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 Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried Ultimate Fellow Posts: 787 Threads: 121 Joined: Aug 2007 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: $\exp(x) = x \cdot f^{\circ 1}(x)\cdot f^{\circ 2}(x) \cdot \ldots$ 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 $\exp(x) = x \cdot f^{\circ 1}(x)\cdot f^{\circ 2}(x)\cdot f^{\circ 3}(x)\cdots$ (where the circle-notation means iteration, and $f^{\circ 0}=x, f^{\circ 1}(x)=f(x)$) - and I ask: what does this function look like? What I'm doing then is this substitution: $\begin{array} {lrll} 1.& \exp(x) & = &x & \cdot f^{\circ 1}(x) & \cdot f^{\circ 2}(x) & \cdot f^{\circ 3}(x) & \cdots \\ 2.& \exp(f(x))&= && f^{\circ 1}(x) & \cdot f^{\circ 2}(x) & \cdot f^{\circ 3}(x) & \cdots \\ \\ \\ 3.& {\exp(f(x))\over \exp(x) } & = & \frac 1x \\ \\ & \exp(f(x)) & = & &{ \exp(x) \over x} \\ \\ \\ 4. & f(x)&=& x & - \log(x) \end{array}$ $\qquad \qquad$ *(From 4. I know, that x is now restricted to $x \gt 0$)* But if I do now the computation with some example *x* I get the result $y = x \cdot f^{\circ 1}(x)\cdot f^{\circ 2}(x)\cdot f^{\circ 3}(x)\cdots \\ y = \exp(x) / \exp(1)$ ***Q:*** Where does this additional factor come from? Where have the above steps missed some crucial information?
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
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 $\begin{array} {r|r} x_k=f^{\circ k}(x) & (x_k-1) \\ \hline 1.50000000000 & 0.500000000000 \\ 1.09453489189 & 0.0945348918918 \\ 1.00420537512 & 0.00420537512103 \\ 1.00000881788 & 0.00000881787694501 \\ 1.00000000004 & 3.88772483656E-11 \\ 1.00000000000 & 7.55720220223E-22 \\ 1.00000000000 & 2.85556525627E-43 \\ 1.00000000000 & 4.07712646640E-86 \\ 1.00000000000 & 8.31148011150E-172 \\ 1.00000000000 & 1.020640763E-202 \\ 1.00000000000 & 1.020640763E-202 \\ \cdots & \cdots \end{array}$ Gottfried Helms, Kassel « Next Oldest | Next Newest »

 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 RE: Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... - by Balarka Sen - 03/12/2013, 10:01 AM RE: Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... - by Gottfried - 03/12/2013, 10:13 AM RE: Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... - by tommy1729 - 03/13/2013, 12:12 AM RE: Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... - by tommy1729 - 03/13/2013, 10:52 PM RE: Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... - by Gottfried - 07/17/2013, 09:46 AM

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