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I don't think it well established if xx is well-defined for x real and negative. One could argue that for y positive, but that leads to further ambiguities in , which would be -4 as reals, but 4 as integers. Perhaps we should just ignore the question of whether xx is defined for x negative. — Arthur Rubin (talk) 14:50, 22 July 2009 (UTC)
Hyper 3 | Hyper 2 |
---|---|
X^5 | X×X×X×X×X |
X^4 | X×X×X×X |
X^3 | X×X×X |
X^2 | X×X |
X^1 | X |
X^0 | X÷X |
X^-1 | X÷X÷X |
X^-2 | X÷X÷X÷X |
X^-3 | X÷X÷X÷X÷X |
X^-4 | X÷X÷X÷X÷X÷X |
X^-5 | X÷X÷X÷X÷X÷X÷X |
f(-1/2) ~ - 1.41421356237*I, f(-3/5) ~ -0.419847540943 - 1.29215786488*I f(-Pi) ~ -0.0247567717233 + 0.0118013091280*I
Starting at the top, how does one make the case that when y is positive? It seems to me that . what am I missing? Cliff ( talk) 07:22, 22 March 2011 (UTC)
A picture of the line function = should probably be shown on this page. The line should be coming from the top right and should start to curl in on itself when it reaches on the X axis and it should then stop at 1. Robo37 ( talk) 17:17, 22 July 2009 (UTC)
I've seen one discussion/application of x^x (and its inverse) outside/independent of the more general tetration and its inverse/negative heights. However, I just cite an old msg of mine here, I think, I'll not go deeper into detail. Perhaps someone else is more interested to follow this thread.
I found a text called "WexZal" , which deals with the x^^2 term. Don't know about the relevance regarding your question. It was some years ago, so I don't know, whether this document was continued, or whether it is still online at all.
cite from Preface of WexZal:
This book is about the solution to and properties of the Coupled
Exponent equation (y=x^x). The solution to this equation is called the
"Coupled Root function". This work details our research efforts since
1975. Included are computers/calculators used, evolution of ideas,
history of our efforts and still outstanding problems. We have organized
the work into different topics such as "Applications", "Solving logarithmic
Equations", "Integration", etc. to make it easier for the reader to find
a topic. This is a work where the appendices and tables are (in some ways)
more important then the text itself. The text is to explain the theory;
the tables have the actual items of interest.
Our goal in writing this book is to show the (in our opinion) interesting
things we found and to encourage research into this topic as we feel this
is one area that has been mostly overlooked. We feel that the Coupled
Root function has many hidden properties that have the potential to be
useful. Two such applications have been found so far: Ballistics (internal
& external) and automobile acceleration. There is no doubt other areas where
the Coupled Root could be used.
HTH -
Gottfried Helms
If I recall right, then the x^x-function (or its inverse) was used to compute pressure in a closed room in which an explosion was initiated. Gottfried --Gotti 16:03, 19 October 2009 (UTC) —Preceding unsigned comment added by Druseltal2005 ( talk • contribs) Well, I should make it more explicite, that this msg is a vote to keep "superroot" as a tetration-independent article because I think there is also more research for this special function only (maybe the title can be changed).
--Gotti 16:09, 19 October 2009 (UTC) —Preceding unsigned comment added by Druseltal2005 ( talk • contribs)
This page has no refs, no explanation of WP:notability, does not suggest who coined the term "super-root", who invented the symbol for it, does not provide citations for properties it claims, etc. Overall, it appears to be WP:Original Research and I have tagged it as such. If this is an established function and these are established notations for it, please cite some sources. Thanks, — sligocki ( talk) 02:07, 20 October 2009 (UTC)
If there is one thing that makes this article hard to read and write it is that there isn't one "standardish" notation for superroots, there are mnemonic notations (srt, sprt, ssqrt, sroot, ...), symbolic notations , , , and the verbose . AJRobbins ( talk) 08:03, 21 October 2009 (UTC)
The section notes how the super square root can be easily written using the Lambert-W, but I believe the converse is more important, as the supersquare root can be used to find the inverse of any ax^a. —Preceding unsigned comment added by 189.61.5.9 ( talk) 22:51, 4 November 2010 (UTC)
The following table shows the square super-roots of the first 27 integers.
