Fatou's work had a very large influence on the development of
analysis in the 20th century. Fatou's PhD thesis
Séries trigonométriques et séries de Taylor 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
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 . 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 of a
Taylor series 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. (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 has experienced a strong revival since 1982 because of the new discoveries of
Dennis Sullivan,
Adrien Douady,
John Hubbard and others. In 1926, Fatou pioneered the study of dynamics of
transcendental entire functions , 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. He was the first to prove rigorously a theorem (conjectured by
Gauss) 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==