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Theorem on tetration.
#1
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
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