Improper integrals at MSE - 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: Improper integrals at MSE ( /showthread.php?tid=943) |

Improper integrals at MSE - tommy1729 - 12/21/2014
Last Friday while playing chess I talked to my MSE friend mick. You might like this : http://math.stackexchange.com/questions/1076074/about-fractional-iterations-and-improper-integrals regards tommy1729 RE: Improper integrals at MSE - tommy1729 - 12/23/2014
Sheldon's answer on MSE is nice. Thank you Sheldon. I made an intresting observation relating things to complex dynamics. The main thing is the mysterious looking change -> My observation can be considered positive or negative , intresting or dissapointing , it depends on taste I guess and the hope for nontrivial analogues. But the idea of having some function for which every real iterate " works " is found , though it might not be as nontrivial as mick hoped. ( not saying a nontrivial case cannot exist ). - Maybe variants of this exist in calculus textbooks / papers but its very " dynamical " in nature - Anyway here it is : => with => => Solve .. => Thus : Which is trivial. Reminds me of this quote : " Young man, in mathematics you don't understand things. You just get used to them. " John von Neumann. Btw I considered doing the things (steps above) in reverse : showing -> is valid from the validity of regards tommy1729 " the master " RE: Improper integrals at MSE - sheldonison - 12/24/2014
(12/23/2014, 11:31 PM)tommy1729 Wrote: ....function for which every real iterate " works " is found , though it might not be as nontrivial as mick hoped. ( not saying a nontrivial case cannot exist ). Hey Tommy, Not sure I understood all of that ... But it inspired me to consider the following sequence of functions ... Does g(x) converge, and is it a solution of interest to Mick? If g(x) converges, and it is analytic, then it has a Taylor/Laurent series.... Update:, by brute force, using a lot of computer cycles to estimate the limit, and then turn the coefficents it back into a fraction with power's of 2's... I get the following Laurent series, as the function that Mick might be looking for. It would probably be normally expressed as update2: This would be compactly expressed via the Abel function as: And then we get: Finally, Mick's desired function in closed form would be as follows. With a little algebra, we generate all of the fractional iterates of g(z) as well. Then, using Mick's notation we have the desired g(z,t) function, which has all fractional iterates defined as: for t=1, this is the same as the Laurent series above RE: Improper integrals at MSE - tommy1729 - 12/28/2014
I once had a few threads here were I discussed the need for limits of the form ( a + f(n)/(bn)) ^[n] = C or similar. This seems very much like your limit, maybe you got inspired from me too. Anyways I must say that thread was looking for tetration type functions / limits so in that sense your limit is more " classical ". I was not able to find the threads again but this one is somewhat similar : http://math.eretrandre.org/tetrationforum/showthread.php?tid=262&highlight=limit regards tommy1729 RE: Improper integrals at MSE - tommy1729 - 01/11/2015
This leads to the question Is there a solution F(x) = super of ( x + a - 2^t/(x+b) ) for some real a,b (super with respect to t) such that integral f(F(x)) dx = integral f(x) ? regards tommy1729 RE: Improper integrals at MSE - tommy1729 - 01/16/2015
I think this is an underrated thread. If it turns out true , this gives an intresting connection between dynamical systems and calculus ! regards tommy1729 |