Help understanding this approximation

[Ryuzaki]

In a paper that I'm reading, the authors write:-

$N_e \approx \frac{3}{4} (e^{-y}+y)-1.04$ ------------ $(4.31)$

Now, an analytic approximation can be obtained by using the expansion with respect to the inverse number of "e-foldings" ($N_e$ is the number of "e-foldings"). For instance, eq. $(4.31)$ yields:-

$e^y = \dfrac{3}{4N_e} - \dfrac{9ln(N_e)}{16(N_e)^2} -\dfrac{0.94}{(N_e)^2} + O(\dfrac{ln^2(N_e)}{(N_e)^3})$

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

 

[pasmith]

In a paper that I'm reading, the authors write:-

$N_e \approx \frac{3}{4} (e^{-y}+y)-1.04$ ------------ $(4.31)$

Now, an analytic approximation can be obtained by using the expansion with respect to the inverse number of "e-foldings" ($N_e$ is the number of "e-foldings"). For instance, eq. $(4.31)$ yields:-

$e^y = \dfrac{3}{4N_e} - \dfrac{9ln(N_e)}{16(N_e)^2} -\dfrac{0.94}{(N_e)^2} + O(\dfrac{ln^2(N_e)}{(N_e)^3})$

Can anyone tell me how this approximation is done?

It's an asymptotic expansion. Finding these is more of an art than a science. Hinch is a good introduction.

I'm not familiar with the $O$ notation either. What does it mean?

See http://en.wikipedia.org/wiki/Big_O_notation.