Fatou studied at the lycée in Lorient. When he was sixteen years old, he was taught at this school by the philosopher Émile-Auguste Chartier (1868-1951) who is better known by the pseudonym Alain which he later adopted. Alain wrote in the Chapter 'Lorient' in his book Histoire de mes pensées Ⓣ (1936):-
I had during those years a simple and modest pupil, who was a mathematical genius. I taught him somehow the philosophy of these things; he easily understood all this and never made any objection. ... he was the author of a thesis which was understood by perhaps two men in the world. His name was Fatou.Leaving Lorient in 1894, Fatou went to Paris where he studied elementary mathematics and special mathematics at the Collège Stanislas for three years. He was awarded the First Prize in Mathematics in the year 1896-97. He was taught mathematics at the College by the Marianist Brother Charles Biehler (1845-1906) who was director of studies at the College. Biehler had been awarded a Ph.D. in 1879 for his thesis Sur les développements en série des fonctions doublement périodiques de troisième espèce Ⓣ which he had dedicated to his professor Charles Hermite. Fatou spent the year 1897-98 at the lycée Saint-Louis in Paris preparing for the entrance examinations to the École Normale Supérieure and the École Polytechnique. He was placed first in the 'concours général'.
Fatou entered the École Normale Supérieure in Paris in 1898 to study mathematics having been ranked first in the entrance examination. He graduated in 1901, being ranked second in the agrégation (which qualified him to teach in a collège or lycée). He would have had little trouble getting a job as a secondary teacher of mathematics but he wanted to carry on working towards his doctorate and another opportunity arose. Jules Tannery was at this time the assistant director of the École Normale Supérieure and he believed that the Paris Observatory, which had declined over the previous years, needed to appoint top mathematics graduates who were intending to undertake research towards doctorates in mathematics. Fatou was just such a graduate and having a position in the Paris Observatory would benefit both the Observatory and allow him to work on his thesis in Paris without having teaching commitments.
Having been appointed to the astronomy post at the Paris Observatory in November 1901, Fatou worked under Maurice Loewy (1833-1907) who had been made director of the Observatory in 1896 and had done much in reorganising it. Fatou made a promising start to his career at the Observatory and was rapidly promoted but by 1906 his colleagues were feeling that his heart wasn't in his work. An Observatory report of that year states (see, for example  or ):-
M Fatou proved to be very active and zealous before his nomination. Student without salary in November 1901, he was, on the suggestion of the Director, promoted "aide-astronome" on 1 January 1904 and, three months later, he became assistant-astronomer; he thus had an exceptional promotion and we thought he was an excellent acquisition. Unfortunately this hope was entirely thwarted. Not only has this civil servant produced almost nothing during these two years, but his very limited contribution to the work has been a real hindrance for his colleagues. It is because of him that the publication of the volume of observations for 1902 was delayed by more than six months ... An abnormal and disturbing situation is established that cannot last.One should treat this with a little caution since Fatou's health was poor and he applied for sick leave from the Observatory from 1 May 1906 to 31 August 1906. He probably suffered from depression which led to lack of sleep and palpitations. He also had stomach problems so health may have been a major factor in what his superiors at the Observatory considered his poor performance. However, Chazy suggests Fatou's work at the Observatory was excellent despite his health problems :-
Fatou's health was often precarious, and the observations put his physical resistance to a severe test; but he was very conscientious in all the tasks that were entrusted to him, in particular in the redaction of the observations, in the discussion of instrumental constants.However it was mathematics rather than astronomy that was Fatou's passion. In December 1904 he joined the French Mathematical Society and was working hard on the mathematical research for his thesis. He began publishing the results of his mathematical research in 1904 with the short paper Sur les séries entières à coefficients entiers Ⓣ appearing in that year. In 1905 he published four short papers: Sur l'intégrale de Poisson et les lignes singulières des fonctions analytiques Ⓣ; Sur quelques théorèmes de Riemann Ⓣ; Sur l'approximation des incommensurables et les séries trigonométriques Ⓣ; and La série de Fourier et la série de Taylor sur son cercle de convergence Ⓣ. Three more papers were published in 1906: Sur le développement en série trigonométrique des fonctions non intégrables Ⓣ; Séries trigonométriques et séries de Taylor Ⓣ; and Sur l'application de l'analyse de Dirichlet aux formes quadratiques à coefficients et à indéterminées conjuguées Ⓣ.
