**Edvard Phragmén** was a Swedish mathematician who worked on the theory of functions.

- Lars Jakob Phragmén had been born on 11 January 1832 and had studied at the University of Uppsala before becoming a teacher at a school in Stockholm.
- Edvard grew up in Örebro, a town about 200 km west of Stockholm.
- Phragmén was asked by Mittag-Leffler to study a letter from Édouard Goursat and then discuss Georg Cantor's definition of the continuum with Mittag-Leffler.
- Phragmén gave a new proof of the Cantor-Bendixson theorem in Beweis eines Satzes aus der Mannigfaltigkeitslehre Ⓣ(Proof of a theorem from the theory of manifolds) (1884) and then published significant results on elliptic functions in Sur un théorème concernant les fonctions elliptiques Ⓣ(On a theorem on elliptic functions) (1885).
- Phragmén became an editor of Acta Mathematica in 1888.
- It was proof-reading papers for Acta Mathematica that first brought Phragmén to international recognition.
- Phragmén found one dubious point after another.
- However Phragmén found more problems with the paper and eventually, after the paper had been printed, it had to be withdrawn and completely rewritten by Poincaré.
- One of the points which Phragmén questioned prompted Poincaré to take the first steps in chaos theory.
- In 1892 Phragmén obtained a permanent position at Stockholm when he was appointed to succeed Sofia Kovalevskaya.
- In 1904 Phragmén resigned his chair at the university and, from that time, he worked for the state run Royal Inspection of Insurance Companies becoming director and head of the department of insurance in the following year.
- The mathematical result for which Phragmén is best remembered today is the Phragmén-Lindelöf principle.
- Phragmén's first progress towards this was his paper Sur une extension d'un théorème classique de la théorie des fonctions (1904) in which he extended Liouville's result that every entire bounded function must be constant to show that the same result could be obtained for entire functions with controlled growth in a given sector that were bounded outside the sector.
- Although this paper was an important contribution, the major advance came in a joint paper Sur une extension d'un principe classique de l'analyse et sur quelques propriétés des fonctions monogènes dans le voisinage d'un point singulier Ⓣ(On an extension of a classical principle of the analysis and on some properties of monogenic functions in the neighborhood of a singular point) of 1908 by Phragmén and the Finnish mathematician Ernst Lindelöf.
- The breakthrough in this paper came about because in it they generalise Phragmén's earlier result by formulating it as an extension of the maximum principle for the absolute values of analytic functions.
- But Phragmén was far ahead of his time, as has been certified by the expert Antoni Zygmund, who rediscovered parts of the theory 26 years later, unaware of Phragmén's paper.
- Phragmén had also made an important contribution to voting systems with the publication of two papers Proportional elections (Swedish) (1895) and On the question of a proportional election method (Swedish) (1899).
- Phragmén's variant on the single transferable vote, proposed in his 1895 paper, was introduced in the Swedish Elections Act as a method of allocating seats within a party.
- Phragmén was a member of the committee during 1903-03 which prepared the legislation which became law in the Swedish Elections Act.
- All were invited to vote in the December election but, because of the dispute, Mittag-Leffler, Bendixson and Phragmén abstained.
- A second election was held at which only professors voted (after a lengthy argument with the docent who eventually left the meeting with Pettersson) and Phragmén was elected rector.

Born 2 September 1863, Örebro, Sweden. Died 14 March 1937, Stockholm, Sweden.

View full biography at MacTutor

Origin Sweden

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive