I am wondering about the properties of "Asymptotic series expansion".
Considering a representative function
$ f(R)=\frac{a+bR+cR^2}{d+eR+fR^2}$ where $ a, b, c , d , e , f $ are constants.
How can we decompose and/or simplify $ f(R)$ before expanding it asimptotically as $R$ goes to $ \infty $ ?
I must simplify or decompose $ f(R)$ in small pieces as much as possible while expanding. Because it is a very very long function.
Thank you..
Every rational function can be written in the form $$ f(n)=\sum_{i=1}^k \frac{P_i(n)}{(n-\alpha_i)^{\beta_i}} + Q(n), \quad \deg P_i < \beta_i. $$ Here the $\alpha_i$ are the roots of the denominator, with multiplicity $\beta_i$, $P_i, Q$ are polynomials, and $P_i(\alpha_i)\neq0$ (assuming the rational function is reduced). Given this unique representation, one can write an exact formula for the coefficients of the generating series, and also obtain the asymptotics.