Wikipedia:Reference desk/Archives/Mathematics/2023 June 29

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June 29

An analytic continuation passing through the origin:

The background of my question is the fact, that every function passing through the origin (of the coordinate system), has a lot of important algebraic properties (following from its value for zero), that are not shared by some functions [having its domain].

Having mentioned all of that, let's assume that a given real analytic function - defined on a limited domain not including zero - has an analytic continuation passing through the origin.

Does this given function have any non-infinitesimal properties (e.g. algebraic ones), some analytic functions [having its domain] don't have? If it doesn't, then having no choice I will have to ask an analogous question - replacing the condition of "non-infinitesimal (e.g. algebraic)" - by "non-obvious". 2A06:C701:7453:7D00:8F6:E1A9:A503:D522 (talk) 18:55, 29 June 2023 (UTC)

Observation. If $f$ is a function meeting the condition, and $a$ is any non-negative number, then the function $g(x)=f(x+a)-f(a)$ also meets the condition. (In fact, we only need a weaker condition on $f$, namely that it is analytic beyond some given number in the positive direction and allows analytic continuation beyond $a$ in the other direction.) So any old boring function gives in this way rise to a whole family of equally boring functions. I find it hard to imagine they all share some non-obvious property.  --Lambiam 23:57, 29 June 2023 (UTC)

So, for the time being, your answer is negative, intuitively only, without proving it yet, right?

Two remarks:

1. Your function $g(x)=f(x+a)-f(a)$ passes through the origin, for every number $a$ and for every function $f,$ rather than only for non-negative numbers $a$ and only for analytic functions $f$ whose analytic continuation passes through the origin.

2. Please notice, that the answer to my question would have been positive, if I had entirely given up the condition of "non-infinitesimal (e.g. algebraic)", so I wonder why you think the answer to my question probably becomes negative - once I add this non-infinitesimal condition.

2A06:C701:7453:7D00:8F6:E1A9:A503:D522 (talk) 03:29, 30 June 2023 (UTC)

The definition of $g$ requires $f(a)$ to be defined. Given our knowledge about $f$ we know it can be analytically continued beyond $0,$ but we have no idea how far – any negative number might be too far. To have a guarantee that $f(a)$ is defined (also using analytic continuation, if need be) we must restrict $a$ to the non-negative reals.

Of course I have not proved anything, since a proof would need definitions of "non-infinitesimal property" or "non-obvious". Can one state any non-obvious properties of a function, given only that it is analytic on $\mathbb {R}$ and that it vanishes at the origin?  --Lambiam 10:55, 30 June 2023 (UTC)

Please notice I didn't mention "non-obvious" in my first question, but rather in the second one only, which is still in standby, and which will only enter into force if the answer to the first question turns out to be negative.

As for "non-infinitesimal", I thought it was a pretty clear concept. Anyway, I have already agreed to replace it by a simpler property: "algebraic", whether obvious or not, provided that it's not shared by some analytic functions having the same domain as the original one. 147.235.210.58 (talk) 11:10, 30 June 2023 (UTC)

I am not sure how to define "algebraic property of a function" either. Do you mean such things as $a^{2}(f(b)-b)=b^{2}(f(a)-a)$? Isn't "non-obvious property" a wider class than "algebraic property", or are you also interested in obvious algebraic properties? Perhaps you can give us a few concrete examples of algebraic properties of specific functions from $\mathbb {R}$ to $\mathbb {R} .$  --Lambiam 15:29, 30 June 2023 (UTC)

As for your first question: Yes. As for your second question: As I have already written, the only condition for the property is to be algebraic, whether obvious or not, provided that it's not shared by some analytic functions having the same domain as the original one. As for your last sentence: Do you agree with the "background" indicated in the first paragraph of this thread? If you do (and I guess you do), then you have got concrete examples of algebraic properties of every function passing through the origin, that are not shared by some functions having its domain - i.e. (per your example) by some functions from the set of real numbers to itself. 2A06:C701:7444:F700:D807:B820:BC7A:D368 (talk) 18:07, 1 July 2023 (UTC)

I cannot think of any algebraic property for $f(x)=\exp x+\log(x+1)-1$ other than $f(0)=0.$ That is not "a lot of important algebraic properties". Likewise for $\sin x+x,$ $\sin(\sinh x)$ and many other functions. I have no clue what makes you think the "background" is a fact or even a reasonable conjecture.  --Lambiam 21:55, 1 July 2023 (UTC)

