Infinite Products, Factorization, and Mittag-Leffler

The residue calculus takes functions apart at their singularities; this page builds them up from prescribed zeros and poles. Infinite products realize an entire function with a given zero set (Weierstrass), order and genus pin down how fast it may grow (Hadamard), Jensen's formula ties growth to the density of zeros, and the Mittag-Leffler theorem prescribes principal parts. This structural theory underwrites the Gamma and zeta functions. It builds on cauchy-integral.md (Liouville, entire functions) and laurent-residues.md.

Infinite products

An infinite product ${\prod }_n(1+a_n)$ converges (to a nonzero limit) iff ${\sum }_n|a_n|<\infty$; then it vanishes exactly when some factor does. The utility is that a product can carry a prescribed zero at each $z=z_n$ — one factor vanishing there — whereas a sum cannot. To keep the product convergent when the $z_n$ march to infinity, Weierstrass damps each factor with an exponential.

Definition (elementary factors). The Weierstrass elementary factors are $E_0(z)=1-z$ and, for $p\ge 1$, $$E_p(z)=(1-z)\exp (z+\frac{z^2}{2}+\cdots +\frac{z^p}{p}).$$ Each has a single zero at $z=1$ and satisfies $|1-E_p(z)|\le |z{|}^{p+1}$ for $|z|\le 1$, so higher $p$ makes $E_p$ closer to $1$ near the origin.

The Weierstrass factorization theorem

Weierstrass factorization theorem. Given any sequence $z_n\to \infty$ (with multiplicities) there is an entire function with exactly those zeros, namely $$f(z)=z^m{e}^{g(z)}\prod _n{E}_{p_n}(z/z_n),$$ with $g$ entire, $m$ the order of the zero at $0$, and $p_n$ chosen so the product converges. Every entire function factors this way.

This is the transcendental analogue of "a polynomial is a product over its roots": zeros no longer determine $f$ (the factor $e^g$ is free), but they determine it up to a nowhere-zero entire function. The theorem globalizes to any domain (prescribing zeros on a discrete set) and, together with Mittag-Leffler below, shows every meromorphic function is a ratio of two entire functions.

Jensen's formula

The bridge between the growth of $f$ and the density of its zeros:

Jensen's formula. If $f$ is holomorphic on $|z|\le R$ with $f(0)\ne 0$ and zeros $a_1,\dots ,a_N$ (with multiplicity) inside, $$\log |f(0)|=\sum _{k=1}^N\log \frac{|a_k|}{R}>+\frac{1}{2\pi }{\int }_0^{2\pi }\log |f(R{e}^{i\theta })|d\theta .$$

Reading the left side as fixed, the boundary average of $\log |f|$ grows precisely as the zeros accumulate — the quantitative link that makes "order" below meaningful and the starting point of Nevanlinna theory (picard.md).

Order, genus, and Hadamard factorization

The order of an entire function measures its maximal growth: $$\rho =\underset{r\to \infty }{\mathrm{limsup}}\frac{\log \log {\max }_{|z|=r}|f(z)|}{\log r}.$$ Jensen's formula bounds the zero-counting exponent by $\rho$, forcing the exponentials $e^g$ and $E_p$ in the factorization to be polynomial-controlled:

Hadamard factorization theorem. An entire function of finite order $\rho$ factors as $$f(z)=z^m{e}^{g(z)}\prod _n{E}_p(z/z_n),$$ with $g$ a polynomial of degree $\le \rho$ and $p\le \rho$ (the genus).

So finite-order entire functions are as rigid as polynomials up to a controlled exponential — e.g. $\sin \pi z=\pi z{\prod }_{n\ge 1}(1-z^2/n^2)$ (order $1$), the identity behind the reflection formula and $\sum 1/n^2={\pi }^2/6$.

The Mittag-Leffler theorem

The dual construction prescribes poles instead of zeros:

Mittag-Leffler theorem. Given points $z_n\to \infty$ and principal parts $p_n(z)={\sum }_{k=1}^{m_n}{c}_{n,k}(z-z_n{)}^{-k}$, there is a meromorphic function on $\mathbb{C}$ with exactly those poles and principal parts, obtained by summing the $p_n$ with convergence-inducing polynomial subtractions: $$f(z)=\sum _n(p_n(z)-q_n(z)),q_n\text{a Taylor polynomial of}{p}_n.$$

The archetype is the partial-fraction expansion $\pi \cot \pi z=\frac{1}{z}+{\sum }_{n\ne 0}(\frac{1}{z-n}+\frac{1}{n})$, the same kernel used to sum series by residues. Weierstrass (zeros) and Mittag-Leffler (poles) are the two halves of the "build a function with prescribed local data" program; on a general domain they are the analytic content of the Cousin problems and, ultimately, of sheaf cohomology (analytic-continuation.md).

References

• Ahlfors, Complex Analysis, Ch. 5 §2 and Ch. 5 §3.
• Stein & Shakarchi, Complex Analysis, Ch. 5 (products, Hadamard) and Ch. 5 §3.
• Conway, Functions of One Complex Variable, Ch. VII.