When infinity stopped being embarrassing: The doubly infinite series of Pierre Alphonse Laurent and the mathematical rehabilitation of singularities

Abstract

For the better part of a century, isolated singularities were treated as pathological obstructions requiring elaborate avoidance strategies. Pierre Alphonse Laurent (1813--1854), a French military engineer at Le Havre, ended this avoidance in 1843 by extending Cauchy's Taylor-type theorem to doubly connected (annular) domains, producing the doubly infinite power series that now bears his name. Negative-power terms in the expansion encode precise geometric information about the singularity rather than signaling a breakdown of the formalism. Laurent's contribution arrived through an unhappy institutional trajectory -- submitted after a prize deadline, subjected to a priority claim by Cauchy, and issued in full only posthumously in 1863 -- yet it became indispensable to every branch of mathematics and mathematical physics that touches on complex function theory. We reconstruct the mathematical problem Laurent solved, place it within Cauchy's analytic program of the 1830s--1840s, examine the institutional failure that prevented publication, document the independent parallel proof by Weierstrass (1841, published 1894), and trace the series' absorption into the standard toolkit via Briot and Bouquet and the residue calculus. Drawing on Laurent's 1843 Comptes rendus notice, Cauchy's Academy report, Bertrand's memorial notice (1890), and the secondary literature (Neuenschwander 1978, 1981; Manning 1975; Bottazzini 1986; Gray 2015), we analyze the philosophical significance of the series, which we term ``exile mathematics'', and survey its reach into perturbation theory, number theory, probability, and quantum field theory. Readers familiar with the theorem but not its institutional history will find here a documented account of why a foundational result was withheld for two decades and how it nevertheless achieved canonical status.