Person: Bendixson, Ivar Otto
Bendixson taught at Stockholm, then from 1913 to 1927 he was rector of Stockholm University. He worked on set theory and differential equations. He is best remembered for the Poincaré - Bendixson theorem.
Mathematical Profile (Excerpt):
- After attending a new primary school, Ivar studied at a secondary school in Stockholm.
- In 1879 Bendixson went to Uppsala University, the oldest institution of higher learning in Sweden, and he graduated with the equivalent of a Master's degree on 27 January 1881.
- By the time Bendixson graduated from Uppsala, Stockholm University was open and he studied there.
- On 10 June 1890 Bendixson was appointed as a docent at Stockholm University.
- In 1899 Bendixson substituted for the Professor of Pure Mathematics at the Royal Technological Institute and then he was promoted to professor there on 26 January 1900.
- Bendixson showed his exceptional talent for mathematics from the beginning of his student days and, as he progressed, these talents became more and more obvious.
- As a young student Bendixson made his name by proving a theorem which he included in a letter which he wrote to Cantor, the letter being published in Volume 2 of Acta Mathematica.
- The perfect set in this partition is, in today's terminology, the Bendixson derivative of the original set.
- The proof of the theorem which Bendixson gave uses Cantor's notion of transfinite numbers.
- Bendixson also made interesting contributions to algebra when he investigated the classical problem of the algebraic solution of equations.
- Bendixson returned to Abel's original contribution and showed that Abel's methods could be extended to describe precisely which equations could be solved by radicals.
- Abel himself had written shortly before his death that he hoped to be able to achieve this, and it is interesting that Bendixson was able to do so using Abel's methods.
- Bendixson published these results in Acta Mathematica and it is worth noticing that he was prepared to accept the new ideas of set theory but not of group theory.
- Bendixson was a frequent contributor to Acta Mathematica, the journal founded by his colleague Mittag-Leffler.
- Bendixson is probably best remembered for the Poincaré- Bendixson theorem.
- We shall say a little about how Bendixson came to prove this result.
- In examining periodic solutions of differential equations Bendixson used methods based on continued fractions.
- The analysis problem which intrigued Bendixson more than all others was the investigation of integral curves to first order differential equations, in particular he was intrigued by the complicated behaviour of the integral curves in the neighbourhood of singular points.
- This important problem was first studied by Briot and his friend Bouquet and, before Bendixson worked on it, had recently been investigated by Poincaré.
- Poincaré had obtained a qualitative description of the integral curves but it was Bendixson who gave a quantitative description near the singular points.
- The Poincaré-Bendixson theorem, which says an integral curve which does not end in a singular point has a limit cycle, was first proved by Poincaré but a more rigorous proof with weaker hypotheses was given by Bendixson in 1901.
- To do this Bendixson used successive approximations.
- As a teacher Bendixson was always well prepared.
- For his outstanding contributions, Bendixson received many honours including an honorary doctorate on 24 May 1907.
Born 1 August 1861, Bergshyddan, Djurgården, Sweden. Died 29 November 1935, Stockholm, Sweden.
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Adapted from other CC BY-SA 4.0 Sources:
- O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive