03/12/2013, 08:58 AM

I http://math.stackexchange.com/questions/327995 I discuss the problem

Problem with infinite product using iterating of a function:

I think, because of the better latex-formatting it is easier to read there, but for completeness I'll copy&paste the problem here too.

Considering the iteration of functions, with focus on the iterated exponentiation, I'm looking, whether the function which I want to iterate can -hopefully with some advantage- itself be expressed by iterations of a -so to say- "more basic" function.

Now I assume a function f(x) such that

(where the circle-notation means iteration, and ) - and I ask: what does this function look like? What I'm doing then is this substitution:

*(From 4. I know, that x is now restricted to )*

But if I do now the computation with some example *x* I get the result

***Q:*** Where does this additional factor come from? Where have the above steps missed some crucial information?

<hr>

A code snippet using Pari/GP:

<hr>

Here is an example which shows the type of convergence; I use *x_0=1.5* and internal precision of 200 decimal digits. Then we get the terms of the partial product as

Problem with infinite product using iterating of a function:

I think, because of the better latex-formatting it is easier to read there, but for completeness I'll copy&paste the problem here too.

Considering the iteration of functions, with focus on the iterated exponentiation, I'm looking, whether the function which I want to iterate can -hopefully with some advantage- itself be expressed by iterations of a -so to say- "more basic" function.

Now I assume a function f(x) such that

(where the circle-notation means iteration, and ) - and I ask: what does this function look like? What I'm doing then is this substitution:

*(From 4. I know, that x is now restricted to )*

But if I do now the computation with some example *x* I get the result

***Q:*** Where does this additional factor come from? Where have the above steps missed some crucial information?

<hr>

A code snippet using Pari/GP:

PHP Code:

`f(x) = x-log(x) // define the function `

x0=1.5

// = 1.50000000000

[tmp=x0,pr=1] // initialize

for(k=1,64,pr *= tmp;tmp = f(tmp)); pr // compute 64 terms, show result

// = 1.64872127070

exp(x0) // show expected value

// = 4.48168907034

pr*exp(1) // show, how it matches

// = 4.48168907034

<hr>

Here is an example which shows the type of convergence; I use *x_0=1.5* and internal precision of 200 decimal digits. Then we get the terms of the partial product as