(06/28/2014, 11:16 PM)tommy1729 Wrote: I wanted to remark that the "fake function" idea is generalizable.

We can find any entire asymptotic with nonnegative derivatives by simply using a rising integer function T(n) :

etc.

This simple idea / equation is very powerfull.

The fundamental theorem of fake function theory.

In combination with the post about the inverse gamma function we could for instance find the asymptotic :

with the method above.

To stay in the spirit of the exp.

regards

tommy1729

To combine 2 ideas of me we can use this to find an alternative asymptotic half-iterate F(z) of exp(z) such that F(z) = F(-z).

Or even the additional F^[2](z) = F^[2](-z) if we take F(0) = 0.

We do that by defining F(z) = a0 + a1 x + a3 x^3 + a5 x^5 + ...

In other words we take T(n) = 2n + 1 and then solve like described above.

The main difference with 2sinh^[0.5](z) is that we get a different region of almost agreement with exp(z) or even 2sinh(z) after 2 iterations.

Also , though related , notice the derivatives of 2sinh^[0.5](z) are not all positive.

Btw we should also get the similar hadamard product for this F(z) as well as for 2sinh^[0.5].

But how do the plots look like ? And where are the zero's ?

This requires some additional work.

An intresting lemma for the 2sinh if true would be this :

A := 2 pi

2sinh^[0.5](z) = a0 + a1 z + a2 z^2 + ...

g(z) := |a0| + |a1| z + |a2| z^2 + ...

Conjecture : |g(z)| < 1 + A 2sinh^[0.5](|z|)^A

Stronger and weaker variants are also of intrest.

regards

tommy1729