28 February 1878|
|Died||10 August 1929
|Known for||Fatou lemma
Pierre Joseph Louis Fatou (28 February 1878 – 10 August 1929) was a French mathematician and astronomer. He is known for major contributions to several branches of analysis. The Fatou lemma and the Fatou set are named after him.
Fatou entered the École Normale Supérieure in Paris in 1898 to study mathematics and graduated in 1901 when he was appointed an observer (stagiaire) in the Paris Observatory. Fatou was promoted to assistant astronomer in 1904 and to astronomer (astronome titulaire) in 1928. He worked in this observatory until his death.
Mathematical work of Fatou
Fatou's work had very large influence on the development of analysis in the 20th century.
Fatou's PhD thesis Séries trigonométriques et séries de Taylor (Fatou 1906) was the first application of the Lebesgue integral to concrete problems of analysis, 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 who invented his integral in 1901.
The famous Fatou theorem, which says that a bounded analytic function in the unit disc has radial limits almost everywhere on the unit circle was published in 1906 (Fatou 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.
In 1917–1920 Fatou created the area of mathematics which is called holomorphic dynamics (Fatou 1919, 1920, 1920b). 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. (The complement of this set is sometimes called the Fatou set). Some of the basic results of holomorphic dynamics were also independently obtained by Gaston Julia and Samuel Lattes in 1918. Holomorphic dynamics experienced a strong revival since 1982 because of the new discoveries of Dennis Sullivan, Adrian Douady, John Hubbard and others. Beautiful pictures illustrating this theory produced by modern computers stimulate great interest not only of mathematicians but also outside of the mathematical community. In 1926 Fatou pioneered the study of dynamics of transcendental entire functions, a subject which is intensively developing at this time (2012).
As a byproduct of his studies in holomorphic dynamcis, Fatou discovered what are now called Fatou–Bieberbach domains. These are proper subregions of the complex space of dimension n that are biholomorphically equivalent to the whole space. (Such regions cannot exist for n=1.)
Fatou did important work in celestial mechanics. He was the first to prove rigorously  a theorem on the averaging of a perturbation produced by a periodic force of short period, conjectured by Gauss, (Fatou 1928). This work was continued by Leonid Mandelstam and Nikolay Bogolyubov and his students and developed in 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.
- Fatou, P. (1906). "Séries trigonométriques et séries de Taylor". Acta Math 30: 335. doi:10.1007/BF02418579. JFM 37.0283.01.
- Fatou, P. (1919). "Sur les équations fonctionnelles, I". Bulletin de la Société Mathématique de France 47: 161–271. JFM 47.0921.02.; Fatou, P. (1920). "Sur les équations fonctionnelles, II". Bulletin de la Société Mathématique de France 48: 33–94. JFM 47.0921.02.; Fatou, P. (1920b). "Sur les équations fonctionnelles, III". Bulletin de la Société Mathématique de France 48: 208–314. JFM 47.0921.02.
- Fatou, P. (1923). "Sur les fonctions holomorphes et bornées à l'intérieur d'un cercle". Bull. Soc. Math. France 51: 191–202. JFM 49.0221.01.
- Fatou, P. (1926). "Sur l'itération des fonctions transcendantes entières". Acta Math 47 (4): 337–370. doi:10.1007/BF02559517.
- Fatou, P. (1928). "Sur le mouvement d'un système soumis à des forces à courte période". Bull. Soc. Math. France 56: 98–139. JFM 54.0834.01.
- Audin 2009, p. 138.
- "Anciens Présidents" (in French). French mathematical society.
- Audin 2009, p. 132.
- Garnett, John B. (1981). Bounded analytic functions. Academic Press.
- Bieberbach, Ludwig (1955). Analytische Fortsetzung. Berlin: Springer Verlag.
- Mitropolsky, Iu. A. (1967). "Averaging method in non-linear mechanics". Intl. J. Non-Lin. Mech. 2: 69–95. Bibcode:1967IJNLM...2...69M. doi:10.1016/0020-7462(67)90020-0.
- Fatou's theorem
- Fatou set
- Fatou–Lebesgue theorem (same as Fatou's lemma)
- Classification of Fatou components
- Fatou–Bieberbach domain
- Holomorphic dynamics
- Audin, Michèle (2009). Fatou, Julia, Montel, le Grand prix des sciences mathématiques de 1918, et après.... Heidelberg: Springer. ISBN 978-3-642-00445-2.
- Notice sur les travaux scientifique de Pierre Fatou (pdf). Paris. 1929.
- Chazy, Jean (1933). "Pierre Fatou". Bulletin astronomique 8: 389–384.
- O'Connor, John J.; Robertson, Edmund F., "Pierre Fatou", MacTutor History of Mathematics archive, University of St Andrews.
- Pierre Fatou, mathématicien et astronome by Michèle Audin, on the site Images des Mathématiques.
- List of publications of Pierre Fatou on Jahrbuch database.
- "Pierre Fatou". Dictionary of Scientific Biography. New York: Charles Scribner's Sons. 1970–80. ISBN 978-0-684-10114-9.
- Pierre Fatou at the Mathematics Genealogy Project
- Early Days in Complex Dynamics: A History of Complex Dynamics in One Variable During 1906-1942 by Daniel S.Alexander, Felice Iavernaro, Alessandro Rosa