Person: Padé, Henri
Henri Padé made important contributions to the theory of continued fractions.
Mathematical Profile (Excerpt):
- 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.
- In 1889 Padé went to Germany to continue his studies, going first to Leipzig and then to Göttingen, studying under Klein and Schwarz.
- 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 Ⓣ(Differential methods) 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.
- Padé approximants appear in Hankel's doctoral thesis Über eine besondere Classe der symmetrischen Determinanten, Ⓣ(On a special class of symmetric determinants) 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.
- 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.
- In his doctoral thesis Padé showed that, in a properly defined sense, the Padé approximant was the best approximant among all the rational ones.
- Padé goes further, and arranges the approximants, expressed each in its lowest terms, into a table ...
- After completing his doctoral studies, Padé taught at the Lycée Faidherbe in Lille, taking up this post in October 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 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.
- Émile Picard read two of the submissions, including the one by Padé, while these other referees read one each of the remaining three entries.
- The contribution by Padé contains two sealed covers.
- 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.
- 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.
- 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.
Born 17 December 1863, Abbeville, Picardy, France,. Died 9 July 1953, Aix-en-Provence, France.
View full biography at MacTutor
Thank you to the contributors under CC BY-SA 4.0!
Adapted from other CC BY-SA 4.0 Sources:
- O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive