Iterated exponential function

Question

Is it possible to infinitely differentiably extend the function defined as $f(x+1,a)=e^{f(x,a)}$, $f(0,a)=a$ to non-integers?

What I’m trying to do is derive a sort of «half logarithm», a function that if applied twice gives the natural logarithm.

Here's a question from MO about half exponentials which you may be interested in. — Antonio Vargas, Jun 01 '13 at 23:14
I am aware that the domain should have a left bound; for example, in the case where $a=1$, the domain is $(-2,\infty)$. — Fernando Martínez, Jun 01 '13 at 23:25
Thanks! I understand that Carleman matrices map function composition to matrix multiplication, but how exactly am I supposed to multiply a fractional number of matrices together? — Fernando Martínez, Jun 02 '13 at 18:05

DarenW · Accepted Answer · 2024-10-18T02:39:25.117

3

Yes, such a "half logarithm" and its inverse "half exponential" functions have been explored in the past. There are multiple ways to define them. The half-logarithm's domain exists only above a certain value.

The best, as in being smoothest, and computationally feasible, is probably by G. Szekeres, "Fractional Iteration of Exponentially Growing Functions" in J. Australian Math. Soc. vol 2 p 301 (1961).

An older work based on fixed points in the complex plane is: Hellmuth Kneser. Reel analytische Losungen der Gleichung φ(φ(x)) = ex und verwandter Funktionalgleichungen. J. f. Reine Agnew. Math. 187, page 56, page 56 (1950).

Then there's my own invention based on symmetries, originally badly written and published in 1981 in a long defunct journal, but currently available as a PDF on my GitHub, https://github.com/darenw/FRITEXP/tree/master/doc and a shorter version as a slide set on SlideShare: https://www.slideshare.net/DarenWilson1/generalizing-addition-and-multiplication-to-an-operator-parametrized-by-a-real-number.

edited Oct 18 '24 at 02:39

answered Mar 16 '14 at 06:06

DarenW

825

Is there some sort of iterative algorithm to calculate the Szekeres function? – Fernando Martínez Apr 12 '14 at 05:01
Yes. While not complicated, it's hard to describe in full here. We use function g(x)=exp(x)-1, which can be fractionally iterated using a power series expansion. Starting with some x, you repeatedly apply exp() until you get a "large" value. Then apply the inverse of g(x) repeatedly until you get a value small enough to apply a truncated power series to good approximation. Then go back up with g(x) and down with the inverse of exp(x), matching in count to undo what was done before. This is too sketchy for numerical work; see the Szekeres paper for more. – DarenW Apr 13 '14 at 07:43
BTW, I'm writing a paper explaining all three methods in more detail. The more I get nudged to finish it, the sooner it'll get done! – DarenW Apr 13 '14 at 07:46
@DarenW I wonder if - 4 years later - your paper has emerged? I would love to read it - is there a link? – Andreas May 30 '18 at 14:40
It's pretty much done, but I'd like to add a few more plots. Then again, maybe I have enough. I can always put additional plots up on my website, make a Youtube video, whatever. Heck, I might was pop that PDF up on my website. I'll be unlikely to have time to refine it fully, and the main part is good. – DarenW Jun 01 '18 at 04:59
FWIW, my paper is done, but unpublished anywhere. Python code for numerical work and a PDF of the paper are at Github, https://github.com/darenw/FRITEXP – DarenW Oct 08 '18 at 21:40
Update from the year 2024: for a quicker lighter read of the topic, I have a slideshow on Slideshare https://www.slideshare.net/slideshow/generalizing-addition-and-multiplication-to-an-operator-parametrized-by-a-real-number/251422084 – DarenW Oct 12 '24 at 06:09
the ".pdf" is a ".html" - I couldn't open it (but a text editor showed the html-content) Opening as *.html showed only the border of the page (with its menu etc), but no contents. – Gottfried Helms Oct 19 '24 at 21:25

Iterated exponential function

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