Infinite products
Definition. An infinite product \(\prod_n(1+a_n)\) converges to a nonzero limit if the partial products converge and the limit is nonzero. Absolute convergence of \(\sum |a_n|\) is a basic sufficient condition.
Canonical factors
Definition. Weierstrass factors \(E_p(w)=(1-w)\exp(w+w^2/2+\cdots+w^p/p)\) make products converge while preserving a prescribed zero at \(w=1\). The exponential correction controls behavior away from the zero.
Weierstrass factorization theorem
Theorem. Given a discrete sequence \((a_n)\) in \(\mathbb C\) with no finite accumulation point and assigned positive multiplicities, there exists an entire function whose zeros are exactly those points with those multiplicities.
Proof strategy
Proof. Choose sufficiently high canonical factors so that the logarithms of the factors converge normally on compact sets. Normal convergence makes the product holomorphic, and the zero set is controlled factor by factor.
Sine product
Example. The classical product \(\sin\pi z=\pi z\prod_{n=1}^{\infty}(1-z^2/n^2)\) realizes the integer zeros of sine and illustrates how zero sets determine entire functions up to nonvanishing entire factors.
Nonuniqueness
Remark. If two entire functions have the same zeros with the same multiplicities, their quotient is a nowhere-zero entire function, hence locally an exponential. Zero data alone does not determine the function uniquely.
Convergence of holomorphic products
Theorem. If holomorphic \(u_n\) satisfy \[ \sum_n\sup_K|u_n|<\infty \] for every compact \(K\Subset\Omega\), then \(\prod_n(1+u_n)\) converges locally uniformly. If no factor vanishes and the limit is not forced to zero by finitely many factors, the limit is holomorphic and nonvanishing.
Proof. On a tail where \(|u_n|\le1/2\), choose the principal logarithm and use \(|\log(1+u_n)|\le2|u_n|\). The logarithm series converges normally, so its exponential gives a nonzero holomorphic product. Multiplying finitely many initial factors completes the result.
Canonical factors
The elementary factors \[ E_p(w)=(1-w)\exp\left(w+\frac{w^2}{2}+\cdots+\frac{w^p}{p}\right) \] satisfy \[ \log E_p(w)=O(w^{p+1})\quad(w\to0). \] Given nonzero points \(a_n\to\infty\), choose integers \(p_n\) so that \(\sum|z/a_n|^{p_n+1}\) converges locally uniformly. Then \[ \prod_nE_{p_n}(z/a_n) \] is entire with zeros exactly at \(a_n\), with prescribed multiplicities after repetition or powers.
Weierstrass factorization
Theorem. If an entire function \(f\) has a zero of order \(m\) at \(0\) and nonzero zeros \(a_n\), repeated by multiplicity, then \[ f(z)=z^me^{g(z)} \prod_nE_{p_n}(z/a_n) \] for an entire \(g\) and suitable \(p_n\).
Proof. Construct the canonical product \(P\) with the same zeros. Then \(f/(z^mP)\) is entire and nonvanishing. Since \(\mathbb C\) is simply connected, it has an entire logarithm \(g\), giving the factorization.
The sine product
The zeros of \(\sin\pi z\) are the integers, and pairing opposite zeros gives the genus-zero product \[ \frac{\sin\pi z}{\pi z} =\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right). \] Logarithmic differentiation yields \[ \pi\cot(\pi z) =\frac1z+\sum_{n=1}^{\infty} \left(\frac1{z-n}+\frac1{z+n}\right), \] with locally uniform interpretation away from the poles.
Factorization is nonunique: multiplying by \(e^{h(z)}\) changes no zeros. Growth restrictions determine how small the canonical genus and exponential factor can be.