# Tetration Forum

Full Version: Searching for an asymptotic to exp[0.5]
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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
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
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
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
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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