Talk:Super-root

Square super-root (or super-square-root) of negative numbers

I don't think it well established if x^x is well-defined for x real and negative. One could argue that $(-y)^{-y}=-y^{y}$ for y positive, but that leads to further ambiguities in $(-2)^{-2}$, which would be -4 as reals, but 4 as integers. Perhaps we should just ignore the question of whether x^x is defined for x negative. — Arthur Rubin (talk) 14:50, 22 July 2009 (UTC)

• • (Note, $-4$ and $4$ should be ${\frac {1}{4}}$and $-{\frac {1}{4}}$ respectively. The principle is still the same; you get a different answer if you treat the numbers as rational numbers, or as real numbers. — Arthur Rubin (talk) 20:22, 22 July 2009 (UTC)

Whenever I'm doing negative powers I always use the following table.

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

Robo37 (talk) 16:05, 22 July 2009 (UTC)

But what are:

$(-1/2)^{-1/2}$

$(-2/3)^{-2/3}$

$(-3/5)^{-3/5}$

$(-\pi )^{-\pi }$

? — Arthur Rubin (talk) 16:26, 22 July 2009 (UTC)

Including to my calculator; -1.141213562... , -1.310370697... , -1.358655183... and -0.027425693... . Robo37 (talk) 16:45, 22 July 2009 (UTC)

Your calculator is making an assumption; that $(-x)^{y}=-\left(x^{y}\right)$, if x is positive. That's a plausible convention, but it's clearly wrong if y is an even integer. — Arthur Rubin (talk) 20:22, 22 July 2009 (UTC)

Okay I see your point, I've just typed in '(-1/2)^(-1/2)' in google calculator now and it came up with the answer '- 1.41421356 i', so I suppose there is no real solution. Robo37 (talk) 21:35, 22 July 2009 (UTC)

I think, for the non-integer or general case, it is defined x^x = exp(log(x)*x) (isn't this also here in wikipedia? ;) ), and inherits the ambiguity of the log(x) because exp(log(x)*x) =/= exp( (log(x)+k*2*Pi*I)*x) . So for x^x a "principal value" may be defined with k=0 in the previous formula.

And thus for the inverse, the "super-square-root" or "iterative square-root of exponentiation".

Using a general math-software like Pari/GP, we get, using f(x)=x^x =exp(log(x)*x) for the "principal value"

           f(-1/2) ~                   - 1.41421356237*I, 
           f(-3/5) ~  -0.419847540943  - 1.29215786488*I
           f(-Pi)  ~  -0.0247567717233 + 0.0118013091280*I

The function itself and its inverse (iterative squareroot) has a powerseries in (x-1) and either this or a root-searching algo can be invoked to find "the principal" solution approximately.

(But that all seems much elementary, so why does this occur here as serious question? Did I overlook something?)

Gotti 19:19, 26 July 2009 (UTC) —Preceding unsigned comment added by Druseltal2005 (talk • contribs)

I'm afraid this is all related to the question of the domain of the hyperoperations, and hence the range (and domain) of their inverses. As has been pointed out many times before, tetration is only clearly defined for the base being positive, and the height being an integer (greater than -2, the singularity). Hence the "principal value" of the super root is clearly defined for positive integer height and domain reals > 1, with some further multi-valued extensions for the range being the positive reals. — Arthur Rubin (talk) 19:44, 26 July 2009 (UTC)

x^x is well-defined for negative integer x. (-2)^(-2) = 1/4. Even if you use the multivalued complex logarithm you can only get 1/4. mike4ty4 (talk) 21:00, 4 August 2009 (UTC)

Starting at the top, how does one make the case that $(-y)^{-y}=-y^{y}$ when y is positive? It seems to me that $(-y)^{-y}=(-1/y)^{y}$. what am I missing? Cliff (talk) 07:22, 22 March 2011 (UTC)

That was a mistake on my part. The problem is that $\left(-y\right)^{-y}=\left({\frac {-1}{y}}\right)^{y}$, but that may be equal to either $\pm \left({\frac {1}{y}}\right)^{y}$, or something else entirely, depending on various properties of y. If y is a rational number with odd numerator and denominator, one can make a case for "−", but if y is a rational number with even numerator and odd denominator, one can make a case for "+". As y is real, one cannot really make a good case for either. If a function is multivalued, its inverse is even more problematical. — Arthur Rubin (talk) 08:46, 22 March 2011 (UTC)

Ok, I wanted to make sure I wasn't missing something. Thanks, Cliff (talk) 15:12, 22 March 2011 (UTC)

Line function

A picture of the line function ${Y}$ = ${\sqrt {x}}_{s}$ 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 $e^{(-1/e)}$ on the X axis and it should then stop at 1. Robo37 (talk) 17:17, 22 July 2009 (UTC)

Application of superroot ("wexzal"(?))

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.

See msg concerning wexzal

 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)

Original Research

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)

I second it. How can they even talk about TWO inverses of a function? It's nonsense, these people need to read analysis 101 to get their terminology straight! And what the hell is with this plot: http://upload.wikimedia.org/wikipedia/en/b/b4/The_graph_y_%3D_%E2%88%9Ax%28s%29.png ? It's not even a function! I'm so confused right now %( Kallikanzarid (talk) 15:03, 1 November 2010 (UTC)

I'm afraid that multiple inverses of a multivariate function is standard notation, although I can't immediately find it here on Wikipedia.

If

$f(x_{1},x_{2},\ldots ,x_{k},\ldots ,x_{n})=y,$

Then the kth inverse of f satisfies:

$g(x_{1},x_{2},\ldots ,y,\ldots ,x_{n})=x_{k}.$

— Arthur Rubin (talk) 15:12, 1 November 2010 (UTC)

The image is a horrible illustration of the concept. The function is bi-variate and as such should be graphed in 3-space, if at all. Cliff (talk) 22:18, 25 March 2011 (UTC)

Too many notations

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 ${}^{n}{\overline {)x}}$, ${}_{4}^{n}{\overline {|x}}$, ${\sqrt[{n}]{x}}_{s}$, and the verbose $(\uparrow \uparrow n)^{-1}(x)$. AJRobbins (talk) 08:03, 21 October 2009 (UTC)

Lambert-W

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)

Table of values

The following table shows the square super-roots of the first 27 integers.

${x}$	${\sqrt {x}}_{s}$	${x}$	${\sqrt {x}}_{s}$	${x}$	${\sqrt {x}}_{s}$
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)