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

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

 

[tacman]

Yes, we can construct the functions a0, a1, a2, ... by knowing G(x) for big x. This is because the integral (1) is a representation of G(x) in terms of a series of functions a0, a1, a2, ... which are multiplied by powers of 1/x. By knowing G(x) for big x, we can determine the behavior of G(x) as x approaches infinity and therefore determine the behavior of the series. This will allow us to determine the functions a0, a1, a2, ... as they are coefficients in the series.

However, it is important to note that this process is not always straightforward and may require some mathematical manipulation and analysis. Additionally, the accuracy of the constructed functions a0, a1, a2, ... will depend on the accuracy of the known values of G(x) for big x. If the known values are not precise enough, the constructed functions may not accurately represent the behavior of G(x) for big x.

In summary, while it is possible to construct the functions a0, a1, a2, ... by knowing G(x) for big x, the accuracy of the constructed functions will depend on the accuracy of the known values of G(x) and may require some mathematical analysis.