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Help understanding this approximationby Ryuzaki
Tags: approximation 
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#1
Jun1914, 12:27 PM

P: 40

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 "efoldings" ([itex]N_e[/itex] is the number of "efoldings"). 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?...ication_detail 


#2
Jun1914, 01:35 PM

HW Helper
Thanks
P: 1,003

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


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