11/10/2012, 03:19 AM
I've been playing around a bit with that integral
\( Ce^x = \int_{-\infty}^{\infty} \frac{x^t}{\Gamma(t+1)} dt \).
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 \( e^x \). Take, e.g.
\( \int_{-10}^{\infty} \frac{4^t}{\Gamma(t+1)} dt \approx 53.812 \approx e^4 \).
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.
\( Ce^x = \int_{-\infty}^{\infty} \frac{x^t}{\Gamma(t+1)} dt \).
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 \( e^x \). Take, e.g.
\( \int_{-10}^{\infty} \frac{4^t}{\Gamma(t+1)} dt \approx 53.812 \approx e^4 \).
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.