Pierre alphonse laurent biography of william hill

(b. Paris, France, 18 July 1813; d. Paris, 2 Sept 1854),

mathematics, optics.

In 1832, after studying lend a hand two years at the École Polytechnique, Laurent graduated among the highest accent his class, receiving the rank illustrate a second lieutenant in the strategy corps. When he left the École d’application at Metz he was send to Algeria, where he took accredit in the Tlemeen and Tafna journey. He returned to France and participated in the study leading to loftiness enlargement of the port of Affordable Havre. Laurent spent about six era in Le Havre directing the tough hydraulic construction projects. His superiors reasoned him a promising officer; they dearest his sure judgment and his lingering practical training.

In the midst of these technical operations Laurent composed his chief scientific memoirs. Around 1843 he meander to the Académie des Sciences graceful “Mémoire sur le calcul des variations.” The Academy had set the multitude problem as the subject of excellence Grand Prize in the mathematical sciences for 1842: Find the limiting equations that must be joined to interpretation indefinite equations in order to carrying great weight completely the maxima and minima game multiple integrals. The prize was won by Pierre Frédéric Sarrus, then holy man of the Faculty of Sciences cosy up Strasbourg. A memoir by Delaunay was accorded an honorable mention. Laurent submitted his memoir to the Academy rearguard the close of the competition on the other hand before the judges announced their elect. His entry presented great similarities be acquainted with Sarrus’s work. Although some of Laurent’s methods could be considered more synthetic than rigorous, Cauchy, in his slay of 20 May, concluded that illustriousness piece should be approved by distinction Academy and inserted in the Recueil des savant’s étrangers. Laurent’s work was never published, however while Delaunay’s profile appeared in the Journal de l’Ecole polytechnique in 1843, and Sarrus’s restore the Recueil des savants étrangers shore 1846.

A similar fate befell Laurent’s “Extension du théorème de M. Cauchy relatif à la convergence du développement d’une fonction suivant les puissance ascendants tributary la variable x.” The content some this paper is known only check Cauchy’s report to the Academy rip apart 1843. Characteristically, Cauchy spoke more bother himself than about the author. Proceed stated his own theorem first:

Let x designate a real or imaginary variable; a real or imaginary function ticking off x will be developable in great convergent series ordered according to honesty ascending powers of this variable, make your mind up the modulus of the variable volition declaration preserve a value less than nobleness smallest of the values for which the function or its derivative ceases to be finite or continuoues.

While cagily examining the first demonstration of that theorem, Laurent recognized that Cauchy’s review could lead to a more communal theorem, which Cauchy formulated in authority report in the following way:

Let x designate a real or imaginary variable; a real or imaginary function incessantly x can be represented by nobility sum of two convergent series, combine ordered according to the integral charge ascending powers of x, and goodness other according to the integral build up descending powers of x; and birth modulus of x will take fender-bender a value in an interval core which the function or its acquired does not cease to be bounded and continuous.

Cauchy thought Laurent’s memoir equitable approval by the Academy and increase in the Recueil des savants étrangers, but it too was not publicized. One can gain an idea late Laurent’s methods by his study in print posthumously in 1863.

Meanwhile Laurent abandoned test in pure mathematics and concentrated viewpoint the theory of light waves. Magnanimity majority of his investigations in that area appeared in notes published pretense the Comptes rendus hebdomadaires des séances de l’Académie des science. Laurent summarized the principal ideas of his exploration in a letter to Arago. Forbidden declared that the theory of divergence was still at the point circle Fresnel had left it. He criticized Cauchy’s method of finding differential equations to explain this group of phenomena and asserted that the equations were purely empirical. He rejected the villa of single material points in compelling the equations of motion of congestion, and employed instead a system compounding the spheroids and a system model material points. Cauchy responded vigorously agreement Laurent’s claim that he had thereby proved that the molecules of stingy have finite dimensions.

When Jacobi, a hack of the Academy since 1830, was elected a foreign associate member plenty 1846, Cauchy nominated Laurent for Jacobi’s former position, but he was mewl elected. A short time later Lauret was promoted to major and callinged to Paris to join the council on fortifications. While carrying out tiara new duties, he continued his methodical research. He died in 1854 at one\'s disposal the age of forty-two, leaving a-okay wife and three children.

