Josip Plemelj

Josip Plemelj
Josip Plemelj in 1920
Born	December 11, 1873(1873-12-11) Grad na Bledu, Slovenia
Died	May 22, 1967(1967-05-22) (aged 93) Ljubljana, Slovenia

Josip Plemelj (December 11, 1873 – May 22, 1967) was a Slovene mathematician, whose main contributions were to the theory of analytic functions and the application of integral equations to potential theory.

1 Life
2 Research
3 A geometrical construction from his schooldays
4 See also
5 External links
6 Sources
7 References

Life

Plemelj was born in the village of Bled near Bled Castle in Austria-Hungary (now Slovenia); he died in Ljubljana, Yugoslavia (now Slovenia). His father, Urban, a carpenter and crofter, died when Josip was only a year old. 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. After leaving and obtaining the necessary examination results he went to 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.

On May 1898 Plemelj presented his doctoral thesis under Escherich's tutelage entitled O linearnih homogenih diferencialnih enačbah z enolično periodičnimi koeficienti (Über lineare homogene Differentialgleichungen mit eindeutigen periodischen Koeffizienten, About linear homogeneous differential equations with uniform periodical coefficients). He continued with his study in Berlin (1899/1900) under the German mathematicians Frobenius and Fuchs and in Göttingen (1900/1901) under Klein and Hilbert.

In April 1902 he became a private senior lecturer at the University of Vienna. In 1906 he was appointed assistant at the Technical University of Vienna. In 1907 he became associate professor and in 1908 full professor of mathematics at the University of Chernivtsi (Russian Черновцы), Ukraine. From 1912 to 1913 he was dean of this faculty. In 1917 his political views led him to be forcibly ejected by the Government and he fled to Bohemia (Moravska). After the First World War he became a member of the University Commission under the Slovene Provincial Government and helped establish the first Slovene university at Ljubljana, and was elected its first rector. In the same year he was appointed professor of mathematics at the Faculty of Arts. After the Second World War he joined the Faculty of Natural Science and Technology (FNT). He retired in 1957 after having lectured in mathematics for 40 years.

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),^[1] 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 $\Gamma$ :^[2]

$f_%2B(z)={1\over2i\pi} \int_\Gamma{\phi(t)-\phi(z)\over{t-z}}\, dt %2B \phi(z)$

$f(z)={1\over2i\pi} \int_\Gamma{\phi(t)-\phi(z)\over{t-z}}\, dt %2B {1\over2} \phi(z)$

$f_-(z)={1\over2i\pi} \int_\Gamma {\phi(t)-\phi(z)\over{t-z}}\, dt \quad z\in\Gamma$

These formulae are variously called the Plemelj formulae, the Sokhatsky–Weierstrass theorem, the Sokhotsky-Plemelj formulae, or sometimes (mainly in German literature) the Plemelj-Sokhotsky formulae, after the Russian mathematician Sokhotsky (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 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".

A geometrical construction from his schooldays

Plemelj had shown his great gift for mathematics early in elementary school. 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. At that time he discovered alone series for sin x and cos x. Actually he found a series for cyclometric function arccos x and after that he just inverted this series and then guessed a principle for coefficients. Yet he did not have a proof for that.

Plemelj had great joy for a difficult constructional tasks from geometry. 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. He started to occupy himself with mathematics in fourth and fifth class of high school. Beside in mathematics he was interested also in natural science and especially astronomy. He studied celestial mechanics already at high school. He liked observing the stars. His eyesight was so sharp he could see the planet Venus even in the daytime.

Let us hear about his early days in school in his own words: "It was the April 1891 in fifth class. The class had two rows of desks with crossing in between and I was sitting on the most side inside chair very rear. I think there were only two desks after me. Professor Borštner did not lecture. He had only given a lection from the book for the next lesson. He called to the blackboard two pupils and there he was discussing the subject and furthermore with this he included the whole class for cooperation. He used to have such a habit gratefully to give geometrical constructional tasks which he dictated from some collection he brought with. Once he gave amongst the other a task: Draught a triangle if one side, its altitude and a difference of two angles along it are given. Classmates had appealed to me before the lesson if the task was a little bit hard. They could not solve this task after several lessons and he had asked them the other. Borštner used to come from before the master's desk and he stopped ahead of me, sat toward me in the desk and hence he examined. After sometime he had said we should solve that task. Perhaps he was suspicious about that we had not yet solved it so he turned to me and asked me if I had tried this construction. I had said to him I could not find any path for the solution. Then he said he would show it in the next lesson. This had plucked up my courage to inspect it again. I had found a solution with subsidiary points, lines and so forth which seems to be inaccessible for human mind if the way which had inevitably led me to my aim is hidden from. Next lesson professor Borštner had sat toward me in the desk again. After customary examination of my classmates he said: "Well, let us work out that construction task." I whispered him I had succeeded till then and he said: "So, let me show how had you done this." He thought I would show him written on paper and he said: "Well, all-right." He had stepped aside and we went before the blackboard. I drew a triangle ΔABC with an ordinary analysis: Given side AB = c, its altitude v_c and the difference 0 < α − β < π.

