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