Besides the property f(0)=0, the functions you have indicated have a lot of additional imprortant algebraic properties, all of which follow from the first property, e.g. the additional property f(x)f(0)=f(0). Please notice, this additional propertry - shared by all functions passing through the origin - is also shared by the constant function f(x)=1 which does not pass through the origin, but this additional propertry is not shared by some other functions - e.g. the constant function f(x)=2. Here is another additional propery: f(x)+f(0)=f(x). Following from the first property, this additional property is only shared by all functions passing through the origin, and not by any other function. That's why I'd written there are "a lot" of important algebraic properties. I'd also written those properties follow from the value given by the function to zero.

Anyway, my original question does not refer to any function that passes throught the origin, but rather to analytic functions whose analytic continuation does - whereas the analytic functions themselves are not defined for zero. 2A06:C701:7444:F700:D807:B820:BC7A:D368 (talk) 22:54, 1 July 2023 (UTC)

Take the function $f$ defined on the real left-bounded interval $(0,\infty )$ by $f(x)=\sin(\sinh x)$ This function is not defined for $x=0.$ It can be extended in many ways, only one of which is analytic. That is also true for all the other functions. Perhaps you might consider adding "non-obvious" or "interesting". Otherwise, letting $P(x,y_{1},...,y_{n})$ stand for any property that becomes a tautology if you set $x=0$ but not in general, such as $xy_{1}+y_{2}=y_{2}/(x+1),$ you can generate heaps of "algebraic" (but obvious and uninteresting) properties by such instantiations as $P(f(0),f(f(1)),f(2){-}f(3)).$ Just give us one algebraic property (following from its value for zero) that is not of this form.  --Lambiam 23:30, 1 July 2023 (UTC)

By stating one "can generate heaps of algebraic (but obvious and uninteresting) properties", you just repeat my claim in the "backround" (I hadn't added the avjective "obvious and uninteresting", though). So, I'm asking WHETHER: just as functions passing through the origin have "heaps of algebraic properties" (you addding "obvious and uninteresting" if you want to), which are not shared by some analytic functions, so analytic functions whose analytic continuation passes the origin - have algebraic properties [not shared etc.] - whereas the analytic functions themselves are not defined for zero. 2A06:C701:7444:F700:D807:B820:BC7A:D368 (talk) 23:59, 1 July 2023 (UTC)

If it is a fact that there are a lot of important algebraic properties (following from such a function's value for zero), why don't you give us one example of such a property not producible by the $P(x,y_{1},...,y_{n})$ scheme (which produces only trivial ones).  --Lambiam 06:43, 2 July 2023 (UTC)

I would have had to give you the example you want, if and only if I had ever considered your scheme to be obvious. That said, I mention I've never discussed the matter about whether your scheme is obvious. However, as long as I don't give you the example you want (whether because of laziness or becuase of any other reason), you are allowed to assume that I don't consider your scheme to be obvious. Further, my original question doesn't care about whether the algebraic property I'm looking for is obvious or not: You are allowed to present any algebraic properties, whether obvious or not, as you wish, provided that they are not shared by some analytic functions [having the same domain as before]. 2A06:C701:7444:F700:D807:B820:BC7A:D368 (talk) 10:03, 2 July 2023 (UTC)

Hopefully, I am also allowed not to engage any further. I tried to be helpful, but this looks more like trolling.  --Lambiam 11:59, 2 July 2023 (UTC)

First, thank you for your efforts to help me, I appreciate them, frankly and honestly. Second, I disagree about your estimation about me: I'm not trolling, frankly and honestly (In my view, honesty is the best policy), so it seems you didn't interpret well my question, so I will try again: I don't consider your scheme to be obvious, because it's not shared by some analytic functions [having the same domain as before]. Third, I'm still waiting for answers of other editors, and you too are invited. My question is pretty simple: Do all functions whose analytic continuations pass through the origin, have any algebraic properties, not shared by some analytic functions [having the same domain as before]? Fourth, thank you again for your answers, I appreciate them. 2A06:C701:7444:F700:D807:B820:BC7A:D368 (talk) 12:43, 2 July 2023 (UTC)