then change order of integration and evaluate ∫0∞tke−t(f(x1)+…+f(xn))dt=Γ(k+1)(f(x1)+…+f(xn))k+1 \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 Γ(z)=∫0∞tz−1e−tdt \Gamma(z) = \int_0^{\infty} t^{z-1} e^{- t} dt of the gamma function. Note that we don't need to assume that kk 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 ff is the hyperbolic cosine and the integral K0(t)=∫0∞e−tcosh⁡(x)dx K_0(t) = \int_0^{\infty} e^{- t \cosh(x)} dx is a modified Bessel function.