# Siméon Denis Poisson

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Siméon Poisson
Siméon Denis Poisson (1781–1840)
Born21 June 1781
Died25 April 1840 (aged 58)
NationalityFrench
Alma mater École Polytechnique
Known for Poisson process
Poisson equation
Poisson kernel
Poisson distribution
Poisson bracket
Poisson algebra
Poisson regression
Poisson summation formula
Poisson's spot
Poisson's ratio
Poisson zeros

Euler–Poisson–Darboux equation
Scientific career
Fields Mathematics
Institutions École Polytechnique
Bureau des Longitudes
Faculté des sciences de Paris  [ fr ]
École de Saint-Cyr
Pierre-Simon Laplace
Doctoral students Michel Chasles
Joseph Liouville
Other notable students Nicolas Léonard Sadi Carnot
Peter Gustav Lejeune Dirichlet

Baron Siméon Denis Poisson FRS FRSE (French: ; 21 June 1781 25 April 1840) was a French mathematician, engineer, and physicist, who made several scientific advances.

Baron is a rank of nobility or title of honour, often hereditary. The female equivalent is baroness.

The President, Council and Fellows of the Royal Society of London for Improving Natural Knowledge, commonly known as the Royal Society, is a learned society. Founded on 28 November 1660, it was granted a royal charter by King Charles II as "The Royal Society". It is the oldest national scientific institution in the world. The society is the United Kingdom's and Commonwealth of Nations' Academy of Sciences and fulfils a number of roles: promoting science and its benefits, recognising excellence in science, supporting outstanding science, providing scientific advice for policy, fostering international and global co-operation, education and public engagement.

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

## Biography

Poisson was born in Pithiviers, Loiret district in France, the son of Siméon Poisson, an officer in the French army.

Pithiviers is a commune (municipality) in the Loiret department in north-central France. It is twinned with Ashby-de-la-Zouch in Leicestershire, England and Burglengenfeld in Bavaria, Germany.

Loiret is a department in the Centre-Val de Loire region of north-central France.

France, officially the French Republic, is a country whose territory consists of metropolitan France in Western Europe and several overseas regions and territories. The metropolitan area of France extends from the Mediterranean Sea to the English Channel and the North Sea, and from the Rhine to the Atlantic Ocean. It is bordered by Belgium, Luxembourg and Germany to the northeast, Switzerland and Italy to the east, and Andorra and Spain to the south. The overseas territories include French Guiana in South America and several islands in the Atlantic, Pacific and Indian oceans. The country's 18 integral regions span a combined area of 643,801 square kilometres (248,573 sq mi) and a total population of 67.0 million. France, a sovereign state, is a unitary semi-presidential republic with its capital in Paris, the country's largest city and main cultural and commercial centre. Other major urban areas include Lyon, Marseille, Toulouse, Bordeaux, Lille and Nice.

In 1798, he entered the École Polytechnique in Paris as first in his year, and immediately began to attract the notice of the professors of the school, who left him free to make his own decisions as to what he would study. In 1800, less than two years after his entry, he published two memoirs, one on Étienne Bézout's method of elimination, the other on the number of integrals of a finite difference equation. The latter was examined by Sylvestre-François Lacroix and Adrien-Marie Legendre, who recommended that it should be published in the Recueil des savants étrangers, an unprecedented honor for a youth of eighteen. This success at once procured entry for Poisson into scientific circles. Joseph Louis Lagrange, whose lectures on the theory of functions he attended at the École Polytechnique, recognized his talent early on, and became his friend. Meanwhile, Pierre-Simon Laplace, in whose footsteps Poisson followed, regarded him almost as his son. The rest of his career, till his death in Sceaux near Paris, was nearly occupied by the composition and publication of his many works and in fulfilling the duties of the numerous educational positions to which he was successively appointed. [1]

École polytechnique ; also known as EP or X), is a French public institution of higher education and research in Palaiseau, a suburb southwest of Paris. It is one of the most prestigious and selective French scientific and engineering schools, called grandes écoles in French. It is known for its ingénieur polytechnicien scientific degree program which is equivalent to both a bachelor and master of science. Its entrance exam, the X-ENS exam, is renowned for its selectivity with a little over 400 admitted students out of the 5,206 students enrolled in the preparatory programs for the French scientific and engineering schools taking the exam.

