01/26/2021, 01:19 PM

Many methods use something like taking n log's of some function f(s+n) or f(exp(exp... n times )).

So basically we take many log's of something that approximates many exp.

Sinh method and the recent NBLR method for instance use this.

If we consider it from the nonreal perspective we end up with a dense set of singularities usually because the exp iterates make many smaller copies close to each other. See sheldon's recent comments.

Not going into details here.

A potential solution is starting from the real line and then doing analytic continuation.

Usually analytic continuation is not done from a line but from a circle or polygon.

So that is a bit controversial.

So what is a logical alternative ?

Perhaps avoiding these many copies by "going in the other direction".

By that I mean exp(exp(... n times g(s+n)).

However this requires a function that grows approximately like slog(x).

Series and products ( that are also interpolation methods usually btw) tend to give nonanalytic solutions.

So I wonder about an infinite composition that grows close like slog(x).

As James Nixon demonstrated infinite compositions exist to approximate sexp sufficiently.

So maybe they also exist for slog(x).

I got stuck trying to find them.

Do they even exist ??

Regards

tommy1729

So basically we take many log's of something that approximates many exp.

Sinh method and the recent NBLR method for instance use this.

If we consider it from the nonreal perspective we end up with a dense set of singularities usually because the exp iterates make many smaller copies close to each other. See sheldon's recent comments.

Not going into details here.

A potential solution is starting from the real line and then doing analytic continuation.

Usually analytic continuation is not done from a line but from a circle or polygon.

So that is a bit controversial.

So what is a logical alternative ?

Perhaps avoiding these many copies by "going in the other direction".

By that I mean exp(exp(... n times g(s+n)).

However this requires a function that grows approximately like slog(x).

Series and products ( that are also interpolation methods usually btw) tend to give nonanalytic solutions.

So I wonder about an infinite composition that grows close like slog(x).

As James Nixon demonstrated infinite compositions exist to approximate sexp sufficiently.

So maybe they also exist for slog(x).

I got stuck trying to find them.

Do they even exist ??

Regards

tommy1729