**Josip Plemelj**was born in Grad on Bled, a village on the shores of Lake Bled. This glacial lake, in the extreme north-west of Slovenia, is situated northwest of Ljubljana and is 475 metres above sea level at the foot of the Julian Alps. Josip's father was Urban Plemelj, a carpenter in the village who farmed a small piece of land, while his mother was Marija Mrak. Urban died was Josip was only one year old and it was left to Marija to bring up her family in very difficult circumstances.

Slovenia was under Austrian Habsburg rule when Plemelj was born, as it had been for most of the preceding 500 years. However it was around this time that the Slovenes, Serbs, and Croats began to seek independence from Austria and various ideas of political union of these groups began to be discussed. It was a political atmosphere which would later have a great influence on Plemelj's career but, given the poor financial circumstances of his family, he was lucky to receive an education which would allow him to display his talents. At primary school he soon showed his mathematical abilities which were quickly recognised. His mother was able to send him to Ljubljana where he attended high school from 1886 to 1894.

After four years at high school Plemelj had covered the whole school mathematics syllabus. He then showed his inclination to teach by tutoring the other pupils at the school for their final examinations although he was much younger than the pupils he helped. From his fifth year he began to study more advanced mathematics. He was soon making his own mathematical discoveries, for example he discovered for himself the series expansion for sin *x* and for cos *x*. The way he did this was to first find the series expansion for arcsin *x* and then invert the series to obtain that for cos *x*. He also developed a love for solving geometrical problems while at high school and some of the research topics he later worked on, such as his construction of a regular seven sided polygon, clearly developed out of the investigations he began to make while at school. Topics other than mathematics also interested him, especially physics and astronomy. He studied both the theoretical aspects of astronomy and also the practical aspects spending many evenings observing the stars and planets.

In 1894 Plemelj took his final school examinations and entered the Faculty of Arts of the University of Vienna to study his three favourite school subjects of mathematics, physics and astronomy. He was lucky to have some excellent lecturers at Vienna, many of whom were newly appointed. He was taught analysis by von Escherich while Gegenbauer (appointed a professor in Vienna in 1893) and Mertens (appointed to Vienna in 1894) taught him algebra and number theory. His physics lecturer was Boltzmann, perhaps the best known of all his teachers who was appointed to the chair of theoretical physics in Vienna in 1894, while his astronomy lecturer was Edmund Weiss. Plemelj undertook research under von Escherich's supervision and in May 1898 was awarded his doctorate for a thesis on linear homogeneous differential equations with uniform periodical coefficients: *über lineare homogene Differentialgleichungen mit eindeutigen periodischen Koeffizienten* Ⓣ.

The years which Plemelj spent at the University of Vienna were ones in which political events were taking place which would lead to Slovenia gaining independence from Austrian rule and the eventual formation of Yugoslavia. The first political parties in Slovenia were formed during this time and Plemelj strongly supported their aims. After his doctorate Plemelj travelled to Germany where he studied with Frobenius and Fuchs for the academic year 1899-1900. He then went to Göttingen where he spent the following academic year studying under Klein and Hilbert.

An important mathematical event occurred while he was at Göttingen, for that was the year in which Holmgren lectured on Fredholm's theory of integral equations at Göttingen. Hilbert immediately saw the he importance of Fredholm's theory and work at Göttingen on this topic began immediately. Plemelj was a major contributor to this work and he was among the first to make major advances. The contributions he made to integral equations and potential theory were brought together in a work he published in 1911 for which he was awarded the Prince Jablonowski Prize.

In April 1902 he was appointed as a privatdozent at the University of Vienna where he remained until 1906 when he was appointed as an assistant professor at the Technical University of Vienna. He only spent one year in this post for in 1907 he was appointed associate professor at the University of Chernivtsi. Chernivtsi is now in south-western Ukraine, but at the time Plemelj was working there it was part of Austria-Hungary. After only one year he was appointed to the position of full professor of mathematics and around this time he produced some of his most outstanding contributions to mathematical research.

Riemann's problem, concerning the existence of a linear differential equation of the Fuchsian class with prescribed regular singular points and monodromy group, had been reduced to the solution of an integral equation by Hilbert in 1905. Plemelj discovered equations relating to boundary values of holomorphic functions which are now called the "Plemelj formulae" and shortly after this was able to solve Riemann's problem in his paper *Riemannian classes of functions with given monodromy group* published in *Monatshefte für Mathematik und Physik* Ⓣ in 1908. The equations are today important in a number of different fields, including neutron transport theory where a singular integral equation is encountered. We should note that the Plemelj formulae are sometimes called the Sokhotsky-Plemelj formulae since, as Kechkic writes in [5]:-

Plemelj's methods for solving the Riemann's problem were further developed by Nikolai Ivanovich Mushelisvili into the theory of singular integral equations. Also important are Plemelj's contributions to the theory of analytic functions which he developed while investigation the problem of uniformization of algebraic functions [2]:-... the so-called Plemelj formulae are ... due to Sokhotsky, who published them in his doctor's thesis in1873, that is to say35years before Plemelj.