1 | 1 | 10 | 2.506184... | 19 | 2.830223... |
2 | 1.559610... | 11 | 2.555604... | 20 | 2.855308... |
3 | 1.825455... | 12 | 2.600295... | 21 | 2.879069... |
4 | 2 | 13 | 2.641061... | 22 | 2.901637... |
5 | 2.129372... | 14 | 2.678523... | 23 | 2.923122... |
6 | 2.231828... | 15 | 2.713163... | 24 | 2.943621... |
7 | 2.316454... | 16 | 2.745368... | 25 | 2.963219... |
8 | 2.388423... | 17 | 2.775449... | 26 | 2.981990... |
9 | 2.450953... | 18 | 2.803663... | 27 | 3 |
This information seems unnecessary. If the square super-root is given by Lambert's function, why do these values need to be posted? They are easily calculable and don't seem to offer much insight or value to the page. Cliff ( talk) 05:36, 26 March 2011 (UTC)
This redirect does not require a rating on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
I don't think it well established if xx is well-defined for x real and negative. One could argue that for y positive, but that leads to further ambiguities in , which would be -4 as reals, but 4 as integers. Perhaps we should just ignore the question of whether xx is defined for x negative. — Arthur Rubin (talk) 14:50, 22 July 2009 (UTC)
Hyper 3 | Hyper 2 |
---|---|
X^5 | X×X×X×X×X |
X^4 | X×X×X×X |
X^3 | X×X×X |
X^2 | X×X |
X^1 | X |
X^0 | X÷X |
X^-1 | X÷X÷X |
X^-2 | X÷X÷X÷X |
X^-3 | X÷X÷X÷X÷X |
X^-4 | X÷X÷X÷X÷X÷X |
X^-5 | X÷X÷X÷X÷X÷X÷X |
f(-1/2) ~ - 1.41421356237*I, f(-3/5) ~ -0.419847540943 - 1.29215786488*I f(-Pi) ~ -0.0247567717233 + 0.0118013091280*I
Starting at the top, how does one make the case that when y is positive? It seems to me that . what am I missing? Cliff ( talk) 07:22, 22 March 2011 (UTC)
A picture of the line function = should probably be shown on this page. The line should be coming from the top right and should start to curl in on itself when it reaches on the X axis and it should then stop at 1. Robo37 ( talk) 17:17, 22 July 2009 (UTC)
I've seen one discussion/application of x^x (and its inverse) outside/independent of the more general tetration and its inverse/negative heights. However, I just cite an old msg of mine here, I think, I'll not go deeper into detail. Perhaps someone else is more interested to follow this thread.
I found a text called "WexZal" , which deals with the x^^2 term. Don't know about the relevance regarding your question. It was some years ago, so I don't know, whether this document was continued, or whether it is still online at all.
cite from Preface of WexZal:
This book is about the solution to and properties of the Coupled
Exponent equation (y=x^x). The solution to this equation is called the
"Coupled Root function". This work details our research efforts since
1975. Included are computers/calculators used, evolution of ideas,
history of our efforts and still outstanding problems. We have organized
the work into different topics such as "Applications", "Solving logarithmic
Equations", "Integration", etc. to make it easier for the reader to find
a topic. This is a work where the appendices and tables are (in some ways)
more important then the text itself. The text is to explain the theory;
the tables have the actual items of interest.
Our goal in writing this book is to show the (in our opinion) interesting
things we found and to encourage research into this topic as we feel this
is one area that has been mostly overlooked. We feel that the Coupled
Root function has many hidden properties that have the potential to be
useful. Two such applications have been found so far: Ballistics (internal
& external) and automobile acceleration. There is no doubt other areas where
the Coupled Root could be used.
HTH -
Gottfried Helms
If I recall right, then the x^x-function (or its inverse) was used to compute pressure in a closed room in which an explosion was initiated. Gottfried --Gotti 16:03, 19 October 2009 (UTC) —Preceding unsigned comment added by Druseltal2005 ( talk • contribs) Well, I should make it more explicite, that this msg is a vote to keep "superroot" as a tetration-independent article because I think there is also more research for this special function only (maybe the title can be changed).
--Gotti 16:09, 19 October 2009 (UTC) —Preceding unsigned comment added by Druseltal2005 ( talk • contribs)
This page has no refs, no explanation of WP:notability, does not suggest who coined the term "super-root", who invented the symbol for it, does not provide citations for properties it claims, etc. Overall, it appears to be WP:Original Research and I have tagged it as such. If this is an established function and these are established notations for it, please cite some sources. Thanks, — sligocki ( talk) 02:07, 20 October 2009 (UTC)
If there is one thing that makes this article hard to read and write it is that there isn't one "standardish" notation for superroots, there are mnemonic notations (srt, sprt, ssqrt, sroot, ...), symbolic notations , , , and the verbose . AJRobbins ( talk) 08:03, 21 October 2009 (UTC)
The section notes how the super square root can be easily written using the Lambert-W, but I believe the converse is more important, as the supersquare root can be used to find the inverse of any ax^a. —Preceding unsigned comment added by 189.61.5.9 ( talk) 22:51, 4 November 2010 (UTC)
The following table shows the square super-roots of the first 27 integers.
1 | 1 | 10 | 2.506184... | 19 | 2.830223... |
2 | 1.559610... | 11 | 2.555604... | 20 | 2.855308... |
3 | 1.825455... | 12 | 2.600295... | 21 | 2.879069... |
4 | 2 | 13 | 2.641061... | 22 | 2.901637... |
5 | 2.129372... | 14 | 2.678523... | 23 | 2.923122... |
6 | 2.231828... | 15 | 2.713163... | 24 | 2.943621... |
7 | 2.316454... | 16 | 2.745368... | 25 | 2.963219... |
8 | 2.388423... | 17 | 2.775449... | 26 | 2.981990... |
9 | 2.450953... | 18 | 2.803663... | 27 | 3 |
This information seems unnecessary. If the square super-root is given by Lambert's function, why do these values need to be posted? They are easily calculable and don't seem to offer much insight or value to the page. Cliff ( talk) 05:36, 26 March 2011 (UTC)