Revitalizing an old idea : estimated fake sexp'(x) = F3(x) tommy1729 Ultimate Fellow     Posts: 1,859 Threads: 402 Joined: Feb 2009 02/27/2022, 10:17 PM Revitalizing an old idea : estimated fake sexp'(x). It's an old idea but I want to put some attention to it again. Perhaps with our improved skills and understanding this might lead us somewhere. As most members and frequent readers know so-called fake function theory was developed by myself and sheldon to  1) create a real entire function 2) that is an asymptotic of a given function for positive reals 3) that has all its taylor or maclauren coefficients positive or at least non-negative. There are some extra conditions such as strictly rising for positive real x imput for the given function and such. But basicly that is what fake function theory is about. We call such created functions - or the attempts - fake functions. We also considered the cases were the given function already satisfied 1 -- 3 and how our algorithms created " fake ones ". Many variants occured , such as replacing sums with analogue integrals and stuff but that is not important here. We got good results for fake exp^[1/2](x).   So far the intro to the fake function theory part. We also discussed base change constants here. And we discussed the " expontential factorial " function ; the analogue of tetration like 2^3^4^... , the analogue of the factorial 1*2*3*4*... . We are also familiar with telescoping sums. And we even discussed converging infinite sums (over natural index n )  of the n th iteration of a function. We also know the derivative of exp^[n](x) = exp^[n](x) * exp^[n-1](x) * exp^[n-2](x)*... And how close this is related to the derivative of sexp(x) ; d sexp(x)/dx = sexp(x) * d sexp(x-1)/dx. Is this all related to an old idea of myself and partially others ?? Yes certainly. Lets estimate the derivative of sexp(x) but without using fake function theory tools directly. Impossible ? Well far from , if you accept brute estimates. So we want something faster than any fixed amount of exp iterations. But slower than sexp(x^2) or sexp(x)^2 or so. The exact speed of the function is something to investigate and discuss. But a logical attempt is this : This function is entire ... F3(x) = (1 + exp(x)/2^3) (1 + 2^3 exp^(x)/ 2^3^4 ) (1 + 2^3^4 exp^(x)/ 2^3^4^5 ) ... and it converges fast. It is also faster than any exp(x)^[k](x) for fixed k. And all the coefficients are positive. Also notice the pseudotelescoping product. This leads to the asymptotic conjecture : Using big-O notation : For 1 < x  integral from 0 to x   F3(t) dt = F4(x) =  O ( fake sexp(x + c) ). For some integration constant , and some constant c. How about that ? *** Next idea  lim v to oo ; ln^[v] F4(x + v) = ?? Which also looks very familiar. Regards tommy1729 Tom Marcel Raes « Next Oldest | Next Newest »

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