Paris is the capital and most populous city of France, with an area of 105 square kilometres and an official estimated population of 2,140,526 residents as of 1 January 2019. Since the 17th century, Paris has been one of Europe's major centres of finance, diplomacy, commerce, fashion, science, as well as the arts. The City of Paris is the centre and seat of government of the Île-de-France, or Paris Region, which has an estimated official 2019 population of 12,213,364, or about 18 percent of the population of France. The Paris Region had a GDP of €709 billion in 2017. According to the Economist Intelligence Unit Worldwide Cost of Living Survey in 2018, Paris was the second most expensive city in the world, after Singapore, and ahead of Zurich, Hong Kong, Oslo and Geneva. Another source ranked Paris as most expensive, on a par with Singapore and Hong Kong, in 2018. The city is a major railway, highway, and air-transport hub served by two international airports: Paris-Charles de Gaulle and Paris-Orly. Opened in 1900, the city's subway system, the Paris Métro, serves 5.23 million passengers daily, and is the second busiest metro system in Europe after Moscow Metro. Gare du Nord is the 24th busiest railway station in the world, but the first located outside Japan, with 262 million passengers in 2015.

Étienne Bézout was a French mathematician who was born in Nemours, Seine-et-Marne, France, and died in Avon, France.

Immediately after finishing his studies at the École Polytechnique, he was appointed répétiteur (teaching assistant) there, a position which he had occupied as an amateur while still a pupil in the school; for his schoolmates had made a custom of visiting him in his room after an unusually difficult lecture to hear him repeat and explain it. He was made deputy professor (professeur suppléant) in 1802, and, in 1806 full professor succeeding Jean Baptiste Joseph Fourier, whom Napoleon had sent to Grenoble. In 1808 he became astronomer to the Bureau des Longitudes; and when the Faculté des sciences de Paris  [ fr ] was instituted in 1809 he was appointed a professor of rational mechanics (professeur de mécanique rationelle). He went on to become a member of the Institute in 1812, examiner at the military school (École Militaire) at Saint-Cyr in 1815, graduation examiner at the École Polytechnique in 1816, councillor of the university in 1820, and geometer to the Bureau des Longitudes succeeding Pierre-Simon Laplace in 1827. [1]

A répétiteur is an accompanist, tutor or coach of ballet dancers or opera singers.

Napoléon Bonaparte was a French statesman and military leader of Italian descent who rose to prominence during the French Revolution and led several successful campaigns during the French Revolutionary Wars. He was Emperor of the French as Napoleon I from 1804 until 1814 and again briefly in 1815 during the Hundred Days. Napoleon dominated European and global affairs for more than a decade while leading France against a series of coalitions in the Napoleonic Wars. He won most of these wars and the vast majority of his battles, building a large empire that ruled over much of continental Europe before its final collapse in 1815. He is considered one of the greatest commanders in history, and his wars and campaigns are studied at military schools worldwide. Napoleon's political and cultural legacy has endured as one of the most celebrated and controversial leaders in human history.

Grenoble is a city in southeastern France, at the foot of the French Alps where the river Drac joins the Isère. Located in the Auvergne-Rhône-Alpes region, Grenoble is the capital of the department of Isère and is an important European scientific centre. The city advertises itself as the "Capital of the Alps", due to its size and its proximity to the mountains.

In 1817, he married Nancy de Bardi and with her, he had four children. His father, whose early experiences had led him to hate aristocrats, bred him in the stern creed of the First Republic. Throughout the Revolution, the Empire, and the following restoration, Poisson was not interested in politics, concentrating on mathematics. He was appointed to the dignity of baron in 1821; [1] but he neither took out the diploma nor used the title. In March 1818, he was elected a Fellow of the Royal Society, [2] in 1822 a Foreign Honorary Member of the American Academy of Arts and Sciences, [3] and in 1823 a foreign member of the Royal Swedish Academy of Sciences. The revolution of July 1830 threatened him with the loss of all his honours; but this disgrace to the government of Louis-Philippe was adroitly averted by François Jean Dominique Arago, who, while his "revocation" was being plotted by the council of ministers, procured him an invitation to dine at the Palais-Royal, where he was openly and effusively received by the citizen king, who "remembered" him. After this, of course, his degradation was impossible, and seven years later he was made a peer of France, not for political reasons, but as a representative of French science. [1]

Fellowship of the UK Royal Society is an award granted to individuals that the Royal Society of London judges to have made a 'substantial contribution to the improvement of natural knowledge, including mathematics, engineering science, and medical science'.

