2

Excuse my lack of expertise, I study natural sciences (physics) and not mathematics so I will be off with my terminology and mathematical vocabulary. Please feel free to poke and build at this idea but I do request at least playing around with it seriously and giving it a chance. enter image description here

Can we describe this by an infinitely repeating function? At every odd position going from 1, 3, 5 and so on fractions we end up with 2. At every even position we get 1/2. This constantly changes values between 2 and 1/2 so it gave me the idea of describing the ith fraction position by a periodic function. An even more interesting idea popped up in my head. What if we have a continuous position for the fraction? Instead of the 2nd position of the fraction giving 1/2 what would it loom like for a fraction to be fractional in position such that the numbers dont take a discrete set of positions (aka numerators or denominators)? Can we give an analytical continuation for this?

Captain HD
  • 57
  • 1
    Here's the thing about analytic continuations: for most discrete valued functions, there are infinitely many analytic continuations, so the sheer existence of a continuation is pretty uninteresting. Rather, we tend to be more interested in continuations that satisfy certain properties (which tend to be extensions of the properties of the original function). – Rushabh Mehta Feb 10 '21 at 18:06
  • The gamma function is a great example of this. It's an analytic continuation of $n-1!$, but it also extends the factorial behavior to its whole domain, i.e, for all $x$, $\Gamma(x+1)=(x+1)\Gamma(x)$. How exactly do you extend the behavior that you are describing to a continuous domain? – Rushabh Mehta Feb 10 '21 at 18:08
  • I am not even ready to do analytic continuation until the end of this semester but I am very inspired by the analytic continuation of the factorial function from integer inputs to real inputs (I think it can even take complex inputs). I would suppose representing the pattern as a function in some way (perhaps a partial function) as a first step but again I came here because I will need a lot of help if I wanted to play around with this and extend the notion of a fraction to take a continuous set of positions rather than having integer positions and what exactly that would mean. – Captain HD Feb 10 '21 at 18:17
  • Notice that there are only discrete positions where in this specific example we get an overall value of 2 for odd values but 1/2 for even values so my question would be, what do we get for the 7.5th position? – Captain HD Feb 10 '21 at 18:19
  • I'd suggest re-reading my comments. I addressed all of the points you've mentioned in the comments. – Rushabh Mehta Feb 10 '21 at 18:38
  • Yes I have read it. If you find anything interesting however, please do come back. – Captain HD Feb 10 '21 at 18:46
  • What did you take from it? I can give you many analytic continuations. That doesn't make the question interesting. – Rushabh Mehta Feb 10 '21 at 18:49
  • All I can do is shrug. Is it not possible to find a single interesting one? Maybe if I saw a few examples of a continuation for these fractions I could have a look at it just to get a “feel” for the for extension and explore what properties it has. Basically, if you could give me a few random ones or ones relatively more interesting than others that would be nice. They don’t have to be as interesting as you describe just anything will do as I am somewhat eager to explore this. – Captain HD Feb 10 '21 at 19:21
  • Ok, here's one: $1.25-0.75\cos(\pi x)$. – Rushabh Mehta Feb 10 '21 at 19:36
  • I don't know whether this meets your demand, but perhaps it does. See http://go.helms-net.de/math/tetdocs/index.htm and from there http://go.helms-net.de/math/tetdocs/FracIterAltGeom.htm where I've discussed the function $f(x)=1/(1+x)$ as iterable, and finally fractionally iterable, function. – Gottfried Helms Mar 09 '21 at 15:43
  • Thank you! I will keep this for future reference hopefully in a month time – Captain HD Mar 09 '21 at 21:00

0 Answers0