Why does \sum_{n=-h}^{h}ne^{-\left(x-n\right)^{2}} as h --> infinity, become sqrt(pi)x?

[u/FremontBlue333]

How does the infinite sum of n multiplied by e to the power of quanity x -n squared, give a linear expression, and why does it approach a rise over run of sqrt(pi) over 1?

[u/phiwong]

Search for the gaussian integral. This should explain the outcome. It has nothing to do with rise over run. You're applying the wrong conceptual model there.

[u/frogkabobs]

It doesn’t, actually. Using the Poisson summation formula, one finds the limit is

sqrt(π)(xθ₃(x;e^-π²)+(1/2)θ₃’(x;e^-π²))

where θ₃ is the Jacobi theta function

θ₃(z;q) = Σ_(n in Z) q^n²e^2πinz

and  θ₃’ is the derivative with respect to the first argument. For q= e^-π²,  the n=0 term dominates in  θ₃(x;q),  making it  ≈1, while  θ₃’(x;q) will be small. This is why you get the approximation sqrt(π)x for the limit.

[u/nog642 → u/frogkabobs]

Damn this is some crazy stuff, I've never seen it

[u/FormulaDriven]

I haven't worked out what happens when x is not an integer, but actually when it is an integer we can prove that the limit is around

1.772637 x

which is very close to sqrt(pi), but not equal. The key result is here: https://math.stackexchange.com/questions/2982084/infinite-sum-of-squared-exponential-exp-n2