The American Academy of Arts and Sciences is one of the oldest learned societies in the United States. Founded in 1780, the Academy is dedicated to honoring excellence and leadership, working across disciplines and divides, and advancing the common good.

The Royal Swedish Academy of Sciences is one of the royal academies of Sweden. Founded on June 2, 1739, it is an independent, non-governmental scientific organization which takes special responsibility for promoting the natural sciences and mathematics and strengthen their influence in society, whilst endeavouring to promote the exchange of ideas between various disciplines.

As a teacher of mathematics Poisson is said to have been extraordinarily successful, as might have been expected from his early promise as a répétiteur at the École Polytechnique. As a scientific worker, his productivity has rarely if ever been equaled. Notwithstanding his many official duties, he found time to publish more than three hundred works, several of them extensive treatises, and many of them memoirs dealing with the most abstruse branches of pure mathematics, [1] applied mathematics, mathematical physics, and rational mechanics. (Arago attributed to him the quote, "Life is good for only two things: doing mathematics and teaching it." [4] )

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.

Mathematical physics refers to the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". It is a branch of applied mathematics, but deals with physical problems.

Dominique François Jean Arago, known simply as François Arago, was a French mathematician, physicist, astronomer, freemason, supporter of the carbonari and politician.

A list of Poisson's works, drawn up by himself, is given at the end of Arago's biography. All that is possible is a brief mention of the more important ones. It was in the application of mathematics to physics that his greatest services to science were performed. Perhaps the most original, and certainly the most permanent in their influence, were his memoirs on the theory of electricity and magnetism, which virtually created a new branch of mathematical physics. [1]

Next (or in the opinion of some, first) in importance stand the memoirs on celestial mechanics, in which he proved himself a worthy successor to Pierre-Simon Laplace. The most important of these are his memoirs Sur les inégalités séculaires des moyens mouvements des planètes, Sur la variation des constantes arbitraires dans les questions de mécanique, both published in the Journal of the École Polytechnique (1809); Sur la libration de la lune, in Connaissance des temps (1821), etc.; and Sur le mouvement de la terre autour de son centre de gravité, in Mémoires de l'Académie (1827), etc. In the first of these memoirs, Poisson discusses the famous question of the stability of the planetary orbits, which had already been settled by Lagrange to the first degree of approximation for the disturbing forces. Poisson showed that the result could be extended to a second approximation, and thus made an important advance in planetary theory. The memoir is remarkable inasmuch as it roused Lagrange, after an interval of inactivity, to compose in his old age one of the greatest of his memoirs, entitled Sur la théorie des variations des éléments des planètes, et en particulier des variations des grands axes de leurs orbites. So highly did he think of Poisson's memoir that he made a copy of it with his own hand, which was found among his papers after his death. Poisson made important contributions to the theory of attraction. [1]

His is one of the 72 names inscribed on the Eiffel Tower.

## Contributions

### Potential theory

Poisson's well-known generalization of Laplace's second order partial differential equation for potential

${\displaystyle \nabla ^{2}\phi =-4\pi \rho \;}$

is known as Poisson's equation after him, was first published in the Bulletin de la société philomatique (1813). [1] If ${\displaystyle \rho =0}$, we retrieve Laplace's equation

${\displaystyle \nabla ^{2}\phi =0.\;}$

If ${\displaystyle \rho (x,y,z)}$ is a continuous function and if for ${\displaystyle r\rightarrow \infty }$ (or if a point 'moves' to infinity) a function ${\displaystyle \phi }$ goes to 0 fast enough, a solution of Poisson's equation is the Newtonian potential of a function ${\displaystyle \rho (x,y,z)}$

${\displaystyle \phi =-{1 \over 4\pi }\int {\frac {\rho (x,y,z)}{r}}dV\;}$

where ${\displaystyle r}$ is a distance between a volume element ${\displaystyle dV}$and a point ${\displaystyle P}$. The integration runs over the whole space.

