|Born||March 12, 1859|
|Died||September 12, 1906 (aged 47)|
Torre Annunziata, Italy
Ernesto Cesàro (12 March 1859 – 12 September 1906) was an Italian mathematician who worked in the field of differential geometry. He wrote a book, Lezione di geometria intrinseca (Naples, 1890), on this topic, in which he also describes fractal, spacefilling curves, partly covered by the larger class of de Rham curves, but are still known today in his honor as Cesàro curves. He is known also for his 'averaging' method for the 'Cesàro-summation' of divergent series, known as the Cesàro mean.
After a rather disappointing start of his academic career and a journey through Europe - with the most important stop at Liege, where his older brother was teaching mineralogy at the local university - Ernesto Cesàro graduated from the University of Rome in 1887, while he was already part of the Royal Science Society of Belgium for his numerous works that he had already published.
The following year, he got a mathematics chair at the University of Palermo, which he kept until 1891. He settled in Rome, where he stayed as a professor at the Sapienza University until his accidental death, while trying to rescue his youngest son Manlio from drowning.
Cesàro's main contributions are in the field of differential geometry. Lessons of intrinsic geometry, written in 1894, explains in particular the construction of a fractal curve. After that, Cesàro also studied the "snowflake curve" of Koch, continuous but not differentiable in all its points.
Among its other remarkable works, the following are worth to cite: Introduction to the mathematical theory of infinitesimal calculus (1893), Algebraic analysis (1894), Elements of infinitesimal calculus (1897). He proposes a possible definition of a divergent sequence, known today as "Cesàro's sum," given by the limit of the mean of the sequence partial terms' sum.
Books by E. Cesàro
- Lezioni di geometria intrinseca (Naples, 1896) ISBN 1-4297-0344-X (trans. into German under the title Vorlesungen über natürliche Geometrie; 1901, 1st edn.; 1926, 2nd edn. trans. and with an appendix by Gerhard Kowalewski)
- Elementi di calcolo infinitesimale con numerose applicazioni geometriche (L. Alvano, Naples, 1905) ISBN 1-4297-0178-1
- Corso di analisi algebrica con introduzione al calcolo infinitesimale (Bocca, Torino, 1894) ISBN 1-4297-0100-5
- Stolz–Cesàro theorem
- Cesàro's theorem
- Cesàro equation
- Cesàro mean
- Cesàro summation
- Cesàro curve
- Lévy C curve
- Kasner, Edward (1930). "Review: Lezioni di geometria intrinseca trans. as Vorlesungen über natürliche Geometrie" (PDF). Bull. Amer. Math. Soc. 36 (1): 32. doi:10.1090/s0002-9904-1930-04873-9.
- O'Connor, John J.; Robertson, Edmund F., "Ernesto Cesàro", MacTutor History of Mathematics archive, University of St Andrews.
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