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I was thinking about constructing functions that have arbitrary values at some integers. One way to do that is by taking the functions $$ f_n = \sum_{0 \leq i \leq m, i \not = n} \frac{x - i}{n-i} $$ This is $ 1 $ at $ x = n $ and $ 0 $ at all other integers $ 1 \leq x \leq m $. One can add these together with the right coefficients to get a function that takes some chosen values at all integers between $ 0 $ and $ m $.

Then I wondered about the infinite case. Let's have a look around $ 0 $: $$ f(x) = \prod_{i=1}^\infty \left(1 - \frac{x^2}{i^2}\right). $$ This seems to converge to zero at all nonzero integers, and to converge to a nonzero value everywhere else. However, I am unable to discern anything more about its behaviour. Is this function known under some name? Do we know anything about what it does? Does it show up in other areas of math or physics?

Tempestas Ludi
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In fact, this is very similar to Euler's representation of $\sin{x}$ as an infinite product:

$$\sin(x) = x\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right)$$

Here's a link to the wikipedia page if it helps. You can obtain the value of $f(x)$ using this formula

Luca Armstrong
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