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Let $f$ be an entire function of order $ρ<∞$. <\infty$. Assume that$f$does not vanish identically on$ℂ$. \mathbb{C}$. Then, we know that $f$ has a Hadamard's product formula

$f(s)=exp(g(s))s^{r}∏$ f(s) =e^{g(s)}s^{r}\prod _ {k=1}^{∞}(((sk=1}^{\infty}\frac{s _ {k}-s)/(s_{k})))exp( (s/(s_{k})))$k}-s}{s _ {k}} e^{s/s _ k}$$the integer$r$is the order of vanishing of$f$at$s=0$, the$s_{k}$are the other zeros of$f$listed with multiplicity,$g$is a polynomial of degree at most$ρ$, and the product converges uniformly in bounded subsets of$ℂ$. My question is how I can deduce directely a Hadamard's product formula for the derivative$f′$f^′$ from the one of the function $f$.

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# Hadamard's product formula for the derivative

Let $f$ be an entire function of order $ρ<∞$. Assume that $f$ does not vanish identically on $ℂ$. Then, we know that $f$ has a Hadamard's product formula

$f(s)=exp(g(s))s^{r}∏{k=1}^{∞}(((s{k}-s)/(s_{k})))exp( (s/(s_{k})))$

the integer $r$ is the order of vanishing of $f$ at $s=0$, the $s_{k}$ are the other zeros of $f$ listed with multiplicity, $g$ is a polynomial of degree at most $ρ$, and the product converges uniformly in bounded subsets of $ℂ$. My question is how I can deduce directely a Hadamard's product formula for the derivative $f′$ from the one of the function $f$.