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Operator similar to tetration
#1
Daniel Geisler referred me to this forum, and I'm surprised how active a forum is on a topic like tetrationSmile

I have a question about an operator that is similar to a tetration that I'm trying to locate. I'm a bit new to this topic so forgive the poor terminology.

An infinite tetriation with exponent (1/2) could be written as a recurrence relation, I believe:

F(n) = square_root( F(n-1) )
F(0) =.5

I'm trying to locate the mathematical operator that describes something very similar, but just multiplying a negative through each iteration (see example).

F(n) = square_root( -1 * F(n-1) )

Please let me know if you could help me out with this. I think this has some very interesting properties. Any guidance would be appreciated. Thanks.

-Ryan G.
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#2
rsgerard Wrote:Daniel Geisler referred me to this forum, and I'm surprised how active a forum is on a topic like tetrationSmile
Smile

Quote:I have a question about an operator that is similar to a tetration that I'm trying to locate. I'm a bit new to this topic so forgive the poor terminology.

An infinite tetriation with exponent (1/2) could be written as a recurrence relation, I believe:

F(n) = square_root( F(n-1) )
F(0) =.5

Thats not infinite tetration. If you expand this, you get







I would guess what you mean is (at least this resembles infinite tetration):







yes, no?

Quote:I'm trying to locate the mathematical operator that describes something very similar, but just multiplying a negative through each iteration (see example).

F(n) = square_root( -1 * F(n-1) )

Ok, taking your original definition a bit more generally:






Now we know that for

hence for



in our case and we get


however the derivation is somewhat sloppy as the exponential laws are generally not applicable for complex numbers (as painfully observed by Gottfried Wink ). So the above should be considered only valid for .

Indeed one sees that the the sequence has no limit, but oscillates between two values, i.e. has two limit points. The task for the reader is to determine these both points Smile
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#3
bo198214,
Quote:Thats not infinite tetration. If you expand this, you get







I would guess what you mean is (at least this resembles infinite tetration):







yes, no?
You are completely correct, the recurrence relation I suggested does not evaluate the exponent from "highest to lowest". I agree that the rewritten expression accurately describes a tetration.

Quote:Ok, taking your original definition a bit more generally:






Now we know that for

hence for



in our case and we get


however the derivation is somewhat sloppy as the exponential laws are generally applicable for complex numbers (as painfully observed by Gottfried Wink ). So the above should be considered only valid for .

Indeed one sees that the the sequence has no limit, but oscillates between two values, i.e. has two limit points. The task for the reader is to determine these both points Smile
First, thanks a lot for expanding the math out here. This is very helpful to see.

When I attempt to evaluate this recurrence, I see the result oscillate between 2 values when the principle root is used and -1 otherwise. So, I get the following solutions:



I believe these are the roots for

My two specific questions are:
1. Is this above correct?
2. Has something similar ever been shown in a more general case:
For example:
seems to be the roots for ???

Sorry for being a bit off topic to tetrations but this operation performed an infinite number of times relates to taking the 1/(n+1) exponent.

Please let me know if this is on the right track, or have I derailed somewhereSmile Thanks.

-Ryan
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#4
rsgerard Wrote:When I attempt to evaluate this recurrence, I see the result oscillate between 2 values when the principle root is used and -1 otherwise. So, I get the following solutions:



I believe these are the roots for

My two specific questions are:
1. Is this above correct?
2. Has something similar ever been shown in a more general case:
For example:
seems to be the roots for ???

First the roots of are
, .

So the roots of are at
, and :
If we look up the values , we indeed get that the roots of are and .

To tackle the general question we first have to know how a root at the complex domain is defined. More precisely which root is chosen from the possible roots of .
The general definition of the power in the complex domain is:
, where is the standard branch of the logarithm, i.e. is chosen such that the .

The oddity is that has a jump (not continuous) at the negative real axis. If we approach -1 from the upper plane we get and if we approach -1 from the lower plane we get . Both values are apart and if we repeat winding around 0 we get all the other branches of the logarithm .

So how applies this to the case of roots?
, where again is chosen. Taking the n-th root divides the argument/angle by n, in the way the angle is chosen it moves the point towards the positive real axis, like a scissor.

If we however consider (for simplicity for ) we first mirror at 0 and then divide the angle by towards the positive real axis. Mirroring at 0 means either to add for or to subtract for .

Say we start with a value in the upper halfplane, i.e. , then , is a point in the lower halfplane, is again in the upper halfplane and so on:





Generally
with and
with .

We dont know yet whether the sequences and have a limit, but if they have then the limit for must satisfy:
hence

, via :

Similarly the limit of would be
.

To be really complete one have to show that is strictly increasing for a starting value and striclty decreasing for a starting value , because then the existence of the limit is guarantied.

Ok, conclusion, if we compare above the roots of we see that the two limit points of are exactly the both roots of that are nearest the positive real axis ( and ). (That was also clear.)
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#5
bo198214,

Thanks for going above and beyond and actually going through the details on how this function behaves. I find it particularly interesting that an infinite number of square roots can relate to the cube root function and so on.

Any suggestions on where to read up on this more? I really haven't found too many places to learn about sequences with roots. Also, have you come across this particular sequence in any of your other work? Thanks again for the thorough response.

-Ryan
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#6
rsgerard Wrote:I find it particularly interesting that an infinite number of square roots can relate to the cube root function and so on.
Yes, thats really surprising.

Quote:Any suggestions on where to read up on this more? I really haven't found too many places to learn about sequences with roots. Also, have you come across this particular sequence in any of your other work?

No, sorry, I dont know about any references. Though I think there must be some, perhaps in a more general context.
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