Pierre Fatou
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Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and
astronomer An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
. He is known for major contributions to several branches of
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
. The Fatou lemma and the
Fatou set In complex dynamics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values ...
are named after him.


Biography

Pierre Fatou's parents were Prosper Ernest Fatou (1832-1891) and Louise Eulalie Courbet (1844-1911), both of whom were in the military. Pierre's family would have liked for him to enter the military as well, but his health was not sufficiently good for him to pursue a military career. Fatou entered the
École Normale Supérieure École or Ecole may refer to: * an elementary school in the French educational stages normally followed by Secondary education in France, secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing i ...
in Paris in 1898 to study mathematics and graduated in 1901 when he was appointed an intern (''stagiaire'') in the
Paris Observatory The Paris Observatory (, ), a research institution of the Paris Sciences et Lettres University, is the foremost astronomical observatory of France, and one of the largest astronomical centres in the world. Its historic building is on the Left Ban ...
. Fatou was promoted to assistant astronomer in 1904 and to astronomer (''astronome titulaire'') in 1928. He worked in this observatory until his death. Fatou was awarded the
Becquerel The becquerel (; symbol: Bq) is the unit of radioactivity in the International System of Units (SI). One becquerel is defined as an activity of one per second, on average, for aperiodic activity events referred to a radionuclide. For applicatio ...
prize in 1918; he was a knight of the
Legion of Honour The National Order of the Legion of Honour ( ), formerly the Imperial Order of the Legion of Honour (), is the highest and most prestigious French national order of merit, both military and Civil society, civil. Currently consisting of five cl ...
(1923). He was the president of the
French mathematical society French may refer to: * Something of, from, or related to France ** French language, which originated in France ** French people, a nation and ethnic group ** French cuisine, cooking traditions and practices Arts and media * The French (band), ...
in 1927. He was in friendly relations with several contemporary French mathematicians, especially,
Maurice René Fréchet Maurice may refer to: *Maurice (name), a given name and surname, including a list of people with the name Places * or Mauritius, an island country in the Indian Ocean *Maurice, Iowa, a city * Maurice, Louisiana, a village * Maurice River, a tr ...
and
Paul Montel Paul Antoine Aristide Montel (29 April 1876 – 22 January 1975) was a French mathematician. He was born in Nice, France and died in Paris, France. He researched mostly on holomorphic functions in complex analysis. Montel was a student of Émile ...
. In the summer of 1929 Fatou went on holiday to Pornichet, a seaside town to the west of Nantes. He was staying in Le Brise-Lames Villa near the port and it was there at 8 p.m. on Friday 9 August that he died in his room. No cause of death was given on the death certificate but Audin argues that he died as a result of a stomach ulcer that burst. Fatou's nephew Robert Fatou wrote: Fatou's funeral was held on 14 August in the church of Saint-Louis, and he was buried in the Carnel Cemetery in Lorient.


Mathematical work of Fatou

Fatou's work had very large influence on the development of
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
in the 20th century. Fatou's PhD thesis ''Séries trigonométriques et séries de Taylor'' was the first application of the
Lebesgue integral In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
to concrete problems of
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
, mainly to the study of analytic and harmonic functions in the unit disc. In this work, Fatou studied for the first time the Poisson integral of an arbitrary measure on the unit circle. This work of Fatou is influenced by
Henri Lebesgue Henri Léon Lebesgue (; ; June 28, 1875 – July 26, 1941) was a French mathematician known for his Lebesgue integration, theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an ...
who invented his integral in 1901. The Fatou theorem, which says that a bounded
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
in the unit disc has radial limits
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
on the unit circle was published in 1906 . This theorem was at the origin of a large body of research in 20th-century mathematics under the name of ''bounded analytic functions''. See also the Wikipedia article on functions of bounded type. A number of fundamental results on the
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ne ...
of a
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
belong to Fatou. In 1917–1920 Fatou created the area of mathematics which is called holomorphic dynamics . It deals with a global study of iteration of analytic functions. He was the first to introduce and study the set which is called now the
Julia set In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of ...
. (The complement of this set is sometimes called the
Fatou set In complex dynamics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values ...
). Some of the basic results of holomorphic dynamics were also independently obtained by Gaston Julia and Samuel Lattes in 1918. Holomorphic dynamics has experienced a strong revival since 1982 because of the new discoveries of
Dennis Sullivan Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the Graduate Center of the City University ...
,
Adrien Douady Adrien Douady (; 25 September 1935 – 2 November 2006) was a French mathematician born in La Tronche, Isère. He was the son of Daniel Douady and Guilhen Douady. Douady was a student of Henri Cartan at the École normale supérieure, and initi ...
, John Hubbard and others. In 1926, Fatou pioneered the study of dynamics of transcendental
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any ...
s , a subject which is intensively developing at this time. As a byproduct of his studies in holomorphic dynamics, Fatou discovered what are now called Fatou–Bieberbach domains . These are proper subregions of the complex space of dimension ''n'', which are biholomorphically equivalent to the whole space. (Such regions cannot exist for ''n=1''.) Fatou did important work in
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
. He was the first to prove rigorously a theorem (conjectured by
Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
) on the averaging of a perturbation produced by a periodic force of short period . This work was continued by Leonid Mandelstam and Nikolay Bogolyubov and his students and developed into a large area of modern applied mathematics. Fatou's other research in celestial mechanics includes a study of the movement of a planet in a resisting medium.


Selected publications

* * ; ; * * * * *


See also

* Fatou conjecture * Fatou's theorem *
Fatou set In complex dynamics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values ...
*
Fatou–Lebesgue theorem In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequality (mathematics), inequalities relating the integrals (in the sense of Lebesgue integration, Lebesgue) of the limit superior and limit inferior, limit inferior and the lim ...
(same as
Fatou's lemma In mathematics, Fatou's lemma establishes an inequality (mathematics), inequality relating the Lebesgue integral of the limit superior and limit inferior, limit inferior of a sequence of function (mathematics), functions to the limit inferior of ...
) *
Classification of Fatou components In mathematics, Fatou components are connected component (analysis), components of the Fatou set. They were named after Pierre Fatou. Rational case If f is a rational function :f = \frac defined in the extended complex plane, and if it is a nonli ...
* Fatou–Bieberbach domain * Holomorphic dynamics


Notes


References

* * * * Daniel Alexander, Felice Iavernaro, Alessandro Rosa
''Early days in complex dynamics: a history of complex dynamics in one variable during 1906-1942''
History of Mathematics 38, American Mathematical Society 2012


External links

*

by Michèle Audin, on the site Images des Mathématiques. * List of publications of Pierre Fatou o
zbMATH
* * {{DEFAULTSORT:Fatou, Pierre 1878 births 1929 deaths 20th-century French mathematicians 20th-century French astronomers École Normale Supérieure alumni