Talk:Half-exponential function

Latest comment: 3 years ago by Cuzkatzimhut in topic Link to Tetration?

Proved only for "positive" combinations?

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"Not expressible in terms of elementary functions", really? Following the link (to Scott) I see much weaker statement, and a question: "And of course, how do we handle subtraction and division?" Or is this question answered already? Boris Tsirelson (talk) 19:58, 22 May 2014 (UTC)Reply

I agree with you - Scott's proof definitely can't handle subtraction and division - so I've changed the wording of the sentence and removed the "factual inaccuracy" template. David9550 (talk) 00:17, 11 April 2016 (UTC)Reply
After a bit more searching, I found a MathOverflow post that points to some references that claim to do the full case. So I've added that in. David9550 (talk) 00:29, 11 April 2016 (UTC)Reply

The article says...

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It has been proven that every function ƒ composed of basic arithmetic operations, exponentials, and logarithms, then ƒ(ƒ(x)) is either subexponential or superexponential:[3] half-exponential functions are not expressible in terms of elementary functions.

Stronger statement that I would like to know if it's true:

The smallest set of all functions that contains all elementary functions as well as being closed under indefinite integration still doesn't have the half-exponential functions.

In other words, if   is a half-exponential function, then no function in the sequence whose first function is   and each function after that is the derivative of the preceding function will be elementary. Georgia guy (talk) 20:25, 22 May 2014 (UTC)Reply

That could be interesting, but for now the statement in the article seems to be too strong, not too week, as compared to what is proved. Boris Tsirelson (talk) 04:58, 23 May 2014 (UTC)Reply

Aspiration for smoothness

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One would like to have a solution f to f(f(x))=exp(x) which is not merely continuous and strictly increasing, but as smooth as possible. One might hope for C, but even that would still leave the possibility of undulations in the rate of increase. Analyticity would be desireable, but the sources suggest that it is not possible. If there were a place where the function was especially smooth, it could be used to extend the definition of such a smooth function to the whole real line using the facts:

 

and

 

Define A=f(0). We can show that as x approaches -∞, f(x) approaches ln(A)<0. This is asymptotic. So we might hope that f will take an especially simple form in this limit. Let us take the derivative using the chain rule as x approaches -∞

 

So for some constant c1,

 

Integrating from -∞ to x gives

 

If we go to the second derivative, we get

 

So for some constants c1 and c2,

 

Integrating from -∞ to x gives

 

Integrating from -∞ to x gives

 

The pattern suggests that

 

where c0=ln(A). In other words, we can hope that there is a function h analytic in a neighborhood of 0 such that

 

since exp(x) approaches 0 as x approaches -∞. Notice that

 

and consequently h cannot be entire because f -1(ln(A)) is undefined. JRSpriggs (talk) 00:32, 26 May 2014 (UTC)Reply

The maximum value of A consistent with it lying within the radius of convergence of

 

is Ω, the Omega constant, where A=−ln(A). JRSpriggs (talk) 01:26, 27 May 2014 (UTC)Reply

For a while, I explored the possibility that A=Ω=0.5671... . Then I realized that that value was too large to be consistent with my belief that the appropriate f should be C and that all its derivatives should be positive on the whole real line. According to the mean value theorem, there should be a point in the interval (ln(A),0) where the first derivative is

 

and there should be a point in the interval (0,A) where the first derivative is

 

If we require the first of these to be less than the second and cross-multiply, we get

 

If A were Ω, then

 

which would imply

 

a clear falsehood. JRSpriggs (talk) 04:46, 29 May 2014 (UTC)Reply

From

 

we get

 

by the chain rule. If we then substitute f−1(x) in place of x, we get

 

from which it follows that

 

This will allow us to calculate the derivative of f at larger arguments, if we know it at smaller arguments. By analogy with

 

we could calculate a series expansion for f near x, if we know a series for its derivative at f−1(x). We begin with

 

for

 

Applying our formula and using analytic continuation, we get

 

for

 

Applying it again

 

for

 

Applying it yet again

 

for

 

JRSpriggs (talk) 06:28, 11 June 2014 (UTC)Reply

For the second derivative, the recursion rule is

 

Here is a table of values

 

JRSpriggs (talk) 11:04, 11 June 2014 (UTC)Reply

We need constraints on the possible values of the parameters A, c0, c1, c2, c3, ... :

 
 
 
 
 

Assuming that the derivative of f is strictly increasing, we get

 

Assuming that the second derivative of f is strictly increasing, we get

 

And so forth. JRSpriggs (talk) 04:19, 12 June 2014 (UTC)Reply

So far, it appears that

 
 

JRSpriggs (talk) 07:15, 14 June 2014 (UTC)Reply

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I think this article should include some mention of tetration. While it defines f(x) as a piecewise, it would make more sense to define it using tetration.

Let's focus on this for example:

 

This can be rewritten using tetration:

 

Here's a proof to show that you indeed get   as a result:

 
 
 
 
 

Unfortunately, I can't figure out a way to get that coefficient in the front, so using the function above only satisfies   when  . Regardless, I believe it's worth mentioning. Not only would using tetration help make it easier to define functions such as  , but the fact that the link exists means that there might be some way to calculate  , although this assumes that the value of a half-exponential function at   is known. Expfac user (talk) 20:05, 20 January 2021 (UTC)Reply

I don't see the point. Functional iteration is a mature, self-standing field, as your can see from the cited bibliography, and no salutary information or techniques can come from tetrations. You might consider linking this one to the iterated function page, however. Cuzkatzimhut (talk) 01:20, 21 January 2021 (UTC)Reply

"linking this one to the iteration page" as in, linking the half exponential function to the iteration page? Because if that's the case, the page already has a link to iterated functions. As for the comment on tetration...yeah, I guess it isn't practical. Although, would it be worth it to mention half-exponentials on the tetration page, assuming it hasn't been done already?
Other than that appreciate the advice, I suppose there's no point mentioning tetration if it's not practical. Expfac user (talk) 16:03, 21 January 2021 (UTC)Reply
Apologies; I took my own advice and linked the iterated function page to this one, after my comment above! Personally, more Kneser stuff is in order, but I lack the energy to do that. Cuzkatzimhut (talk) 22:00, 21 January 2021 (UTC)Reply