Can we construct the functions a0,a1,a2,... by knowing G(x) for big x?

[eljose]

Let be the integral:

\int_{a}^{x}dtF(t)/t (1)

Let,s suppose we can find a divergent asymptotic series for it in the form:

\int_{a}^{x}dtF(t)/t=a0(x)/x+a1(x)/x^{2}+.....

where of course the a0,a1,... are also function of x, let,s also suppose that we could calculate the integral (1) exactly and that was equal to the function G(x), then my question is if we could construct the functions a0,a1,a2,... by knowing G(x) for big x

 

[ReyChiquito]

i don't understand the question, but i know this much

\begin{array}{l}<br /> a_0(x)=\int_a^x F(t)dt\\<br /> \\<br /> a_1(x)=\int_a^x \int_a^{x_1} F(t) dt dx_1\\<br /> \\<br /> a_2(x)=2\int_a^x \int_a^{x_2} \int_a^{x_1} F(t)dtdx_1dx_2\\<br /> \vdots\\<br /> a_n(x)=n! \int_a^x\int_a^{x_{n-1}}\dots\int_a^{x_1}F(t)dt dx_1\dots dx_{n-1}<br /> \end{array}