Simplified form for $\int_{\lambda}^\infty \int_{0}^\infty e^{-(s+t)}(st)^{x-1}\ln \frac{s}{t} ds dt$

Question

I'm interested in a simplification for the integral given by $$\int_{\lambda}^\infty \int_{0}^\infty e^{-(s+t)}(st)^{x-1}\ln \frac{s}{t} ds dt$$

which arises as a definition for the density of a random variable following the "continuous Poisson" distribution.

Answer 1

This isn't a complete answer.

Here is a way to convert the double integral into two simple integrals (that are numerically more tractable).

Write the initial integral

$$\int_{\lambda}^\infty \int_{0}^\infty e^{-(s+t)}(st)^{x-1}\ln \frac{s}{t} ds dt$$

into

$$\int_{\lambda}^\infty e^{-s}s^{x-1}\int_{0}^\infty e^{-t}t^{x-1}(\ln s - \ln t) dt ds$$

$$=\int_{\lambda}^\infty e^{-s}s^{x-1}\ln s \underbrace{\left(\int_{0}^\infty e^{-t}t^{x-1}dt\right)}_{\Gamma(x) \ \text{independent of} s} ds - \int_{\lambda}^\infty e^{-s}s^{x-1}\underbrace{\left(\int_{0}^\infty e^{-t}t^{x-1}\ln t dt\right)}_{I=\Gamma^{\ \prime}(x)} ds$$

For the proof of $I=\Gamma'(x)$, see for example here.

Sorry to stop there, wishing you further advances by different techniques (integration by parts, etc...)

Answer 2

There's not much simplification you can really do, but there is a nice connection to the Wronskian determinant.

Let $$F(x)=\int_{\lambda}^\infty\int_0^\infty e^{-s-t}(st)^{x-1}\log(s/t)\mathrm{d}s\mathrm{d}t$$ First separate the log- $$=\int_\lambda^\infty\int_0^\infty e^{-s-t}(st)^{x-1}\left(\log s-\log t\right)\mathrm ds\mathrm dt$$ Which we break up, and then separate variables. $$=\int_\lambda^\infty e^{-t}t^{x-1}\int_0^\infty e^{-s}s^{x-1}\log(s)\mathrm ds\mathrm dt-\int_{\lambda}^\infty e^{-t}t^{x-1}\log t\int_0^\infty e^{-s}s^{x-1}\mathrm ds\mathrm dt$$ Recognizing the gamma function and its derivative, as well as the upper incomplete Gamma function, this is $$=\Gamma'(x)\gamma_{\text{up}}(x,\lambda)-\Gamma(x)~\partial_x\gamma_{\text{up}}(x,\lambda)$$ This is in fact the Wronskian of the two: $$F(x)=W(\gamma_{\text{up}},\Gamma)(x)$$ So there are certain ways you can relate it to solutions of second order linear ODEs.