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 Is the following expression for tetration useful? mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 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. « Next Oldest | Next Newest »

 Messages In This Thread Is the following expression for tetration useful? - by JmsNxn - 11/07/2012, 08:23 PM RE: Is the following expression for tetration useful? - by mike3 - 11/07/2012, 09:19 PM RE: Is the following expression for tetration useful? - by JmsNxn - 11/07/2012, 10:11 PM RE: Is the following expression for tetration useful? - by mike3 - 11/07/2012, 10:38 PM RE: Is the following expression for tetration useful? - by mike3 - 11/07/2012, 10:52 PM RE: Is the following expression for tetration useful? - by JmsNxn - 11/08/2012, 02:40 PM RE: Is the following expression for tetration useful? - by mike3 - 11/10/2012, 03:19 AM RE: Is the following expression for tetration useful? - by tommy1729 - 11/13/2012, 11:50 PM RE: Is the following expression for tetration useful? - by tommy1729 - 11/12/2012, 04:12 PM RE: Is the following expression for tetration useful? - by tommy1729 - 11/14/2012, 11:29 PM

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