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I have trouble wrapping my head around this equality. I am looking for an intuitive (perhaps geometrical if possible) insight as opposed to an abstract one. I can follow proofs that compute the two sides of the equation. However, surely:

$$\int^{\infty}_{- \infty} \frac{\sin(x)}{x} dx = \sum^{\infty}_{n = - \infty} \frac{\sin(n)}{n} $$

implies that when taking the Riemann summation as an approximation of the integral (with vanishingly small error) that the rectangles everywhere except at the integer values on the number line perfectly cancel each other. However just looking at the graph for $\frac{\sin(x)}{x}$ :

graph of sinc(x)

I don't see how this can be true since there is clearly an overall positive area. I suppose with this particular function I just don't see how Riemann summation can be used to approximate it.

hahahasan
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  • Use \ before sin to get sin as $\sin$ instead of $sin$. – 19aksh Aug 07 '19 at 17:11
  • Almost seems like they cancel everywhere except around zero where the maximum occurs, I'm guessing that that area is equal to $\pi$? –  Aug 07 '19 at 22:25
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    Note that $\sum_{n=-\infty}^\infty \sin(a n)/n = \pi$ for all $a\in(0,2\pi)$. So there isn't anything special about the $a = 1$ case. – eyeballfrog Aug 07 '19 at 22:40
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    Exactly what does "when taking the Riemann summation as an approximation of the integral (with vanishingly small error) that the rectangles everywhere except at the integer values on the number line perfectly cancel each other" mean? – David C. Ullrich Aug 08 '19 at 22:10
  • @DavidC.Ullrich I mean to split the area enclosed by the curve into an arbitrary number of rectangles. As the number of rectangles tends to infinity, from what I understand, the area they encompass tends to the exact value of the integral of the function. However, with the sinc function, the equality I mentioned above appears to imply that just the contributions from rectangles located at integer values are sufficient to give the final result. This would then imply that contributions from rectangles not at integer values must sum to 0 (or "cancel each other out" as I worded it before). – hahahasan Aug 09 '19 at 09:57

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