11/10/2012, 03:19 AM

I've been playing around a bit with that integral

.

The thing is, this integral doesn't seem to converge directly. Namely, the reciprocal gamma function blows up faster-than-exponentially toward the left. But, I found that if we make x sufficiently large, and then the lower bound not too large, it seems this gives a sort of "asymptotic" integral that gives . Take, e.g.

.

This behavior makes me wonder whehter it's not possible to somehow regularize this integral with some form of "divergent integration" technique, analogous to divergent summation for sums.

.

The thing is, this integral doesn't seem to converge directly. Namely, the reciprocal gamma function blows up faster-than-exponentially toward the left. But, I found that if we make x sufficiently large, and then the lower bound not too large, it seems this gives a sort of "asymptotic" integral that gives . Take, e.g.

.

This behavior makes me wonder whehter it's not possible to somehow regularize this integral with some form of "divergent integration" technique, analogous to divergent summation for sums.