Poisson's integral is the solution for the Green function for Laplace's equation with Dirichlet boundary condition over a circular disk:

${\displaystyle \phi (\xi \eta )={1 \over 4\pi }\int _{0}^{2\pi }{R^{2}-\rho ^{2} \over R^{2}+\rho ^{2}-2R\rho \cos(\psi -\chi )}\phi (\chi )\,d\chi ,\;}$

where ${\displaystyle \xi =\rho \cos \psi }$, ${\displaystyle \eta =\rho \sin \phi }$, and ${\displaystyle \phi }$ is a boundary condition holding on the disk's boundary.

In the same manner, we define the Green function for the Laplace equation with Dirichlet condition, ∇² φ = 0 over a sphere of radius R. This time the Green function is:

${\displaystyle G(x,y,z;\xi ,\eta ,\zeta )={1 \over r}-{R \over r_{1}\rho }\;,}$

where

${\displaystyle \rho ={\sqrt {\xi ^{2}+\eta ^{2}+\zeta ^{2}}}}$ is the distance of a point (ξ, η, ζ) from the center of a sphere,

r is the distance between points (x, y, z) and (ξ, η, ζ), and

r1 is the distance between the point (x, y, z) and the point (Rξ/ρ, Rη/ρ, Rζ/ρ), symmetrical to the point (ξ, η, ζ).

Poisson's integral now has a form:

${\displaystyle \phi (\xi ,\eta ,\zeta )={1 \over 4\pi }\iint _{S}{R^{2}-\rho ^{2} \over Rr^{3}}\phi \,ds\;.}$

Poisson's two most important memoirs on the subject are Sur l'attraction des sphéroides (Connaiss. ft. temps, 1829), and Sur l'attraction d'un ellipsoide homogène (Mim. ft. l'acad., 1835). In concluding our selection from his physical memoirs, we may mention his memoir on the theory of waves (Mém. ft. l'acad., 1825). [1]

In 1812, Poisson discovered that Laplace's equation is valid only outside of a solid. A rigorous proof for masses with variable density was first given by Carl Friedrich Gauss in 1839. Both equations have their equivalents in vector algebra. Poisson's equation for the divergence of the gradient of a scalar field, ${\displaystyle \phi }$ in 3-dimensional space is:

${\displaystyle \nabla ^{2}\phi =\rho (x,y,z)\;.}$

Consider for instance Poisson's equation for surface electrical potential, ${\displaystyle \Psi }$ as a function of the density of electric charge, ${\displaystyle \rho _{e}}$ at a particular point:

${\displaystyle \nabla ^{2}\Psi ={\partial ^{2}\Psi \over \partial x^{2}}+{\partial ^{2}\Psi \over \partial y^{2}}+{\partial ^{2}\Psi \over \partial z^{2}}=-{\rho _{e} \over \varepsilon \varepsilon _{0}}.\;}$

The distribution of a charge in a fluid is unknown and we have to use the Poisson–Boltzmann equation:

${\displaystyle \nabla ^{2}\Psi ={n_{0}e \over \varepsilon \varepsilon _{0}}\left(e^{e\Psi (x,y,z)/k_{B}T}-e^{-e\Psi (x,y,z)/k_{B}T}\right),\;}$

which in most cases cannot be solved analytically. In polar coordinates the Poisson–Boltzmann equation is

${\displaystyle {1 \over r^{2}}{d \over dr}\left(r^{2}{d\Psi \over dr}\right)={n_{0}e \over \varepsilon \varepsilon _{0}}\left(e^{e\Psi (r)/k_{B}T}-e^{-e\Psi (r)/k_{B}T}\right),\;}$

which also cannot be solved analytically. If a field, φ is not scalar, the Poisson equation is valid, as can be for example in 4-dimensional Minkowski space

