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- 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

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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

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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

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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

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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

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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

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My friend mick used fake function theory at MSE, so I guess maybe I should share it here :

https://math.stackexchange.com/questions...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