Did we study this before ? - Printable Version +- Tetration Forum ( https://math.eretrandre.org/tetrationforum)+-- Forum: Tetration and Related Topics ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)+--- Forum: Mathematical and General Discussion ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)+--- Thread: Did we study this before ? ( /showthread.php?tid=927) |

Did we study this before ? - tommy1729 - 10/02/2014
Im not sure if we studied this before or not. dexp(x) = exp(x) - 1. exp^[1/2](x) - dexp^[1/2](x) = a_1 x^-1 + a_2 x^-2 + ... That last equation is conjectured. A much weaker conjecture is that it holds asymptotically for large x. ( and yes I could involve fake function theory now , but I prefer not , its not the main issue here ) In fact , it is not difficult to show that IF exp^[1/2](x) - dexp^[1/2](x) is a laurent series then the x^0 , x^1 , ... terms are all 0. This all looks pretty familiar and Im not sure if we considered this before or not. Since dexp has a parabolic fixpoint , I assume the equation only holds asymptotically for large x. The reason for that is that if exp^[0.5] is laurent and dexp^[0.5] is not then their difference cannot be laurent. One could also study exp^[0.5] - 2 sinh^[0.5] but that difference grows to 0 very fast. exp^[1.5] fast ?? Anyway those differences between half-iterates of similar functions fascinates me. But in this specific dexp case its quite likely that my fascination comes from stuff I have temporarily forgotten and/or from not understanding dexp^[0.5] well. So back to dexp^[0.5]. the fixpoint is parabolic. Now recently I mentioned using fake function theory to get a Taylor anyway. But again the focus here is not on fake function theory. What I was intrested in was something like this : dexp^[0.5] = f ( g(x) ). where f is a Taylor series. Preferably entire. And g is an analytic function that is not entire. Or similar. For instance dexp^[0.5](x) = f ( sqrt(x) ) or dexp^[0.5](x) = f ( ln(x^3 + 1) ) I think that the number of pettals gives A such that parabolicfixpointfunction^[0.5](x) = f ( x^(1/A) ) or something. Maybe I need to reconsider parabolic fixpoints again. Notice the once again returning idea of fake function theory : parabolic fixpointfunction^[0.5](x) = f ( fake x^(1/A) ). thereby giving a systematic way of computing fake half-iterates at parabolic fixpoints. Im probably typing more than thinking and should start thinking and perhaps reading now. But even if I bump my head in 5 min from now , saying it was trivial , I still want to share this. regards tommy1729 RE: Did we study this before ? - tommy1729 - 10/02/2014
5 min have passed. Still no aha moment so I must be either good or bad today. regards tommy1729 |