Another contribution that we should mention was Plemelj's simple proof of the n = 5 case of Fermat's Last Theorem which he published in 1912. During 1912-13 Plemelj was Dean of the Faculty at the University of Chernivtsi. This was a period during which he received many honours, for his book on potential theory not only led to him receiving the Prince Jablonowski Prize, which we mentioned above, but also the Richard Lieben Prize in 1912. This latter prize was awarded to him in Vienna for:-He was the first to discover the sharp formulation of Koebe's distortion theorem. Within the theory of differential equations he worked mostly on equations of the Fuchs type and on Klein's theorems.

Plemelj certainly qualified as an Austrian mathematician but it was ironical that he would be described as such during a period when the various nations which formed the Austrian empire were beginning to look towards independence and Plemelj himself was strongly associated with such aims. He continued to work at the University of Chernivtsi through the first years of World War I, but the movement for independence of Slovenia from Austria continued to strengthen. With Austria allied with Germany against the Allies, the independence movement grew in strength with the Yugoslav Committee, consisting of exiles in Paris and London, founded in April 1915. In 1917 the government forced Plemelj to leave Chernivtsi because of his political views and he fled north into Bohemia. On 20 July 1917, the Yugoslav Committee in conjunction with the exiled Serbian government issued the Corfu Declaration which paved the way for a South Slav state of Serbs, Croats, and Slovenes. The Kingdom of Serbs, Croats, and Slovenes (which would became Yugoslavia in 1929) was formed on 1 December 1918. In fact at the end of World War I, among the many changes to the map of Europe, Chernivtsi became Romanian.... the most outstanding work in the field of pure and applied mathematics written by an Austrian mathematician in the previous three years.

The Slovene Provincial Government set up a University Commission to oversee the reopening of the University of Ljubljana as a Slovene University and Plemelj was appointed a member of this Commission. In fact the University of Ljubljana had been founded in 1595 but now an important step towards Slovenian independence from Austrian rule was the creation of a Slovene University. The University of Ljubljana reopened as a Slovene university in 1919 with Plemelj as its first Rector. He was also appointed Professor of Mathematics in the Faculty of Arts. Vidav, who was a student of Plemelj at Ljubljana, writes in [2]:-

However his time in Ljubljana was very beneficial for his country [2]:-Accepting a chair at the University of Ljubljana, Plemelj in fact sacrificed his scholarly career. What with the post-war lack of contact with the scientific world, his professional loneliness, and his illness, Plemelj taught, rather than created much new during his Ljubljana years.

These courses became the basis of three of his textbooks published by the Slovene Academy of Sciences and Arts. They werePlemelj's presence in Ljubljana was of paramount importance for the development of mathematics and other exact sciences among Slovenes. Several generations of mathematicians benefited from his teaching. He used to hold a general course of mathematics and a three-year cycle of lectures on differential equations, the theory of analytic functions, and algebra including number theory.

*The theory of analytic functions*(1953),

*Differential and integral equations. The theory and the application*(1960), and

*Algebra and the number theory*(1962). His final work,

*Problems in the Sense of Riemann and Klein*appeared in 1964 and described those parts of mathematics to which Plemelj had been a major contributor.

Plemelj received many honours in addition to the Prince Jablonowski Prize and the Richard Lieben Prize which we mentioned above. He was elected to the Yugoslav Academy in Zagreb in 1923, the Serbian Academy in 1930, and the Slovene Academy of Sciences and Arts when it was founded in 1938. In 1954 he received the Presernova nagrada Prize and was elected to the Bavarian Academy in the same year. The University of Ljubljana awarded him an honorary doctorate in 1963 on the occasion of his 90^{th} birthday.

We end this biography with a few comments about Plemelj's style as a lecturer. He always lectured without notes, seemingly without having prepared the lecture at all. He did prepare each lecture, however, thinking carefully about it as he walked between his home and the university. He was interested in calligraphy, writing his lectures in beautifully drawn characters. Language also fascinated him and he chose his words carefully, expecting his students to do the same. Although much is lost from the following quotation in the translation, it does illustrate the importance he attached to shades of meaning:-

Plemelj retired in 1957 when he was 83 years old and died in Ljubljana in his 94The engineer who does not know mathematics never needs it. But if he knows it, he uses it frequently.

^{th}year. He is buried in his home town of Bled where his Villa was bequeathed by him to the Society of Mathematicians, Physicists and Astronomers of Slovenia. Today it contains a memorial to one of the most famous mathematicians of Slovenia.

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

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