${\displaystyle {\sqrt {\phi _{ik}}}=\rho (x,y,z,ct).\;}$

### Optics

Poisson was a member of the academic "old guard" at the Académie royale des sciences de l'Institut de France, who were staunch believers in the particle theory of light and were skeptical of its alternative, the wave theory. In 1818, the Académie set the topic of their prize as diffraction. One of the participants, civil engineer and opticist Augustin-Jean Fresnel submitted a thesis explaining diffraction derived from analysis of both the Huygens–Fresnel principle and Young's double slit experiment. [5]

Poisson studied Fresnel's theory in detail and looked for a way to prove it wrong. Poisson thought that he had found a flaw when he demonstrated that Fresnel's theory predicts an on-axis bright spot in the shadow of a circular obstacle blocking a point source of light, where the particle-theory of light predicts complete darkness. Poisson argued this was absurd and Fresnel's model was wrong. (Such a spot is not easily observed in everyday situations, because most everyday sources of light are not good point sources.)

The head of the committee, Dominique-François-Jean Arago, performed the experiment. He molded a 2 mm metallic disk to a glass plate with wax. [6] To everyone's surprise he observed the predicted bright spot, which vindicated the wave model. Fresnel won the competition.

After that, the corpuscular theory of light was dead, but was revived in the twentieth century in a different form, wave-particle duality. Arago later noted that the diffraction bright spot (which later became known as both the Arago spot and the Poisson spot) had already been observed by Joseph-Nicolas Delisle [6] and Giacomo F. Maraldi [7] a century earlier.

### Mathematics

In pure mathematics, his most important works were his series of memoirs on definite integrals and his discussion of Fourier series, the latter paving the way for the classic researches of Peter Gustav Lejeune Dirichlet and Bernhard Riemann on the same subject; these are to be found in the Journal of the École Polytechnique from 1813 to 1823, and in the Memoirs de l'Académie for 1823. He also studied Fourier integrals. We may also mention his essay on the calculus of variations (Mem. de l'acad., 1833), and his memoirs on the probability of the mean results of observations (Connaiss. d. temps, 1827, &c). [1] The Poisson distribution in probability theory is named after him.

### Mechanics

In his Traité de mécanique (2 vols. 8vo, 1811 and 1833), which was written in the style of Laplace and Lagrange and was long a standard work, [1] he showed many novelties such as an explicit usage of momenta:

${\displaystyle p_{i}={\partial T \over {\partial (\partial q_{i}/\partial t})},}$

which influenced the work of Hamilton and Jacobi. A translation of Poisson's Treatise on Mechanics was published in London in 1842.

### Other works

Besides his many memoirs, Poisson published a number of treatises, most of which were intended to form part of a great work on mathematical physics, which he did not live to complete. Among these may be mentioned: [1]

In 1815 Poisson studied integrations along paths in the complex plane. In 1831 he derived the Navier–Stokes equations independently of Claude-Louis Navier.[ citation needed ]

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## References

1.  One or more of the preceding sentences incorporates text from a publication now in the public domain : Chisholm, Hugh, ed. (1911). . Encyclopædia Britannica . 21 (11th ed.). Cambridge University Press. p. 896.
2. "Library and Archive Catalogue". The Royal Society. Retrieved 4 October 2010.
3. "Book of Members, 1780–2010: Chapter P" (PDF). American Academy of Arts and Sciences. Retrieved 9 September 2016.
4. François Arago (1786–1853) attributed to Poisson the quote: "La vie n'est bonne qu'à deux choses: à faire des mathématiques et à les professer." (Life is good for only two things: to do mathematics and to teach it.) See: J.-A. Barral, ed., Oeuvres complétes de François Arago ..., vol. II (Paris, France: Gide et J. Baudry, 1854), page 662.
5. Fresnel, A.J. (1868), OEuvres Completes 1, Paris: Imprimerie impériale
6. Fresnel, A.J. (1868), OEuvres Completes 1, Paris: Imprimerie impériale, p. 369
7. Maraldi, G.F. (1723), 'Diverses expèriences d'optique' in Mémoires de l'Académie Royale des Sciences, Imprimerie impériale, p. 111