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Searching for an asymptotic to exp[0.5]
- 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
Reply
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
Reply
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
Reply


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