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 Searching for an asymptotic to exp[0.5] tommy1729 Ultimate Fellow Posts: 1,454 Threads: 350 Joined: Feb 2009 09/06/2016, 03:47 PM Back to basics In addition to post 17,18 notice that D exp^[1/2](x) = D exp^[1/2] (exp(x) ) * exp(x) / exp^[1/2](exp(x)). That follows from ln exp^[a] ( exp(x) ) = exp^[a](x) and the chain rule for derivatives. By induction / recursion this gives a nice way ( product ) to compute the derivative. This strenghtens the conclusions from post 17 , 18 and shows that 1 + o(1) <<_n 2. ( smaller after only a few iterations n ) We conclude by noting that the Taylor T T = Sum_{K=4}^{oo} d_k x^k With d_k = exp( - k^2 ) grows slower then exp^[1/2](x) , yet faster then any polynomial. T has growth 0 , like exp( ln^2 (x) ) and similar. Did we meet T yet ?? I believe T was a fake of Some elementairy like exp( ln^2 ) or such ... This again leads to the desire of inverse fake or its related integral transforms ... --- Regards Tommy1729 tommy1729 Ultimate Fellow Posts: 1,454 Threads: 350 Joined: Feb 2009 03/15/2018, 01:23 PM I request complex plots of f(x) = fake exp^[1/3](x) , f(f(x)) and f(f(f(x))). Like sheldon did for fake exp^[1/2](x) in one of the early posts in this thread. It is very important !! ( potentially new results/conjectures based on those plots ! ) Regards Tommy1729 Ps make sure to make backups of this website/content ? Bo ? tommy1729 Ultimate Fellow Posts: 1,454 Threads: 350 Joined: Feb 2009 07/18/2021, 11:47 PM Im currently into this  f(x) = integral from 1 to +oo [ t^x g(t) dt ] and the related f(x) = a_0 + a_1* 2^x + a_2* 3^x + a_3 4^x + ... ( for suitable f(x) ) This is ofcourse similar to finding taylor series and fake function theory ( so far ). The idea floating around of finding approximate entire dirichlet series with a_n >=0 is ofcourse tempting. However maybe the inequality a_n < min ( f(x)/n^x ) might be less efficient here ?? What do you guys think ? NOTICE the integral transform is NOT the mellin transform. Anyone knows an inverse integral transform for this ? I think f(x) = gamma(x,1) + (constant) might make an interesting case ... regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,454 Threads: 350 Joined: Feb 2009 07/21/2021, 05:11 PM (07/18/2021, 11:47 PM)tommy1729 Wrote: Im currently into this  f(x) = integral from 1 to +oo [ t^x g(t) dt ] and the related f(x) = a_0 + a_1* 2^x + a_2* 3^x + a_3 4^x + ... ( for suitable f(x) ) This is ofcourse similar to finding taylor series and fake function theory ( so far ). The idea floating around of finding approximate entire dirichlet series with a_n >=0 is ofcourse tempting. However maybe the inequality a_n < min ( f(x)/n^x ) might be less efficient here ?? What do you guys think ? NOTICE the integral transform is NOT the mellin transform. Anyone knows an inverse integral transform for this ? I think f(x) = gamma(x,1) + (constant) might make an interesting case ... regards tommy1729 As a small example :  integral from 1 to +oo [ t^x g(t) dt ] with g(t) = exp(- ln(t)^2 )  equals : (1/2) * ( erf((x+1)/2) +1) *  sqrt(pi) * exp( (1/4)* (x+1)^2 ). I find this fascinating. btw g(t) reminds me again of the binary partition function. Differentiation under the integral sign (dx) will probably be a usefull trick too. More math must exist. regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,454 Threads: 350 Joined: Feb 2009 07/21/2021, 06:51 PM (07/21/2021, 05:11 PM)tommy1729 Wrote: (07/18/2021, 11:47 PM)tommy1729 Wrote: Im currently into this  f(x) = integral from 1 to +oo [ t^x g(t) dt ] and the related f(x) = a_0 + a_1* 2^x + a_2* 3^x + a_3 4^x + ... ( for suitable f(x) ) This is ofcourse similar to finding taylor series and fake function theory ( so far ). The idea floating around of finding approximate entire dirichlet series with a_n >=0 is ofcourse tempting. However maybe the inequality a_n < min ( f(x)/n^x ) might be less efficient here ?? What do you guys think ? NOTICE the integral transform is NOT the mellin transform. Anyone knows an inverse integral transform for this ? I think f(x) = gamma(x,1) + (constant) might make an interesting case ... regards tommy1729 As a small example :  integral from 1 to +oo [ t^x g(t) dt ] with g(t) = exp(- ln(t)^2 )  equals : (1/2) * ( erf((x+1)/2) +1) *  sqrt(pi) * exp( (1/4)* (x+1)^2 ). I find this fascinating. btw g(t) reminds me again of the binary partition function. Differentiation under the integral sign (dx) will probably be a usefull trick too. More math must exist. regards tommy1729 Ofcourse im not trying directly to solve the integral transform for exp^[0.5]. What I am trying here is to solve for standard functions ... and then use them. As in TT(exp^[0.5](s)) = TT (sum over standard functions) = sum over TT (standard functions) Where TT stands for " tommy-transform " the integral transform mentioned above. If this is the best strategy , I do not know.  It just felt natural. regards tommy1729 « Next Oldest | Next Newest »

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