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Searching for an asymptotic to exp[0.5]
#45
The final chapter on fake exp^[0.5] into a Hadamard product is the consideration of the zero's off the real line.

Thats the tricky part and Its not immediate how to take care of it.
Perhaps a further generalization of Carlson ?

Note that I do NOT intend to say we already understand the zero's on the real line completely.

But I wonder what would happen if we " forget " these zero's in our product ? What kind of function would we get ?
I have some ideas , but its mainly just handwaving at the moment.

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A visual satisfaction would be a video showing fake exp^[1/k](z)^[k] for k slowly growing from 0 to 99 , the analogue of sheldon's pictures.

Talking about that picture of fake exp^[0.5](z)^[2] - exp(z) it seems that for z near the negative line the function grows faster than polynomial.

I cant help comparing to 2sinh^[0.5](z) where we got that 2sinh^[0.5](z)^[2] - exp(z) grows like exp(|z|) near the negative real line.

There seem to be many ideas from those type of comparisons.
For instance some type of conservation law saying that if an approximation is better somewhere it must be worse somewhere else , and you cannot have one that is worse everywhere (or better everywhere).

Such ideas seem complex right now.

regards

tommy1729
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RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 07/10/2014, 11:48 PM

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