# Help understanding this approximation

by Ryuzaki
Tags: approximation
 P: 40 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?...ication_detail
HW Helper
Thanks
P: 1,003
 Quote by 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?

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.

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