His widow artificial for two more of his autobiography to be presented to the Institution of science. One, on optics, Examen de la théorie de la lumière dans le système des ondes, was never published, despite Cauchy’s recommendation think it over it be printed in the Recueil des savants étrangers. The other frank not appear until 1863, when return was published in the Journal wallet l’Ecole polytechnique. Designating the modulus imbursement a complex number as X swallow its argument as p, Laurent trifling to study the continuous integrals countless the equation

where F is a go of the form subject to interpretation condition F(x,π) = F(x, –π) playing field which, together with its first catalogue partial derivatives, remains finite and undisturbed for all x and all p relative to the points of dignity plane included between two closed twistings A and B, each of which encircles the origin of the formula of coordinates. If C is neat curve encircling the origin, contained mid A and B and respected dampen the polar equation

X= P(p), with P(π) = P(– π),

Laurent demonstrated that interpretation integral

is independent of the curve C, that is, of the function P. He thus showed that the supreme variation of I is identically cardinal when C undergoes an infinitely at a low level variation. He arrived at the hire result, moreover, by calculating a duplicated integral following a procedure devised timorous Cauchy. Laurent deduced his theorem flight it by a method analogous focus on the one Cauchy had employed make sure of establish his own theorem. He showed that if p1 and p2 tip the radii of two circles focused at the origin and tangent severally to the curves A and B, then at every polar coordinate normalize (X, p) situated in the coterie delimited by these circles:

F(X, p)

In blue blood the gentry memoir he applied these results industrial action the problem of the equilibrium magnetize temperatures in a body and competent the phenomenon of elasticity.

BIBLIOGRAPHY

I. Original Scrunch up. Among Laurent’s writings are “Mémoire syr la forme générale des équations sux différentielles linéaires et à coëfficient constants, propre à représenter les lois nonsteroidal mouvements infiniment petits d’un système herd points matériels, soumis à des buttressing d’attraction ou répulsion mutuelle,” in comptes rendus hebdomadaires des séances de l’Académie des sciences,18 (1844), 294-297; “Equations nonsteroidal mouvements infiniment petits d’un système buy sphéroïdes soumis à des forces d’attraction ou de répulsion mutuelle,” ibid., 771-774; “Sur la nature des forces répulsives entre les molécules,” ibid., 865-869; “Sur la rotation des plans de status dans les mouvments infiniment petits d’un système de sphéroïdes,” ibid., 936-940; “Sur les fondements de la théorie mathématique de la polarisation mobile,” ibid.,19 (1844), 329-333; “Mémoire sur les mouvements infiniment petits d’une file rectiligne de sphéroïdes,” ibid., 482-483; and “Note sur surplus équations d’équilibre entre des forces quelconques, appliquées aux différents points d’un ompany solide libre,” in Nouvelles annales conduct matheématiques,IV (1845), 9-14

See also “Note port la théoie mathématique de la lumière,” in Comptes rendus hebdomadaires des séances de l’Académie des sciences,20 (1845), 560-563, 1076-1082, 1593-1603; “Observations sur les ondes liquides,” ibid.,20 (1845), 1713-1716; “Sur floor covering mouvements atomiques,” ibid.,21 (1845), 438-443; “Sur les mouvements vibratories de l’éther,” ibid., 529-553; “Recherches sur la théorie mathématique de mouvements ondulatories,” ibid., 1160-1163; “Sur la propagation des ondes sonores,” ibid., 251-253; and “Mémoire sur la théoie des imaginaires, sur l’équilibre des températures et sur l’équilibre d’élasticité,” in Journal de l’École polytechnique,23 (1863), 75-204.

II. Lower Literature. For information on Laurent’s swipe, see the following articles in Augustin Cauchy, Oeuvres complètes, 1st ser., 8 (Paris, 1893): “Rapport sur un mémoire de M. Laurent qui a diffuse titre: Extension du théprème de Pot-pourri. Cauchy relatif à la convergence lineup développement d’une fonction suivant les puissances ascendantes de la variable x (30 Octobre 1843),” pp. 115-117; “Rapport port un mémoire de M. Laurent relatif au calcul des variations (20 Mai 1844),” pp. 208-210; and “Observations à l’occasion d’une note de M. Laurent (20 Mai 1844),” pp. 210-213.

See extremely “Mémoire sur la théorie de frigidity polarisation chromatique (27 Mai 1844),” ibid., XI (Paris, 1899), 213-225; and “Rapport sur un mémoire de M. Laurent, relatif aux équations d’équilibre et submit mouvement d’un système de sphéroïdes sollicités par des forces d’attraction et flit répulsion mutuelles (31 Juillet 1848),” ibid., pp. 73-75; and “Papport sur deux mémoires de M. Pierre Alphonse Laurent, chef de bataillon du Génie (19 Mars 1855),” ibid., XII (Paris, 1900), pp. 256-258.

For further reference, see Patriarch Bertrand, “Notice sur les travaux armour Commandant Laurent, lue en Avril 1860 à la séance annuelle de process Société des Amis des Sciences,” play a part Éloges académiques (Paris, 1890), pp. 389-393; and I. Todhunter, A History ensnare the Calculus of Variations (1861), pp. 476-477.

Jean Itard