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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 |