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If you want to calculate the Taylor expansion of a function, you only need to know the derivatives of the function at the point of expansion.

Is there a similar algorithmic approach that can be implemented, from which the Weierstrass factorization of the function can be obtained, provided that the zeros are known?

Is there a general approach available at all? How can you find out the exponentials that are required for convergence?

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  • $\begingroup$ As explained on the Wikipedia page about the product expression, we need to know some thing about the rate at which $|a_n|$ grow. Once we know that an explicit formula is given on that page. The expression, unlike the Taylor expansion, is not unique. $\endgroup$
    – Kapil
    Commented Nov 19, 2021 at 17:46
  • $\begingroup$ If zeros "are known" then their exponent of convergence is also "known" :-) Speaking of the exponential factor, its coefficients are simply related to the Taylor series at $0$. So if the genus is finite, and you know zeros, there is no problem. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 19, 2021 at 18:48

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