(1) The limit as a Gaussian's width (1/α) goes to zero is 0 (2) The same limit of the "Gaussian Sum" however is 1pic.twitter.com/3U7I4zsQ1r
Show this thread
This result surprised me, since the two expressions actually are quite close to one another.
Δ ≈ 1.83 × 10⁻⁴
The Gaussian Sum (2) has no closed formpic.twitter.com/HpVZEySlev
-
-
Couldn’t you use that theorem from complex analysis for transforming infinite sums of functions into contour integrals for the second one ?
- 3 more replies
New conversation -
-
-
Yeah, surprises me too! How did you come across this result?
-
just was wondering the discrete sum of a gaussian looks like, then realized it was close to root-π and then tried to see if they had similar limiting behaviors
End of conversation
New conversation -
-
-
This Tweet is unavailable.
-
thanks! :) i may need a couple days off in the future though haha
- 1 more reply
-
-
-
The fact that the Gaussian sum and integral are almost identical is an immediate consequence of the "modularity" of the Jacobi theta function (or, more generally, the Poisson resummation theorem). When α=1, the result is basically that π² is big.pic.twitter.com/IGLeYDc7rv
Thanks. Twitter will use this to make your timeline better. UndoUndo
-
-
-
The difference is that measure can be concentrated at a point in the discrete case. It cannot in the continuous case. When you interchange the limit and integral, you get 0 everywhere except x=0 (or n=0).
Thanks. Twitter will use this to make your timeline better. UndoUndo
-
-
-
Poisson summation on the second gives an almost closed form :)
Thanks. Twitter will use this to make your timeline better. UndoUndo
-
-
-
A lot of these are great! Would you consider sharing your steps on a blog or something for some of these that are most interesting/important? Can just be phone shots of your hand-written notes ... or something simple.
Thanks. Twitter will use this to make your timeline better. UndoUndo
-
Loading seems to be taking a while.
Twitter may be over capacity or experiencing a momentary hiccup. Try again or visit Twitter Status for more information.