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+--- Thread: Mick's differential equation (/showthread.php?tid=981)
Mick's differential equation - tommy1729 - 04/05/2015
Mick posted an intresting question here :
http://math.stackexchange.com/questions/1221034/f-x-fx-x1t-1
It reminds of the binary partition function we discussed before which was strongly related by the similar equation F ' (x) = F(x/2).
Maybe the method to solve F ' (x) = F(x/2) can be used/modified to solve Mick's differential equation.
Anyway I think its intresting.
regards
tommy1729
RE: Mick's differential equation - tommy1729 - 04/06/2015
Although far from an answer , using the Mittag-Leffler function to Get a fake (1+x)^t will probably get us a good approximation in terms of a Taylor series.
Although getting these Taylor coëfficiënts is Nice , its not a closed form asymtotic.
Unless we get a simple rule for these coef , its hard to get THE asym from the Taylor.
Regards
Tommy1729
RE: Mick's differential equation - tommy1729 - 04/10/2015
Clearly this function grows slower than any exponential but faster than any polynomial or even exp(x^t).
This implies that if our asymptotic is entire - or the function itself ?? - then it is determined by its zero's completely.
Otherwise fake function theory can be applied.
And then the fake is compl determined by its zero's.
A quick brute estimate is exp(x^T + T x^{tT}) , where T is between t and 1.
ITS not accurate i know.
All of this is ofcourse Up to a multip constant.
Regards
Tommy1729
RE: Mick's differential equation - tommy1729 - 04/10/2015
About post 2 i use truncated carleman matrices for (fake) entire functions.
And then simply solve THE equation.
Regards
Tommy1729