After about ten years in Cheltenham, England, the family returned to France. Pierre Alphonse entered the École Polytechnique in Paris in 1830, in the year of the July Revolution which forced King Charles X from the throne and led to the rule of Louis-Philippe. Laurent graduated from the École Polytechnique in 1832, being one of the best students in his year, and entered the engineering corps as second lieutenant. He then attended the École d'Application at Metz until he was sent to Algeria.
Up until 1830 Algeria was an autonomous province of the Ottoman Empire. France had in the couple of years up to 1830 tried to control the country using various political moves and a naval blockade. However they decided early in 1830 to invade and French troops landed in Algeria on 5 July 1830. They won quick victories since the Algerian people detested their rulers and there was no united forces to oppose the French invasion. However the July Revolution in France led to a period when desire for foreign conquests vanished but they retained the foothold in Algeria that they had established. Two leaders, Ahmad Bey and Abdelkader, established themselves rallying support against the French invaders. The French decided to send forces against these resistance leaders and Laurent took part in two of these expeditions. One was to Tlemcen, in northwestern Algeria near the Moroccan border, the other was the Tafna expedition against Abdelkader which led to the French signing the Treaty of Tafna in 1837.
On 11 February 1840, Laurent's mother, who had been living at 14 rue du Regard, Paris, died. Her name appears in the French version of Eléonore-Françoise Chesser on her death certificate. Laurent's father outlived his wife by about a year and Pierre Michel Laurent died in Paris on 25 February 1841. Laurent had returned to France from Algeria around 1840 and spent six years directing operations for the enlargement of the port of Le Havre on the English Channel coast. Rouen had been the main French port up to the nineteenth century but the hydraulic construction projects on which Laurent worked in Le Havre turned it into France's main seaport. It is clear that Laurent was a good engineer, putting his deep theoretical knowledge to good practical use. Jean Itard writes :-
His superiors considered him a promising officer; they admired his sure judgement and his extensive practical training.In Hautmont, in north west France near the Belgium border, on 18 October 1841, Laurent married Palmyre Angélique Bernardine Depreux (born 1821), the daughter of Jean Baptiste Depreux and Adrienne Leroy. Jean Baptiste Depreux was a farmer and the owner of hotels in Hautmont and Briastre, about 45 km west of Hautmont. Laurent and his wife had three sons: Pierre Georges Laurent (1843-1914), Charles Pierre Laurent (1845-1901), and Émile Laurent. Pierre Georges, born in the Military District of Le Havre, was educated in the schools in Avesnes and Douai, studied in Paris at the École Polytechnique from 1861 to 1863 and became a military engineer.
It was while Laurent was working on the construction project at Le Havre that he began to write his first mathematical papers. He submitted a memoir for the Grand Prix of the Academy of Sciences of 1842, unfortunately after the final date for submission in 1843. The topic proposed was (see for example ):-
Find the limiting equations that must be joined to the indefinite equations in order to determine completely the maxima and minima of multiple integrals.Cauchy reported on Laurent's entry Mémoire sur le calcul des variations , which contains the Laurent series for a complex function, on 20 May 1843. Laurent's theorem for a complex function generalises Taylor's theorem :-
In modern terminology the theorem defined in an annulus (a ring bounded by two concentric circles) and on its boundary can be developed in a general, power series in increasing and decreasing powers of the variable. There exist functions, such as certain Bessel functions, which cannot be expanded into Taylor series, and Laurent's theorem would be applicable to these functions.Being late, the memoir was never seriously considered for the Grand Prix, which was won by Pierre Frédéric Sarrus (1798-1861), a mathematician working at Strasbourg, with Charles Delaunay's entry receiving an honourable mention, but Cauchy and Liouville were asked to review Laurent's paper and consider it for publication. They proposed that Laurent's memoir should be approved and published in the Recueil des savants étrangers . The Academy of Sciences published the entries of Sarrus and Delaunay but they ignored Cauchy and Liouville's recommendation concerning Laurent and his memoir was not published. Cauchy, however, in rather typical fashion, had put the Academy of Sciences in a difficult position by claiming that he had priority over Laurent :-
