In a paper that I'm reading, the authors write:- [itex]N_e \approx \frac{3}{4} (e^{-y}+y)-1.04[/itex] ------------ [itex](4.31)[/itex] Now, an analytic approximation can be obtained by using the expansion with respect to the inverse number of "e-foldings" ([itex]N_e[/itex] is the number of "e-foldings"). For instance, eq. [itex](4.31)[/itex] yields:- [itex]e^y = \dfrac{3}{4N_e} - \dfrac{9ln(N_e)}{16(N_e)^2} -\dfrac{0.94}{(N_e)^2} + O(\dfrac{ln^2(N_e)}{(N_e)^3})[/itex] Can anyone tell me how this approximation is done? I'm not familiar with the $O$ notation either. What does it mean? How do the authors arrive at that expression? If anyone should require it, the original paper can be found here: https://arxiv.org/pdf/1001.5118.pdf?origin=publication_detail
It's an asymptotic expansion. Finding these is more of an art than a science. Hinch is a good introduction. See http://en.wikipedia.org/wiki/Big_O_notation.