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 Searching for an asymptotic to exp[0.5] tommy1729 Ultimate Fellow Posts: 1,645 Threads: 369 Joined: Feb 2009 09/16/2021, 11:13 PM - quote - 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. - end quote - another useful example is   integral from 1 to +oo [ t^x g(t) dt ] with g(t) = ln(t)^v  equals : v! (x-1)^(-v-1) for Re(x)>1. thereby connecting to laurent series and more. This might be well known but for completeness and relevance I had to add it. regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,645 Threads: 369 Joined: Feb 2009 12/12/2021, 10:02 PM A nice example of what fake function theory can do and seems nontrivial without it is the following result. * notice sums and their related integrals can be close * For positive $x$ sufficiently large we get  $\int_1^{\infty} \exp(xt-t^3) dt <\frac{2\exp(\frac{2\sqrt3x^{3/2}}{9})}{ln(x)}$ Comments, sharper bounds or alternative methods are welcome.  regards tommy1729 Tom Marcel Raes tommy1729 Ultimate Fellow Posts: 1,645 Threads: 369 Joined: Feb 2009 05/01/2022, 11:36 PM I talked to my friend Mick friday and that resulted in this MSE post where alot of ideas from here are used. (The integral transformation , the asymptotics , zero's , and taylors with positive coefficients.) https://math.stackexchange.com/questions...r-all-real Guess you would like to know. Regards tommy1729 Tom Marcel Raes tommy1729 Ultimate Fellow Posts: 1,645 Threads: 369 Joined: Feb 2009 08/07/2022, 10:42 PM Maybe I mentioned this before but it seems a related idea is the Wiman-Valiron theory. In particular for the related TPID problem. btw where are the TPID questions gone too ?!? I do not see them anymore !! regards tommy1729 « Next Oldest | Next Newest »

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