Searching for an asymptotic to exp[0.5]

[tommy1729]

- 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]

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 sufficiently large we get 



Comments, sharper bounds or alternative methods are welcome. 

regards

tommy1729

Tom Marcel Raes

[tommy1729]

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]

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

[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