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Actual formulas for tetration and its derivative

I have played around with the concept of tetration for probably 10 years now and never really achieved much in that field. I studied Andrew Robbins paper quite a while ago and played around with numerical functions constructed from the values he computed, to finally create a HyperLog function (hyper-logarithm to base E) and a HyperExp function (hyperexponential function to base E) in the full complex plane.

Now I always had some, let's call it "feelings" about what properties such functions had to have, tested that numerically and was stunned that it came out to be true using that numerical approximations.

Here are my most valuable formulas regarding tetration. I don't know whether that new or not, so I post it here anyway.

First of all, a little notation, because I don't use what's probably around. The following will be in a Mathematica-remniscent code.

Let's define functions:

E = euler's constant e
Exp[x] = E*E*E*...*E (x times)
TetraExp[x] = E^E^E^...^E (x times)
TetraExpPrime[x] = first derivative of TetraExp[x]
ProductLog[x] = lamberts W function = u in v==u*Exp[u]

Here are formulas that define TetraExp using neighbors of TetraExpPrime:

TetraExp[x] == TetraExpPrime[x] / TetraExpPrime[x-1]

TetraExp[x] == ProductLog[TetraExpPrime[x+1] / TetraExpPrime[x-1]]

And here are recurrence relations of TetraExpPrime:

TetraExpPrime[x] == TetraExpPrime[x-1] * Exp[TetraExpPrime[x-1]/TetraExpPrime[x-2]]

TetraExpPrime[x] == TetraExpPrime[x-1] * ProductLog[TetraExpPrime[x+1]/TetraExpPrime[x-1]]

TetraExpPrime[x] == TetraExpPrime[x+1] / Log[TetraExpPrime[x+2]/TetraExpPrime[x+1]]

Of course I don't have any means of proving that. Any comments still appreciated.

Messages In This Thread
Actual formulas for tetration and its derivative - by hyper4geek - 08/31/2007, 03:10 PM

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