Question on Cosmological Constant paper

[zaybu]

In the paper, http://arxiv.org/pdf/1205.3365v1.pdf, page 21, the author argues that if:
t →∞(1-iϵ), all the terms in equation (193) goes to zero, except the first term.

Can anyone explain this to me?

Thanks

 

[clamtrox]

Well, it's pretty simple I think. Suppose you have a sum
S = \sum_n \exp(-i E_n t). Let t = (1-i \epsilon) s where s is real. Then S = \sum_n \exp(-i E_n s) \exp(-E_n s). Now you want to evaluate this sum when s \rightarrow \infty. In that limit, the first term oscillates rapidly and the second term of the sum goes to zero.

The dominating component of the sum is the one that goes to zero the slowest. If we order the E:s so that E_0 < E_1 < E_2 < ..., then the leading order behaviour is given by the E_0-term as the rest go to zero even faster than that one.

 

[zaybu]

Thanks