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+--- Thread: Searching for an asymptotic to exp[0.5] (/showthread.php?tid=863)
RE: Searching for an asymptotic to exp[0.5] - tommy1729 - 09/16/2021
- 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
RE: Searching for an asymptotic to exp[0.5] - tommy1729 - 12/12/2021
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
RE: Searching for an asymptotic to exp[0.5] - tommy1729 - 05/01/2022
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/4440852/is-fx-v-sum-n-0-infty-xn-over-gammav-n-1-0-for-all-real
Guess you would like to know.
Regards
tommy1729
Tom Marcel Raes
RE: Searching for an asymptotic to exp[0.5] - tommy1729 - 08/07/2022
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
RE: Searching for an asymptotic to exp[0.5] - tommy1729 - 09/10/2022
Integral from 0 to oo
Exp(t x) f(t) dt
Is related to all Posts above.
And I tend to use this “fakelaplace” to prove Some things About parabolic ficpoints.
Regards
Tommy1729
RE: Searching for an asymptotic to exp[0.5] - tommy1729 - 06/18/2023
I wanted to point out a kind of reverse of fake function searching.
Suppose we want to find the fake function for f(x).
then we suspect a candidate
f(x) = sum f_n x^n.
Now we try the following :
1) estimate max_n : max f_n x^n.
To do that set
f_a x^a = f_(a+1) x^(a+1)
This a = g(x) is an indicatator for the growth rate and dominant terms of the taylor series.
Notice it depends on x.
Now compute another similar indicator , this time for the inferior terms of the taylor series :
2) estimate max_b : f_b x^b < 1
you might want to set f_n x^n = (x/h_n)^n for convenience.
it follows x around h_n is getting close.
b = h(x) is again an indicator of the growth rate and number of dominant terms we at least need for our taylor.
We now found the " start of the tail " of the taylor series.
3) Using a = g(x) and b = h(x) , we are now kinda ready to estimate
I = integral f_n x^n d n from 0 to infinity.
by comparing to the estimate
(f_c + f_d) x^c = min f(x)
what is the basis for fake function theory :
=> min f(x) / x^c = (f_c + f_d)
and compare that f_c , f_d , c , d , a , b and the integral I to f(x) ,
giving us a good idea of how good our original estimate was and a way to improve it.
If f_c/(f_c + f_d) is close to *constant* you know fake function estimate was quite decent.
Or if lim f_c/(f_c + f_d) is close to *constant* you know fake function estimate was quite decent asymptotically.
This is a sketch of the idea , ofcourse calculus tricks can and should be applied !
But that application is case specific , since we might be working with complicated functions.
Also I used integral but ofcourse any approximation of the sum that is good will do, in particular - if possible - actual sum formulas.
Iterating these estimate methods always results in a good estimate.
I wanted to prove that formally but it might take some time.
On the other hand I am not sure of the conjectures made so far are correct.
Im also thinking about what tetration methods are best for the fake functions of tetration.
I have ideas but nothing I am very convinced of yet, let alone a proof.
It might take me some time.
But I can say the used fixpoints matter.
regards
tommy1729
RE: Searching for an asymptotic to exp[0.5] - tommy1729 - 06/29/2023
My friend mick used fake function theory at MSE, so I guess maybe I should share it here :
https://math.stackexchange.com/questions/4714585/o-exp-lnx-ln-lnx2-sum-n-0-infty-a-n-xn-and-0-a-n-asym
@MISC {4724883,
TITLE = {\(O(\exp(\ln(x) \ln(\ln(x))^2)) = \sum_{n=0}^{\infty} a_n x^n\) and \(0 < a_n\) asymptotics?},
AUTHOR = {mick (https://math.stackexchange.com/users/39261/mick)},
HOWPUBLISHED = {Mathematics Stack Exchange},
NOTE = {URL:https://math.stackexchange.com/q/4724883 (version: 2023-06-24)},
EPRINT = {https://math.stackexchange.com/q/4724883},
URL = {https://math.stackexchange.com/q/4724883}
https://math.stackexchange.com/a/4724883
If anyone can improve it or show how good or bad it is plz do so.
I have not tried all methods we have here for the problem.
I might come back to that.
If anyone wants more details of what he wrote or has questions I can probably explain.
I checked and his computation is correct.
Wonder about the integral methods.
regards
tommy1729