Let us draw a perpendicular AA' from A to side AB and let AA' = 2 v_c. Let us bind A' with C in the way to be A'C = AC = b and draw A'B = m. Along the side A'C let us gather out along A' an angle α and on the left side of a triangle A'B' = c. On this way the risen triangle is ΔA'B'C ~ ΔABC. By B lies an angle <A'B'C = β and an angle <A'CB' = γ.

A triangle ΔBCB' is isosceles and <BCB' = 2 α, so we have < BB'C = π/2 − α. Now it is the angle <A'B'B = π/2 − α + β and we can construct over the side BA' a circumferential angle of a certain circle. We get a point B' at once because it is A'B' = c. Because a triangle ΔBCB' is isosceles a point C lies on a symmetric of the side BB' where it intersects a parallel with AB in a given altitude v_c. With this the triangle ΔABC is drawn. Professor Borštner was gazing when he saw this curious solution and he held of his head: "Aber um Hergottswillen, das ist doch harsträubend, das ist doch doch menschenunmöglich auf so einen Einfall zu kommen; sagen Sie mir doch, was hat Sie zu dieser Idee geführt!" I said to him I had not guessed this strange solution but I had asked myself about a trigonometric determination of a triangle because I could not find the solution in the other way. Geometric interpretation of this solution had led me up to this pure geometrically understandable construction. We did not spoke anymore about this with professor Borštner, but he had after that shown another easier solution, which I could imitate from my own construction and which I had not perceived because I had precisely traced way.

Trigonometric solution is easy: with altitude v_c from point C to side AB it breaks in two parts v_c cot α and v_c cot β. Then we have:

$v_c (\cot\alpha %2B \cot\beta) = c\quad \mbox{or}\quad v_c \sin(\alpha %2B \beta) = c \sin\alpha \sin\beta.$

We can then write:

$2v_c \sin(\alpha %2B \beta) = c (\cos(\alpha - \beta) - \cos (\alpha %2B \beta)).\,$

Because α + β = π − γ, this equation is:

$2 v_c \sin\gamma - c \cos\gamma = c\cos (\alpha - \beta).\,$

From this equation we have to obtain an angle γ. The easiest way is if we introduce a certain subsidiary angle μ. Namely we raise:

2 v_c = m cos μ c = m sin μ.

Both equations give us for μ a uniform certain acute angle and for m a certain positive length. The equation for γ is then:

$m \sin(\gamma - \mu) = c \cos(\alpha - \beta).\,$

We can consider this equation as a theorem of the sine for a certain triangle in which c and m are its sides and their opposite angles are γ − μ and π/2 ± (α − β) respectively ( lightgreen triangle on the picture below ). In this quoted construction this triangle is ΔBA'B', where BA' = m and A'B' = c, the angle <BB'A' = π/2 − α + α and the angle <A'BB' = γ − μ, as we can easily see. In my construction this requested triangle is drafted twice. I saw later on we can interpret above equation with a triangle which has a side AB already. This leads us to very beautiful and short construction. Requested triangle is ΔK'AB.

Straight line CK = a is symmetrically displaced in CK' = a and at the same time is AK = AK' = m. I had spoken as a professor in Černovice with two of my students about this elementary geometrical problem and I said that my high school teacher had dictated this task from a certain collection. They brought me a collection indeed where there was the exact picture from Borštner's construction. Unfortunately I had not written down a title of that book which was with no doubt professor Borštner's collection. Our teacher's library at classical gymnasium in Ljubljana does not have this book at present. But I got from there a Wiegand's book entitled Geometrijske naloge za višje gimnazije (Geometrische Aufgaben für Obergymnasien, Geometric exercises for upper gymnasiums), which does not have this exercise. I found in it a task: construct a triangle if we know one length of an angle symmetrical from one point, perpendicular on this symmetrical from the other point and an angle by the third different solutions. The last has annotation: at the night of the January 1, 1940 after the New Year's Eve 1939".

External links

Sources

Josip Plemelj, Iz mojega življenja in dela (From my life and work), Obzornik mat. fiz. 39 (1992) pp 188–192.

References

^ J. Plemelj, Potentialtheoretische Untersuchungen. Leipzig: B.G. Teubner, 1911, pp XIX+100. Volume 40 of the series Preisschriften der fürstl. Jablonowskischen Gesselschaft in Leipzig
^ J. Plemelj, Riemannsche Funktionenscharen mit gegebener Monodromiegruppe, Monatshefte für Mathematik und Physik, 19, W 1908, 211–246.

Persondata
Name	Plemelj, Josip
Alternative names
Short description
Date of birth	December 11, 1873
Place of birth	Grad na Bledu, Slovenia
Date of death	May 22, 1967
Place of death	Ljubljana, Slovenia