# Henri Eugène Padé

### Born: 17 December 1863 in Abbeville, Picardy, France,

Died: 9 July 1953 in Aix-en-Provence, France

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**Henri Padé**was born in Abbeville which is a town northwest of Amiens in the Picardy region of northern France. He attended school in his home town and obtained his baccalaureate in 1881 at the age of seventeen. He then went to Paris to continue his education at the Lycée St Louis where he spent two years preparing to sit the university entrance examinations.

After completing his studies at Lycée St Louis, Padé sat the entrance examination for the École Normale Supérieure in Paris, entering the École in 1883. Three years later he graduated with his Agrégation de Mathématiques and began a career teaching in secondary schools. He did begin publishing papers on his mathematical research during this period, however, and his first publication appeared in 1888.

In 1889 Padé went to Germany to continue his studies, going first to Leipzig and then to Göttingen, studying under Klein and Schwarz. He returned to France in 1890 and continued teaching in secondary schools while he worked for his doctorate in mathematics under Hermite's supervision. In 1892 he presented his doctoral thesis

*Sur la representation approchee d'une fonction par des fractions rationelles*Ⓣ to the Sorbonne in Paris. Padé defended his thesis on 21 June 1892, the examiners being his supervisor Hermite, together with Émile Picard and Paul Appell.

In his thesis Padé made the first systematic study of what we call today Padé approximants, which are rational approximations to functions given by their power series. He proved results on their general structure and also clearly set out the connection between Padé approximants and continued fractions. Of course, although Padé's thesis was the first systematic study, the ideas had been around for some time although not systematically developed. Daniel Bernoulli studied a Padé-type approximation in 1730 and James Stirling gave a similar method in

*Methodus differentialis*Ⓣ published in the same year. At around the same time Euler used Padé-type approximation to find the sum of a series. In 1758 Lambert found approximants which are Padé approximants, but developed no general theory.

The first who seemed to realise the full significance of the method of Padé approximants was Lagrange in a paper of 1776 where he related them to continued fractions. The method continued to be used from time to time by various mathematicians, for example Kummer in 1837 used Padé approximants to sum series which only converged very slowly. Jacobi deduced a formula for the approximants in terms of determinants in 1845. Padé approximants appear in Hankel's doctoral thesis

*Über eine besondere Classe der symmetrischen Determinanten*, Ⓣ written in 1861, while in his thesis of 1870, supervised by Weierstrass, Frobenius discovered identies between the approximants which he developed more fully in a paper he published twenty years later. It would be fair to say that this work is the first systematic study of Padé approximants. Between these two contributions by Frobenius, Darboux had looked at Padé approximants of the exponential function. Other contributions were made by Laguerre and Chebyshev. Padé's doctoral supervisor Hermite had used approximants and continued fractions in his work of 1873 on proving the transcendence of e.

How much of this earlier work was known to Padé is less obvious and he certainly seemed to be unaware of the contributions of Frobenius. The greatest influence on him, as is to be expected, was the contributions of his supervisor Hermite, who had developed a general theory of interpolation by rational functions. In his doctoral thesis Padé showed that, in a properly defined sense, the Padé approximant was the best approximant among all the rational ones. Van Vleck, at a meeting of the American Mathematical Society in Boston in 1903 said (see for example [1]):-

Padé established various properties of this table in his thesis and developed the ideas further in later papers, particularly in 1899 when he studies the exponential series and in 1901 when he considered (1+The existence of approximants was, of course, well-known before Padé, but no systematic examination of them had been made except by Frobenius, who determined the important relations which normally exist between them. Padé goes further, and arranges the approximants, expressed each in its lowest terms, into a table ...

*x*)

^{m}, for

*m*not an integer.

After completing his doctoral studies, Padé taught at the Lycée Faidherbe in Lille, taking up this post in October 1893. He continued to investigate approximants, and in 1894 he published a memoir in which he generalised the continued fraction algorithm which Hermite had studied in 1863 and again in 1893. The approximants which Padé introduced in this paper are now known as the Padé-Hermite approximants. In January 1897, a little over three years after taking up his appointment at the Lycée Faidherbe, Padé became Maître de Conférences at the University of Lille. In this post he succeeded Émile Borel who had just left Lille to take up an appointment at the École Normale Supérieure in Paris. In 1899 Padé published another major work on Padé approximants which, as we noted above, looked in depth at approximants of the exponential function.

After four years in the post of Maître de Conférences at the University of Lille, Padé left to go to Poitiers where he was appointed as Professor of Rational and Applied Mechanics in June 1902. Only a little more than a year later he moved to Bordeaux where he took up an appointment at the University. In 1906 he received the Grand Prix of the French Academy of Sciences and in the same year was appointed Dean of the Faculty of Science at the University of Bordeaux. Let us look briefly at this Grand Prix competition.

The subject proposed for the Grand Prix of the Paris Academy of 1906 was on the convergence of algebraic continued fractions. Five submissions were received and four referees were appointed, Émile Picard, Painlevé, Poincaré, and Appell. Émile Picard read two of the submissions, including the one by Padé, while these other referees read one each of the remaining three entries. Brezinski writes [1]:-

The second sealed envelope, dated 22 June 1903, contains a paper entitledThe work of Padé consists of a presentation of his previous results concerning the Padé table. He studied the convergence problem for the exponential function. This led him to work on the connection bewteen Sylvester's formulas on the polynomials arising in the application of Sturm's theorem and the theory of continued fractions.6614

The contribution by Padé contains two sealed covers. The first one, numbered, is dated2February1903and has been published in1907in the "Annales de l'École Normale Supérieure". It deals with the development into a continued fraction of the generating function of a sequence satisfying a difference equation.

*On a new method for studying the development of certain functions in continued fractions*.

Three of the five submissions received a prize, with Padé receiving the first prize together with half the total prize money, with smaller amounts going to the submissions judged to be worthy of second and third place. He had reached a high point in his career in universities which he would leave two years later.

By 1908 Padé had written 41 papers, 29 of which were on continued fractions and Padé approximants. Although the theory of Padé approximants which he had developed in his thesis, and in many later papers, was not quick to be taken up by many other mathematicians, it did become well known after Borel presented Padé approximants in his 1901 book on divergent series. Padé had made other significant contributions, however, such as publishing an elementary algebra book and translating Klein's Erlangen programme from German into French. His translation had appeared as

*Le programme d'Erlangen*Ⓣ in the

*Annals of the École Normale Supérieure*in 1891, shortly after he returned from his studies in Germany.

Having achieved high standing at the University of Bordeaux, Padé left universities in 1908, when he was 44 years old, to became Rector of the Academy in Besançon. This too was a high distinction for Padé who became the youngest Rector in France when he was appointed. In 1917 he became Rector of the Academy of Dijon and from 1923 until he retired in 1934, at the age of 70, he was Rector at Aix-Marseille.

**Article by:** *J J O'Connor* and *E F Robertson*

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