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Closed-form derivatives
#1
Looking around the forum, I can't find good references for these, although there are lots of references to these formulas, I think it would be good to have these together, so here they are:


where


I also found a descent approximation to T(a):



Andrew Robbins
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#2
The I use this thread for giving the general formula for a superfunction of :


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#3
(08/23/2009, 01:12 AM)andydude Wrote:

Doesnt it need to read

?
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#4
@Ansus
Thank you so much for finding that mistake! I fixed it in my original post.

Also, I noticed in the MathFacts page that it is a very complete overview, except for two things: the base-sqrt(2) tetration approximation: , and intuitive/natural tetration, for which i would say that the matrix encoding of

is

which can be solved for "intuitively" despite the fact that is a noninvertible matrix.
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#5
(08/24/2009, 05:37 AM)andydude Wrote: @Ansus
Thank you so much for finding that mistake! I fixed it in my original post.

Hey, I also discovered that mistake! Only my solution suggestion was different from Ansus'.
Wink
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#6
I really have to put my foot down on this one. The lower limit is is not zero as it appears in the Tetration_Summary page. It is 1. I have re-derived a more general formula for this that accentuates this lower index:

as you can see from this, if the final derivative in the denominator is evaluated at (), then this means , which means the lower index of the product is (k=1), not (k=0).

@Ansus
Your derivations are based on the (k=0) formula (which is wrong), but other than that, they are quite clever! I never thought to do that. I think there would be less room for error if we use the "P" function to simplify things. Starting with the basic derivatives:




combining them gives:



which is about as rigorous as I can make it, so that should be right.

Andrew Robbins
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#7
(09/02/2009, 10:09 AM)Ansus Wrote: Note that this formula ... works both with lower limit 0 and 1 because .

True, but it does have unfortunate misunderstandings later on, like the extra in the final formula, which is incorrect. Using index substitution, the right formula is:



Our formulas are identical except for the , which should be . This extra logarithm comes from the wrong index.
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#8
(09/03/2009, 01:51 AM)Ansus Wrote: Only because it gives extra ln(a)?
Yes, but wait. I'm wrong now. It is in the denominator... I'm sorry. You're right. All of this off-by-one stuff is hard to keep track of.
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