... at the next meeting of the Academy Cauchy claimed that his papers [written in 1837] gave him priority over Laurent in this matter. Only on 30 October did he present his (and Liouville's) report on Laurent's memoir to the Academy. Cauchy then went on to argue that the main theorem in Laurent's paper could be deduced from a theorem he (Cauchy) had stated in 1840. To support his claims Cauchy added to the report a note of his own to show that "the easiest way" to obtain Laurent's theorem was by reformulating the results in his 1840 paper. ... Sadly, Cauchy's less than generous response, coupled with his extraordinary rapidity of thought, was to prove typical of the way Cauchy would take the work of others as an opportunity to promote his own.Given Cauchy's attempt to claim the result of Laurent's paper it is not surprising the Academy of Sciences chose not to publish it but, despite this, the theorem and series are named for Laurent. However, even if we ignore Cauchy's attempt to claim the theorem, Laurent was not the first to prove the result we today call Laurent's theorem. In fact in 1841, while at Münster, Karl Weierstrass proved this theorem in the paper Darstellung einer analytischen Functionen einer complexen Veränderlichen, deren absoluter Betrag zwischen zwei gegeben Grenzen leigt . However, he did not try to publish the paper at this time and it only appeared as a published paper in Weierstrass's collected works of 1894.
A second paper by Laurent, Extension du théorème de M Cauchy relatif à la convergence du développement d'une fonction suivant les puissances ascendantes de la variable , submitted to the Academy of Sciences around the same time was also considered by Cauchy. This paper presented an extension of one of Cauchy's theorems and again Cauchy proposed that Laurent's memoir should be approved and published in the Recueil des savants étrangers . Cauchy's report quotes first his own theorem that Laurent had extended:-
Let x designate a real or imaginary variable; a real or imaginary function of x will be developable in a convergent series ordered according to the ascending powers of this variable, while the modulus of the variable will preserve a value less than the smallest of the values for which the function or its derivative ceases to be finite or continuous.Cauchy then quoted Laurent's extension:-
Let x designate a real or imaginary variable; a real or imaginary function of x can be represented by the sum of two convergent series, one ordered according to the integral and ascending powers of x, and the other according to the integral and descending powers of x; and the modulus of x will take on a value in an interval within which the function or its derivative does not cease to be finite and continuous.Again the Academy of Sciences decided not to publish the work and it has been lost and is now only known through Cauchy's report from which we quoted above.
After this Laurent, disappointed that his papers had not been accepted for publication, decided that he had better change the topic of his research. He began to study the theory of light waves, in particular examining the theory of polarisation. He published a number of papers on the topic, for example a number of notes entitled Note sur la théorie mathématique de la lumière and papers such as Observations sur les ondes liquides, et remarques relative aux assimilations que l'on a faites de ces ondes aux ondulations lumineuses . He was critical of some of Cauchy's methods of attacking these problems which, as one might expect, produced a strong response from Cauchy. He also published on sound waves in papers such as Sur la propagation des ondes sonores .
Cauchy proposed him for a vacant position in the Academy of Sciences in 1846. The position of corresponding member had become vacant since Carl Jacobi, who had held that position, was promoted to foreign associate member. However he was not elected and soon after this he was promoted to major and sent to Paris to become a member of a committee set up to look at the problems of fortification. He continued to undertake research into applied mathematical topics.
Laurent died at the young age of 41 probably as a consequence of overwork over many years when he was carrying out a demanding job yet producing a large number of mathematical papers at the same time. After his death his widow arranged for two more of his memoirs to be presented to the Academy of Sciences. Examen de la théorie de la lumière dans le système des ondes was considered by Cauchy who proposed that Laurent's memoir should be approved and published in the Recueil des savants étrangers but again it was never published. The second, Mémoire sur la théorie des imaginaires, sur l'équilibre des températures et sur l'équilibre d'élasticité , was published in Journal de l'École Polytechnique but it did not appear until 1863.
Article by: J J O'Connor and E F Robertson