We know (eq. 23) that the sine can be represented by an infinite product: $$\sin(x) = x \prod_{n=1}^{\infty} \Big{(} 1 - \frac{x^{2}}{n^{2}{\pi}^{2}} \Big{)} .$$ I was wondering whether the function $f$ such that $f(f(x)) = \sin(x)$ has an infinite product representation as well. The function $f$ is called the functional square root of the sine. Perhaps it can also be obtained by means of the Weierstrass Factorization Theorem?
2026-03-26 21:34:57.1774560897
Does the functional square root of the sine function have an infinite product representation?
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