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(changed the title to make it, I hope, more clear)
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Qfwfq
Qfwfq
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An Is this entire function which is a square?

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LSpice
LSpice
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Let $f$ be the entire function on $\mathbb C$ defined by $$ f(z)=\frac{z-\sin z}{z}. \tag{1}$$$$ f(z)=\frac{z-\sin z}{z}. \tag{1}\label{1}$$ It is easy to see that $f$ is positive on $\mathbb R^*$ and has a zero of order 2 at 0. Does there exist an entire function $g$ such that $f=g^2$?

Let $f$ be the entire function on $\mathbb C$ defined by $$ f(z)=\frac{z-\sin z}{z}. \tag{1}$$ It is easy to see that $f$ is positive on $\mathbb R^*$ and has a zero of order 2 at 0. Does there exist an entire function $g$ such that $f=g^2$?

Let $f$ be the entire function on $\mathbb C$ defined by $$ f(z)=\frac{z-\sin z}{z}. \tag{1}\label{1}$$ It is easy to see that $f$ is positive on $\mathbb R^*$ and has a zero of order 2 at 0. Does there exist an entire function $g$ such that $f=g^2$?

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Bazin
Bazin
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An entire function which is a square?

Let $f$ be the entire function on $\mathbb C$ defined by $$ f(z)=\frac{z-\sin z}{z}. \tag{1}$$ It is easy to see that $f$ is positive on $\mathbb R^*$ and has a zero of order 2 at 0. Does there exist an entire function $g$ such that $f=g^2$?