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!

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# Approximating a sum of exponentials

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