 Theorem on tetration. - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Theorem on tetration. (/showthread.php?tid=883) Theorem on tetration. - JmsNxn - 06/09/2014 Hey everyone. I thought I'd post this theorem, perhaps someone has some uses for it. Theorem: A.) If is holomorphic for for some and for and . B.) for some and we have Then, for we have Proof: Well this is rather easy: Which follows by cauchy's residue formula and the bounds of F (the gamma function along with x small enough pulls the arc next to our line integral to zero at infinity). For those who don't see, where the right term is entire in z and only contribute asymptotics, observe stirlings asymptotic formula Therefore this holds. Now observe that by a similar argument: And of course, by another similar argument: Therefore since the kernel of this integral transform is zero (its a modified fourier transform). On the line we have . Therefore since both functions are analytic we get the desired. I'm wondering, does anyone see any uses for this? I know with some formal manipulation we can say that, if and and is holo and is invertible which satisfies the bounds above. Then