**Dimitrie Pompeiu** was a Romanian mathematician who worked in mathematical analysis, complex function theory and rational mechanics.

- After being given leave of absence from school teaching, Pompeiu went to Paris in 1898 to continue his mathematical studies.
- For the moment let us continue to outline Pompeiu's career.
- The second of the pair of famous first generation of Romanian mathematicians was David Emmanuel and, when Emmanuel retired in 1930, Pompeiu was named Professor of the Theory of Functions to succeed him.
- A small selection of the papers published by Pompeiu following his doctoral thesis are: Sur les fonctions dérivées Ⓣ(On derived functions) (1907), Sur un Exemple de Fonction Analytique Partout Continue Ⓣ(On an example of an everywhere continuous analytical function) (1910), Sur une équation intégrale Ⓣ(On an integral equation) (1913), Sur les équations fonctionnelles des polynômes à variables réelles Ⓣ(On functional equations of real variables polynomials) (1934), Du point à l'infini comme point singulier isolé Ⓣ(The point at infinity as isolated singular point) (1938), Remarques sur l'équation de Riccati Ⓣ(Notes on the Riccati equation) (1940), La géométrie et les imaginaires: démonstration de quelques théorèmes élémentaires Ⓣ(Geometry and the imaginary: Demonstration of some elementary theorems) (1940), and De la définition du pôle en théorie des fonctions Ⓣ(On the definition of a pole in complex analysis) (1940).
- We promised to return to discuss Pompeiu's doctoral thesis.
- However, Pompeiu's doctoral thesis written in the same year proved the existence of certain analytic functions which could be extended continuously on their set of singularities even though this set had positive measure.
- Clearly both results could not be correct and the difficulty was resolved in 1909 when Denjoy confirmed that Pompeiu's results were correct, and he found the error in Zoritti's theorems.
- Pompeiu's examples had been constructed using ideas due to Koepcke and they were difficult to understand.
- However, in 1907 Pompeiu had clarified the whole situation by constructing simpler examples in his paper Sur les fonctions dérivées Ⓣ(On derived functions) which we mentioned above.
- The functions which he constructed in this paper are now called 'Pompeiu functions'.
- allows Pompeiu to see the compact subsets in the plane as the elements of another set and to define in a natural way limits, closure, etc.
- Consequently, Pompeiu is also considered as one of the founders of the theory of hyperspaces.
- In this work Hausdorff gives a slightly different definition of distance between sets but he also credits Pompeiu's work and shows that the two definitions give the same topology.
- Pompeiu initiated the theory of polygenus functions as a natural extension of analytic functions.
- He introduced the notion of a special type of derivative, the areolar derivative of a complex function, extending the Cauchy formula which today is sometimes called the Cauchy-Pompeiu formula.
- This simple remark has led to many interesting problems in analysis known as the problem of Pompeiu.
- Among other topics on which Pompeiu published research articles we mention interpolation theory.
- Finally let us mention that Pompeiu, along with Petru Sergescu, founded the journal Mathematica (Cluj) and he was its first editor.
- Pompeiu was honoured with election to the Romanian Academy of Sciences in 1934.

Born 22 September 1873, Brosca, near Dorohoi, Romania. Died 8 October 1954, Bucharest, Romania.

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Origin Romania

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