help a /b/rother out

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help a /b/rother out Anonymous Fri May 21 14:45:48 2010 No.988785 [Reply] [Original]

alright /sci/fags,
i have to study a research paper for my bachelor's thesis. in the technical appendix of this paper a certain approximation is used and i simply cannot manage to derive this approximation. it looks very much like a second order taylor expansion, there are some problems. here it is:
the scalar Xt is the state of a dynamic system at time t, X is this variable's steady-state. imagine whatever you want, current, temperature, in my case it is consumption. the authors give the following approximation:

Xt/X = 1 + ln(Xt/X) + 0.5 * (ln(Xt/X))^2

how do you derive that?
i thought of the following:

if you write down the 2nd order taylor expansion for ln(Xt) around X you get:

ln(Xt) = ln(X) + 1/X * (Xt - X) - 0.5 * 1/(X)^2 * (Xt - X)^2

(Xt - X) can be rewritten as X(Xt/X - 1) and Xt/X - 1 is the 1st order taylor expansion of ln(Xt/X). therefore the equation above can be transformed to:

ln(Xt) = ln(X) + ln(Xt/X) - 0.5 * (ln(Xt/X))^2

substracting ln(X) gives ln(Xt/X) on the left hand side. using the 1st oder tayler expansion only on this left hand side (i.e. rewriting it as Xt/X -1) lets you transform this equation to:

Xt/X = 1 + ln(Xt/X) - 0.5 * (ln(Xt/X))^2

which is almost the formular provided by the authors of my paper. you might think the just misprinted the sign before 0.5. i plotted the two approximation formulars, one with the positive sign, one with the negative sign. X was arbitrarily decided to be 5. so Xt/X is Xt/5 (red), the shit approximation (green) is the one derived above with the negative sign, and the rather good one (blue) is the one given by the authors.

what am i missing?

TL;DR i will keep posting nudes till someone answers my question!

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/sci/ - Science & Math