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Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*...
#1
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:
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
Gottfried Helms, Kassel
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Messages In This Thread
Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... - by Gottfried - 03/12/2013, 08:58 AM

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