0
$\begingroup$

I am trying to prove equations (3) given in this paper

http://users.cecs.anu.edu.au/~thush/publications/vtc_final.pdf.

The authors use taylor series to approximate function $f(q) = \left(1 + q \dfrac{w_s}{w_0}\right)^{\alpha}$.

The resulting approximation is

$f(q) = \left(1 + q \dfrac{w_s}{w_0}\right)^{\alpha} \approx exp\left(q \alpha \dfrac{w_s}{w_0} - \dfrac{1}{2} \alpha \left(q \dfrac{w_s}{w_0}\right)^2 + \dfrac{1}{3} \alpha \left(q \dfrac{w_s}{w_0}\right)^3 + \dots\right)$.

I am not able to get the same results. My result is similar but not exactly the same.

I would appreciate if someone would give a try to prove this approximation.

Thanks in advance!

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ This appears to just be using the Maclaurin expansion of $\ln(1+x)$: en.wikipedia.org/wiki/Taylor_series#Natural_logarithm $\endgroup$
    – Gabe Conant
    Commented Jul 31, 2019 at 15:38
  • $\begingroup$ Thx Gabe for help. Indeed by transforming function to $ln(f(q)) = \alpha \left(1+q \dfrac{w_s}{w_0}\right)$ and finding its approximation and transforming back to $f(q)$ I get same results as in the paper. $\endgroup$
    – Emil Bjelski
    Commented Jul 31, 2019 at 16:15

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .