Timeline for Saddle point approximation of terms in a sum
Current License: CC BY-SA 4.0
5 events
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| Nov 23, 2020 at 15:29 | comment | added | Michael Engelhardt | It depends of course on how $b_n $ behaves. But I think one thing we can say is that one has to keep track of the behavior of each term with $N$ and only drop parts that fall off more quickly than $1/N$ - so one can't be too naive in disregarding the behavior of $g(x)$ in the saddle-point approximation. Note that terms of order $1/N$ end up yielding a contribution $(1/N)\sum_{n=0}^{N} \rightarrow \int dn$. | |
| Nov 23, 2020 at 14:58 | comment | added | thedude | Well, the title says "approximation for terms in a sum". I would like to understand how to control the approximation for the terms, if I should divide them in two parts, or not, how so, etc. Doing the finite sum first misses the point. | |
| Nov 23, 2020 at 14:46 | comment | added | Michael Engelhardt | Welllll ... then you should have asked that. And you'd have to say more about $b_n $ - if it depends on $N$, then we have practically nothing to go on anymore. I'd still try to do the finite sum first - after all, you only need the leading-$N$ behavior, which you might still be able to get analytically, even if the exact sum is intractable. | |
| Nov 23, 2020 at 14:40 | comment | added | thedude | Sure, thank you, but suppose I can't do the finite sum first. Say, because actually $a_n=b_n\int e^{Nf(x)+ng(x)}dx$ for some pesky $b_n$. | |
| Nov 23, 2020 at 14:34 | history | answered | Michael Engelhardt | CC BY-SA 4.0 |