(02/17/2013, 01:27 PM)Balarka Sen Wrote: Hi,
I was reading the article and found that the author describes a method to extend the tetration to reals. It uses the recursive relation and the differentiability requirement but a piecewise argument, hence not following the continuity requirement(?). It was a nice method, I am thinking about using it in practice. But I wonder if there are any way to extend the a^^x to complex values of a and x. Do we already have such analytic continuation? I am mainly interested in the recursion and differentiability requirement.
Balarka
.
What were you reading? I can think of the Wikipedia article, Hooshmand's paper, and Andrew Robbins' paper. These all mention a piecewise method, and the Wiki article mentions a "differentiability requirement".
As for the question of how to extend to complex values, there are a few that seem to give the same result, namely Kneser's method and the Cauchy integral. There is also the "continuum sum", the "matrix method" and the above mentioned Robbins' method (which you may be referring to). All five of these appear to yield the same result, but it is not proven, as far as I can tell. In theory, these should be able to define \( ^z a \) for any complex a and z. But the currently-available implementations only work for more limited ranges of bases a.
(Ed: The Cauchy integral may not work for bases on the real interval \( 1 < a < e^{1/e} \), at least if we want it to be analytically compatible with the rest of the bases. Tetration, it seems, as given by these methods has branch points at \( e^{1/e} \), \( 1 \), and \( 0 \) (and possibly more) in the base. A cut is taken on \( (-oo, e^{1/e}] \) to yield a principal branch, and we can define in there (at the cost of discontinuity) via the "continuity from above" convention. Tetration is not real-valued on this interval for non-integer "heights". So far, I'm not sure any luck has been had at solving the Cauchy integral equation on a complex base of any sort, though -- the simple algorithm seems not to want to converge, at least that was my experience trying to toy around with it. There may be more sophisticated methods out there that could be used.)
See here:
https://bitbucket.org/bo198214/bunch/downloads/main.pdf
for a description of these methods. You'll want to pay attention to the "regular iteration", for it serves as the basis for the Kneser solution, and also because it gives a real-valued extension of tetration for bases in \( 1 < a \le e^{1/e} \) -- though not one that can be analytically continued to most of the complex plane in terms of bases.
Other methods have been proposed which give different answers, however the Kneser solution has a simple uniqueness condition based on the inverse function (see here: http://eretrandre.org/rb/files/Trappmann2009_82.pdf, also see sheldonison's post here: http://math.stackexchange.com/questions/...-tetration -- there's another uniqueness condition given, though I'm not sure if this one has a proof.). Some of these other methods include the "sinh" method, which does not seem to even yield an analytic solution (so cannot be extended to complex heights, and perhaps not even complex bases either), and recently, another method based on divergent summation of "iterate series". The sinh method does appear to be at least smooth, but again, this is not proven.
So yes, a lot of methods and proposals, some of which agree with each other, others which don't. And there's a whole lot that's conjecture and not proven. But one can make a case for the Kneser solution due to the simple uniqueness condition, and Robbins' method computes it for at least a limited range of heights and bases in the complex plane. "sheldonison" and I have also written Pari/GP codes that implement the Kneser methods, which give tetration to any complex height, and for possibly (though the current code doesn't allow for it) any complex base:
Description of Kneser solution:
http://math.eretrandre.org/tetrationforu...hp?tid=213
The algorithm:
http://math.eretrandre.org/tetrationforu...hp?tid=486
http://math.eretrandre.org/tetrationforu...hp?tid=487
The code:
http://math.eretrandre.org/tetrationforu...hp?tid=486
http://math.eretrandre.org/tetrationforu...hp?tid=729
(sheldonison's)
http://math.eretrandre.org/tetrationforu...hp?tid=664
(my own implementation -- you'll want the code from the last message -- I had some difficulties with the first code)
http://math.eretrandre.org/tetrationforu...hp?tid=687
(my program for some complex bases)
This would, I believe, provide a fair positive answer to "is there a way to extend a^^x to complex a and x".
