Can Analytical Continuation Extend Asymptotic Expansions to Lower Values of x?

[eljose]

Let,s suppose we have the asymptotyc development of the integral:

\int_{x}^{\infty}F(t)=g(x)[1+a/x+b/x^{2}+c/x^{3}+...]

where a,b,c,.. are known constants and g(x) is a known function then you all will agree that this expression could be useful to compute the integral when x-------->oo, my question is if this expression can be analytically continued to calculate the integral for low x for example x=1,2,3...

 

[matt grime]

Why would I want to compute the integral

\int_{x}^{\infty}F(t)dt

as x tends to infinity? If that integral exists for all x, then obviously I know that the limit, as x tends to infinity must be zero without doing any computation.

 

[eljose]

yes but perhaps you are interested in knowing the values of the integral for big x x=100,1000,100000000000000 or for low x x=1,2,3,4,...

 

[matt grime]

but that is strictly different from evaluating a limit as x--->infinty.