I came across the following statement:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\sum_n p(n)e^{-in\theta} \approx exp[-i\theta \langle n\rangle - \theta^2 \langle ( \delta n)^2 \rangle / 2][/tex]

where [itex]\theta[/itex] is small, [itex]\sum_n p(n) = 1[/itex], [itex]\langle n \rangle = \sum_n p(n)n[/itex], and [itex]\langle ( \delta n)^2 \rangle = \sum_n p(n)(n-\langle n \rangle)^2[/itex].

I am pretty stumped trying to figure out how this asymptotic expression is derived. I tried writing out the exponents as sums to no avail. I can see that [itex](-i\theta)^2 = -\theta^2[/itex] but I am pretty confused regarding the presence of [itex]\langle n \rangle[/itex] and [itex]\langle (\delta n)^2 \rangle[/itex] in the exponential. Any suggestions are greatly appreciated!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Approximating a sum of exponentials

**Physics Forums | Science Articles, Homework Help, Discussion**