---
tag: X9SRCEENV3
kind: concept
name: Laguerre–Pólya class
msc: ["30D15", "30C15", "26C10"]
sources:
  - url: https://en.wikipedia.org/wiki/Laguerre%E2%80%93P%C3%B3lya_class
same_as:
  wikidata: Q6472739
  wikipedia: Laguerre–Pólya class
---

## Summary

The Laguerre–Pólya class is the class of entire functions that are limits, uniformly on compact sets, of real polynomials with only real zeros.
Its members are described by an explicit Hadamard-type product with real zeros and a Gaussian factor.
The class is closed under differentiation, so it is the natural home of real-rootedness for entire functions.
It appears in the study of the zeros of the Riemann zeta function and of special functions.

## Explanations

### L1 human

A real polynomial whose roots are all real has a graph that meets the real axis at each of its roots.
The functions in the Laguerre–Pólya class are the entire functions that are limits of such polynomials.
Examples are $\sin z$, $\cos z$ and $e^{-z^2}$, the last being a limit of $(1 - z^2/n)^n$.
The function $e^{z^2}$ is not in the class, even though it has no zeros at all.