Timeline for Why is it the case that $\sum\limits_{x \in \mathbb{Z}} e^{-(x + \frac{1}{4})^2} = \sqrt{\pi}$?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 1 at 2:40 | vote | accept | dotdashdashdash | ||
| Jun 11 at 18:41 | review | Close votes | |||
| Jun 17 at 1:51 | |||||
| Jun 11 at 18:23 | comment | added | Nemo | Duplicate of Sum of Gaussian pdfs | |
| Jun 11 at 1:54 | vote | accept | dotdashdashdash | ||
| Jun 11 at 1:58 | |||||
| Jun 11 at 1:52 | vote | accept | dotdashdashdash | ||
| Jun 11 at 1:54 | |||||
| Jun 10 at 21:49 | comment | added | TheSimpliFire | This is a duplicate of math.stackexchange.com/questions/3562092/…. When $x=1/4$, the first error term occurs when $n=2$ (since $n=1$ yields $\cos(n\pi/2)=0$) which explains the magnitude of the error of $|\log_{10}(e^{-4\pi^2})|\approx17$ digits. This also explains why $1/4$ is the only near miss, as $n=1$ yields a nonzero term for other choices like $1/3$ or $1/5$, so the magnitude of the error there is at most $|\log_{10}(e^{-\pi^2})|\approx5$ digits. | |
| Jun 10 at 21:26 | history | edited | GH from MO | CC BY-SA 4.0 |
added 4 characters in body; edited tags
|
| Jun 10 at 21:24 | answer | added | Dave Benson | timeline score: 12 | |
| Jun 10 at 21:23 | comment | added | GH from MO | @PeterHumphries This also shows that the LHS is smaller than $\sqrt{\pi}$. I suggest that you turn your comment into an answer so that this question can be closed. | |
| Jun 10 at 21:21 | comment | added | Peter Humphries | The Poisson summation formula states that $$\sum_{n = -\infty}^{\infty} f(n) = \sum_{n = -\infty}^{\infty} \widehat{f}(n).$$ Take $f(y) = e^{-(y + 1/4)^2}$, so that $\widehat{f}(y) = \sqrt{\pi} e^{-\pi^2 y^2} e^{\pi i y}$, which implies that $$\sum_{n = -\infty}^{\infty} e^{-(n + 1/4)^2} = \sqrt{\pi} + 2\sqrt{\pi} \sum_{n = 1}^{\infty} (-1)^n e^{-\pi^2 n^2}.$$ | |
| Jun 10 at 21:16 | comment | added | Christian Remling | This might help (or not): en.wikipedia.org/wiki/Theta_function#Explicit_values | |
| Jun 10 at 21:08 | comment | added | TheSimpliFire | The equality does not hold in general. As @DaveBenson mentioned, the Jacobi theta function identities yield a small residual. Here is an example. | |
| Jun 10 at 20:49 | history | asked | dotdashdashdash | CC BY-SA 4.0 |