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 their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, natural satellite, moons, comets and galaxy, g ...
. 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 the context of complex dynamics, a branch of mathematics, 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 wit ...
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 may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi ...
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 (french: Observatoire de Paris ), a research institution of the Paris Sciences et Lettres University, is the foremost astronomical observatory of France, and one of the largest astronomical centers in the world. Its histor ...
. 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 the activity of a quantity of radioactive material in which one nucleus decays per second. For applications relatin ...
prize in 1918; he was a knight of the
Legion of Honour The National Order of the Legion of Honour (french: Ordre national de la Légion d'honneur), formerly the Royal Order of the Legion of Honour ('), is the highest French order of merit, both military and civil. Established in 1802 by Napoleon, ...
(1923). He was the president of the French mathematical society in 1927. He was in friendly relations with several contemporary French mathematicians, especially,
Maurice René Fréchet Maurice may refer to: People *Saint Maurice (died 287), Roman legionary and Christian martyr *Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor *Maurice (bishop of London) (died 1107), Lord Chancellor and Lo ...
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 of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
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 In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriv ...
of an arbitrary
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
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 theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
who invented his integral in 1901. The
Fatou theorem In mathematics, specifically in complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk. Motivation and statement of ...
, 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 an ...
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 new ...
of a Taylor series belong to Fatou. In 1917–1920 Fatou created the area of mathematics which is called
holomorphic dynamics Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Montel's theorem **Po ...
. 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 the context of complex dynamics, a branch of mathematics, 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 wit ...
. (The complement of this set is sometimes called the
Fatou set In the context of complex dynamics, a branch of mathematics, 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 wit ...
). Some of the basic results of holomorphic dynamics were also independently obtained by
Gaston Julia Gaston Maurice Julia (3 February 1893 – 19 March 1978) was a French Algerian mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals are cl ...
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 City University of New York Graduate Ce ...
,
Adrian Douady Adrien Douady (; 25 September 1935 – 2 November 2006) was a French mathematician. Douady was a student of Henri Cartan at the École normale supérieure, and initially worked in homological algebra. His thesis concerned deformations of comple ...
, 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 fin ...
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 domain In mathematics, a Fatou–Bieberbach domain is a proper subdomain of \mathbb^n, biholomorphically equivalent to \mathbb^n. That is, an open set \Omega \subsetneq \mathbb^n is called a Fatou–Bieberbach domain if there exists a bijective holomorphi ...
s . 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 (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
) on the averaging of a
perturbation Perturbation or perturb may refer to: * Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly * Perturbation (geology), changes in the nature of alluvial deposits over time * Perturbatio ...
produced by a periodic force of short period . This work was continued by
Leonid Mandelstam Leonid Isaakovich Mandelstam or Mandelshtam ( be, Леанід Ісаакавіч Мандэльштам; rus, Леонид Исаакович Мандельштам, p=lʲɪɐˈnʲit ɨsɐˈakəvʲɪtɕ mənʲdʲɪlʲˈʂtam, a=Ru-Leonid_Mande ...
and
Nikolay Bogolyubov Nikolay Nikolayevich Bogolyubov (russian: Никола́й Никола́евич Боголю́бов; 21 August 1909 – 13 February 1992), also transliterated as Bogoliubov and Bogolubov, was a Soviet and Russian mathematician and theoretica ...
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 the context of complex dynamics, a branch of mathematics, 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 wit ...
*
Fatou–Lebesgue theorem In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit superior ...
(same as
Fatou's lemma In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's lemma ...
) *
Classification of Fatou components In mathematics, Fatou components are 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 nonlinear function (degree > 1) : ...
* Fatou–Bieberbach domain *
Holomorphic dynamics Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Montel's theorem **Po ...


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