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If we allow some $g(x) = \frac{\sin(x)}{x}$ such that $g(0) = 1$ and also some $f(x)$ where:

$$f(x) = \prod_{k=1}^{\infty} 1 - \frac{x^2}{\pi^2k^2}$$

Then it is said f and g are equal. But my question is, what are the specifics besides the fact that:

Both have the same value at $x=0$. Both have the same roots, and an infinite number of them of course.

I feel like those two criteria definitely don’t justify equality, so I would like to know what legitimately justifies it.

Martin Sleziak
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MichaelCatliMath
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  • Concretely, yes. But, I want to understand why people insist that demonstrating the fact that they have the same roots and same value at x=0 justifies the equality, even though the legitimate proof is much more complicated. Is there something I’m missing? Thanks though. – MichaelCatliMath May 09 '20 at 05:05
  • @MichaelCatliMath: Having the same roots does not justify equality (you can multiply $g$ by an arbitrary non-zero function $e^h$. Without a concrete reference where “it is said that f and g are equal” or “why people insist that demonstrating the fact that they have the same roots and same value at x=0 justifies the equality” it is difficult to say more. – Martin R May 09 '20 at 05:39

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