Research
Plemelj's main research interests were the theory of linear differential equations, integral equations, potential theory, the theory of analytic functions, and functional analysis. Plemelj encountered integral equations while still a student at Göttingen, when the Swedish professor Erik Holmgren gave a lecture on the work of his fellow countryman Fredholm on linear integral equations of the 1st and 2nd kind. Spurred on by Hilbert, Göttingen mathematicians attacked this new area of research and Plemelj was one of the first to publish original results on the question, applying the theory of integral equations to the study of harmonic functions in potential theory.
His most important work in potential theory is summarised in his 1911 book Researches in Potential Theory (Potentialtheoretische Untersuchungen), which received the Jablonowski Society award in Leipzig (1500 marks), and the Richard Lieben award from the University of Vienna (2000 crowns) for the most outstanding work in the field of pure and applied mathematics written by any kind of 'Austrian' mathematician in the previous three years.
His most original contribution is the elementary solution he provided for the Riemann–Hilbert problem f+ = g f− about the existence of a differential equation with given monodromy group. The solution, published in his 1908 article "Riemannian classes of functions with given monodromy group", rests on three formulas that now carry his name, which connect the values taken by a holomorphic function at the boundary of an arc Γ:
These formulae are variously called the Plemelj formulae, the Sokhotsky-Plemelj formulae, or sometimes (mainly in German literature) the Plemelj-Sokhotsky formulae, after the Russian mathematician Yulian Vasilievich Sokhotski (Юлиан Карл Васильевич Сохоцкий) (1842–1927).
From his methods on solving the Riemann problem had developed the theory of singular integral equations (MSC (2000) 45-Exx) which was entertained above all by the Russian school at the head of Nikoloz Muskhelishvili (Николай Иванович Мусхелищвили) (1891–1976).
Also important are Plemelj's contributions to the theory of analytic functions in solving the problem of uniformization of algebraic functions, contributions on formulation of the theorem of analytic extension of designs and treatises in algebra and in number theory.
1912 Plemelj published a very simple proof for the Fermat's last theorem for exponent n = 5, which was first given almost simultaneously by Dirichlet in 1828 and Legendre in 1830. Their proofs difficult, while Plemelj showed how to use the ring we get if we extend the rational numbers by √ 5.
His arrival in Ljubljana 1919 was very important for development of mathematics in Slovenia. As a good teacher he had raised several generations of mathematicians and engineers. His most famous student is Ivan Vidav. After the 2nd World War Slovenska akademija znanosti in umetnosti (Slovene Academy of Sciences and Arts) (SAZU) had published his three year's course of lectures for students of mathematics: Teorija analitičnih funkcij (The theory of analytic functions), (SAZU, Ljubljana 1953, pp XVI+516), Diferencialne in integralske enačbe. Teorija in uporaba (Differential and integral equations. The theory and the application).
Plemelj found a formula for a sum of normal derivatives of one layered potential in the internal or external region. He was pleased also with algebra and number theory, but he had published only few contributions from these fields — for example a book entitled Algebra in teorija števil (Algebra and the number theory) (SAZU, Ljubljana 1962, pp XIV+278) which was published abroad as his last work Problemi v smislu Riemanna in Kleina (Problems in the Sense of Riemann and Klein) (edition and translation by J. R. M. Radok, "Interscience Tract in Pure and Applied Mathematics", No. 16, Interscience Publishers: John Wiley & Sons, New York, London, Sydney 1964, pp VII+175). This work deals with questions which were of his most interests and examinations. His bibliography includes 33 units, from which 30 are scientific treatises and had been published among the others in a magazines such as: "Monatshefte für Mathematik und Physik", "Sitzungsberichte der kaiserlichen Akademie der Wissenschaften"; in Wien, "Jahresbericht der deutschen Mathematikervereinigung", "Gesellschaft deutscher Naturforscher und Ärzte" in Verhandlungen, "Bulletin des Sciences Mathematiques", "Obzornik za matematiko in fiziko" and "Publications mathematiques de l'Universite de Belgrade". When French mathematician Charles Émile Picard denoted Plemelj's works as "deux excellents memoires", Plemelj became known in mathematical world.
Plemelj was regular member of the SAZU since its foundation in 1938, corresponding member of the JAZU (Yugoslav Academy of Sciences and Arts) in Zagreb, Croatia since 1923, corresponding member of the SANU (Serbian Academy of Sciences and Arts) in Belgrade since 1930 (1931). 1954 he received the highest award for research in Slovenia the Prešeren award. The same year he was elected for corresponding member of Bavarian Academy of Sciences in Munich.
1963 for his 90th anniversary University of Ljubljana granted him title of the honorary doctor. Plemelj was first teacher of mathematics at Slovene university and 1949 became first honorary member of ZDMFAJ, (Yugoslav Union of societies of mathematicians, physicists and astronomers). He left his villa in Bled to the DMFA where today is his memorial room.
Plemelj did not do extra preparation for lectures; he didn't have any notes. He used to say that he thought over the lecture subject on the way from his home in Gradišče to the University. Students are said to have got the impression that he was creating teaching material on the spot and that they were witnessing the formation of something new. He was writing formulae on the table beautifully although they were composited from Greek, Latin or Gothic letters. He requested the same from students. They had to write distinct.
Plemelj is said to have had very refined ear for language and he had made a solid base for the development of the Slovene mathematical terminology. He had accustomed students for a fine language and above all for a clear and logical phraseology. For example, he would become angry if they used the word 'rabiti' (to use) instead of the word 'potrebovati' (to need). For this reason he said: "The engineer who does not know mathematics never needs it. But if he knows it, he uses it frequently".
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