<s>[INST] Sine-related infinite products How do you find the infinite product for functions which share the same roots?
$$\frac{\sin(\pi x)}{\pi x}=\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{4\pi^2}\right)\left(1-\frac{x^2}{9\pi^2}\right)\dotsb .$$
However, this function has the exact same roots, but clearly doesn't have the same infinite product:
$$\frac{(3+\cos x)\sin(\pi x)}{\pi x}.$$
Any ideas? Thanks in advance. :)
 [/INST] They don't have the same roots: $\cos{z}+3=0$ has complex solutions: suppose $x,y$ are real. Then
$$ \cos{(x+iy)} = \cos{x}\cosh{y}+i\sin{x}\sinh{y}, $$
so if this is $3$, we need $\sin{x}=0$ and $\cosh{y} = 3/\cos{x}$. $\sin{x}=0 \implies x = n\pi$ for some integer $n$, so $\cos{x}=(-1)^n$. $\cosh{y}$ is positive, so $n$ is even, and $x=2k\pi$ for some integer $k$. $\arg\cosh{3} = \log{(3+\sqrt{3^2-1})} = \log{(3+2\sqrt{2})} $, so $\cos{z}+3$ is zero when
$$ z = 2k\pi + i\log{(3+2\sqrt{2})}. $$
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