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Searching for an asymptotic to exp[0.5]
I considered exp(exp(x)) by studying the Bell Numbers and Lambert W.
If my calculations are correct, the fake coëfficiënts and the derivatives match very well.

In fact the correcting factor is sqrt( 2 pi n ) (1 + log(n)^C).

I have yet to determine C but it Will be close to -4 < C < 4.

Very similar like with exp(x) !

And like exp^[1/2].

The difficulty with these computations is having good enough estimates for the analogues of Bell and Lambert W.

Fake function ideas seem to show these estimates relate , without computing the estimates first !

I noted that the gaussian method gives a slightly different result. More specific it adds a log factor.

The value of C Will determine if the gaussian beats S9, but my bet is it does.

This also explains the ln part of tpid 17.
Let me explain :

Notice that any function f bounded by exp^[A] above and exp^[B] below , Will satisfy on average

f ' ' (x) / f(x) < O ( ln(f(x)) )^(2+ eps)

Hence we get the ln part.

In fact by this argument we need

O ( sqrt(n) ln(n) ln^2(n) ln^3(n) ... )

As error in tpid 17.
And add the bounds too.

Regards

Tommy1729
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RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/30/2015, 12:25 PM

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