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I'm interested in considering the properties of tetration from the perspective of hyperreal algebra and Non-Standard analysis. I would also like to consider analyzing tetration in complex spaces using non-standard complex analysis. I believe there is a great deal of potential in using this approach because relationships are easier to see using the infinitesimal approach than they are using the classical analytic approach.

Can anybody help me get started?

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(01/01/2011, 10:31 PM)JungleJesus Wrote: I'm interested in considering the properties of tetration from the perspective of hyperreal algebra and Non-Standard analysis. I would also like to consider analyzing tetration in complex spaces using non-standard complex analysis. I believe there is a great deal of potential in using this approach because relationships are easier to see using the infinitesimal approach than they are using the classical analytic approach.

I dont think non-standard analysis helps here.

The problems lie on a different level.

And there is anyway no monograph about tetration yet.

But perhaps you can explain first why you see potential in that approach. One never knows

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I see potential in this approach because the hyperreal number system is easier to work with than the real numbers when performing analysis. With the hyperreals, problems in Calculus become expressible in terms of logic and abstract algebra.

The analytic continuation of tetration and its inverse-operations may seem more forthcoming when using this approach. It may even be possible to identify certain properties that make the continuation unique (ie: Logarithmic Convexity was used to analytically continue the factorial via the construction of the Gamma Function, which is a unique continuation).

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(01/06/2011, 01:14 AM)JungleJesus Wrote: ie: Logarithmic Convexity was used to analytically continue the factorial via the construction of the Gamma Function, which is a unique continuation

Ya but thats exactly the point: Whether you use standard or non-standard analysis, this uniqueness criterion is similarly hard/easy to find. And I guess it is the case for most other problems that arise with tetration.

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Non-standard analysis appeals to me because it is more intuitive. It has a modern "feel" about it. Hyperreal algebra is well defined in terms of abstract algebra, so the addition of a new operator (tetration) should be easier to construct than it would be in the standard approach.

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(01/07/2011, 02:59 AM)JungleJesus Wrote: Hyperreal algebra is well defined in terms of abstract algebra, so the addition of a new operator (tetration) should be easier to construct than it would be in the standard approach.

I dont think there is anything written about it. It seems you have construct this operator yourself. I am happy to to hear about this construction when you succeed.