In 1906 he submitted his thesis, Séries trigonométriques et séries de Taylor Ⓣ which was on integration theory and complex function theory. Fatou proved that if a function is Lebesgue integrable, then radial limits for the corresponding Poisson integral exist almost everywhere. This result led to generalisations by Ivan Privalov, Abraham Plessner and Marcel Riesz. Although not giving a complete solution, Fatou's work also made a major contribution to finding a solution to the related question of whether a conformal mapping of Jordan regions onto the open disc can be extended continuously to the boundary. Lebesgue, who valued Fatou's work very highly, read his thesis and reported on 26 February 1906 (see ):-
In general, it seems to me that Fatou does not highlight very well the interest in the questions he deals with and the results he obtains ... Fatou seems also to quite readily suppose that everybody knows or sees the same thing as he does and gives rather few explanations.This report is quite in keeping with the modesty we know that Fatou displayed. He had not stressed the importance of his work and he had assumed many things without explanation because they were "obvious" to him. Paul Painlevé also read the thesis and reported on it. After considering Lebesgue's suggestions, he resubmitted the thesis and, on 14 February 1907, Fatou received his doctorate for this important work.
We have a description of Fatou by his nephew Robert Fatou (1895-1981), the son of Pierre's older brother Louis Fatou:-
In a family where no doubt many members were unable to understand him, or even to suspect the extent of his intelligence, it is easy to be astonished at his appearance, the neglect of his clothes, or that of his house, his total indifference to worldly matters like the tone of his conversation, which remained dull, slow, even hesitant, as long as the subject matter did not arouse his curiosity. Concerning his passion for mathematics, of course, none of his family was able to follow it. He was given confidence from his exceptional scientific knowledge, because of his qualifications and the brilliant course of his career through the École Normanle Supérieure. His family were very flattered, but some would have preferred that Pierre Fatou had been given a more modest intellectual capacity, less abstract and hence more accessible, but with a less original exterior more consistent with the traditions of his community.At the time he joined the French Mathematical Society in late 1904, Fatou was living with his mother at 172 Boulevard Montparnasse, Paris, close to the Paris Observatory. He continued to live with his mother at this address until her death in 1911. His health continued to cause him problems and he again asked for sick leave for the months of March and April in 1912. Certainly his health was not good enough to see him given military duties during World War I and he continued to work at the Observatory. Following the death of his mother, Fatou continued to live at 172 Boulevard Montparnasse.
The book  presents a beautiful historical account of the global theory of iteration of complex analytic functions. Fatou enters this history in a rather complicated way and the book does an excellent job in explaining an interesting episode in the history of mathematics. In 1915, the Académie des Sciences in Paris gave the topic for its 1918 Grand Prix. The prize would be awarded for a study of iteration from a global point of view. The author of  suggests that mathematicians such as Paul Appell, Émile Picard, and Gabriel Koenigs had put forward the idea to the Académie des Sciences because they were hoping for developments of Paul Montel's concept of normal families. Fatou wrote a long memoir which did indeed use Montel's idea of normal families to develop the fundamental theory of iteration in 1917. Although we do not know for certain that he was intending to enter for the Grand Prix, it seems almost certain that he undertook the work with that in mind.
Given that the topic had been proposed for the prize, it is not surprising that another mathematician would also work on the topic, and indeed Gaston Julia also produced a long memoir developing the theory in a similar way to Fatou. The two, however, chose different ways to go forward. During the later half of 1917 Julia deposited his results in sealed envelopes with the Académie des Sciences. Fatou, on the other hand, published an announcement of his results in the note Sur les substitutions rationnelles Ⓣ in the December 1917 part of Comptes Rendus. It later became evident that they had discovered very similar results. Julia wrote a letter to Comptes Rendus concerning priority which was published on 31 December 1917. Julia had asked the Académie des Sciences to inspect his sealed envelopes and Georges Humbert had been asked to carry out the task. In the same 31 December 1917 part of Comptes Rendus Georges Humbert has a letter reporting on Julia's papers. Almost certainly as a result of these letters Fatou did not enter for the Grand Prix and it was awarded to Julia. Fatou did not lose out completely, however, and even though he had not entered for the prize, the Académie des Sciences gave him an award for his outstanding 280-page paper on the topic, Sur les équations fonctionnelles Ⓣ published in 1920. Whether Julia or Fatou deserves the credit of having priority matters little since their work was certainly totally independent. At Fatou's funeral Henri Mineur delivered a speech in which he said:-
To one of his most remarkable discoveries, i.e., to the iteration of rational functions, the name of Fatou is inalterably tied: he was the first to dare to attack the problem, the first one to solve it.In 1921, following the death of Georges Humbert, the mathematics chair at the Collège de France became vacant. Among the candidates were Fatou and Lebesgue. It was decided that the Academy of Sciences would be asked to recommend which candidate should be appointed and the Academy cast 35 votes for Lebesgue and 29 for Fatou.
