His mother Marija, née Mrak, found bringing up the family alone very hard, but she was able to send her son to school in Ljubljana, where Plemelj studied from 1886 to 1894.
Due to a bench thrown into Tivoli Pond by him or his friends, he could not attend the school after he finished the fourth class and had to pass the final exam privately.
[2] After leaving and obtaining the necessary examination results he went to the University of Vienna in 1894 where he had applied to Faculty of Arts to study mathematics, physics and astronomy.
His professors in Vienna were von Escherich for mathematical analysis, Gegenbauer and Mertens for arithmetic and algebra, Weiss for astronomy, Stefan's student Boltzmann for physics.
He mastered the whole of the high school syllabus by the beginning of the fourth year and began to tutor students for their graduation examinations.
From his high school days originates an elementary problem — his later construction of regular sevenfold polygon inscribed in a circle otherwise exactly and not approximately with simple solution as an angle trisection which was yet not known in those days and which necessarily leads to the old Indian or Babylonian approximate construction.
His most important work in potential theory is summarised in his 1911 book Potentialtheoretische Untersuchungen (Studies in Potential Theory),[3] 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.
Based on his methods of solving the Riemann problem he had developed the theory of singular integral equations (MSC (2000) 45-Exx) which was used above all by the Russian school at the head of Nikoloz Muskhelishvili (Николай Иванович Мусхелишвили) (1891–1976).
After the Second World War Slovenska akademija znanosti in umetnosti (Slovene Academy of Sciences and Arts) (SAZU) had published his three-year 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.
Plemelj found a formula for a sum of normal derivatives of one-layered potential in the internal or external region.
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.
Plemelj was first teacher of mathematics at Slovene university and 1949 became first honorary member of the ZDMFAJ (Yugoslav Union of Societies of Mathematicians, Physicists, and Astronomers).
Plemelj had a very refined ear for languages and created a solid base for the development of Slovene mathematical terminology.