In contrast to the negative report of 1916 about his contributions to the work of the Observatory, he had made important contributions to astronomy. Using existence theorems for the solutions to differential equations, Fatou was able to prove rigorously certain results on planetary orbits which Gauss had suggested but only verified with an intuitive argument. He also studied the motion of a planet in a resistant medium in Sur le mouvement d'une planète dans un milieu résistant Ⓣ (1922) with the intention of explaining how twin stars would form with the capture of one moving in the atmosphere of the other. The position of "astronomer" at the Observatory became open in 1927 and Pierre Salet (1875-1936), Armand Lambert (1880-1944) and Fatou competed for the position. Salet was appointed in 1927 but the position of "astronomer" again became vacant in 1928 and Armand Lambert and Fatou competed again. The Observatory voted to appoint Lambert but the final decision was made by the Academy of Sciences which reversed the decision, voting in favour of Fatou on 25 June 1928. He was appointed as "astronomer" on 7 July 1928.
From the time that Fatou joined the French Mathematical Society in 1904 he took a major role in its operations. He held a number of important positions in the Society, culminating in his election as President for the year 1926-27.
We have mentioned some of his important mathematical work above. We should also mention his work on permutable functions Sur l'itération analytique et sur les substitutions permutables Ⓣ (1923-24), one major paper Substitutions analytiques et équations fonctionnelles à deux variables Ⓣ on iterating functions of two complex variables, published in 1924, as well as ten further papers on iterating functions of a complex variable. His famous work involves Taylor series where he examined the convergence and the analytic extension of the series. Perhaps Fatou's most famous result is that a harmonic function u > 0 in a ball has a non-tangential limit almost everywhere on the boundary.
Edouard Goursat invited Fatou to prepare a revised second edition of Appell and Goursat's Théorie des Fonctions Algébrique Ⓣ (1895) and add material on automorphic functions. Fatou wrote such a major exposition of the subject that it was decided to publish Fatou's material on automorphic functions as a separate volume, becoming Volume 2 of the new edition of Théorie des Fonctions Algébrique Ⓣ which was published in 1930, the year after Fatou's death, as a three author work. You can read a review of Fatou's volume at THIS LINK.
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 ( or ) that he died as a result of a stomach ulcer that burst. Fatou's nephew Robert Fatou wrote:-
Having never thought it useful during his life to consult a doctor, my dear uncle died suddenly in a hotel room in Pornichet.Fatou's funeral was held on 14 August in the church of Saint-Louis, and he was buried in the Carnel Cemetery in Lorient.
We have already quoted from Fatou's nephew Robert Fatou saying a little of Fatou's character. Let us give some further quotes on this. Robert Fatou wrote (see for example ):-
... I started to see, under the extreme originality of my uncle, a heart of gold with an imperturbable frankness and freedom of judgement, hindered by neither convention nor the opinions of others. Nevertheless, his natural gentleness forbade him to offend his neighbour, and he preferred to remain quiet rather than to display contrary feelings to others.One passion in Fatou's life, other than mathematics, was music. Léon Bloch writes :-
Musical passion led him, when he was quite young, to make the pilgrimage to Bayreuth where he knew the historic evenings of the performance of the Ring Cycle. In those last years, he was little by little conquered by the tragic charm of Russian music, by the bright colour of the Spanish, by all this exoticism, so moving that it is sometimes unfair to our French music.About Fatou's love of music, his nephew Robert Fatou wrote:-
Although he never "learned" music, Pierre Fatou certainly found in it a huge pleasure ... [He] had the gift of being able to read music and he savoured it. He used to buy the scores of his favourite composers, the German romantics first, then the Russians for whom he conceived a great admiration. ... In concert, he used to follow the interpretation carefully on his staves and, back home, he "heard" again his favourite arias while reading the pages of his notebooks. Sometimes, when his memory was unfaithful, he helped himself with a piano or viola to remember a melody. Then a smile would brighten his face, showing his deep satisfaction.
Article by: J J O'Connor and E F Robertson