Il teorema di Liouville ovvero perchè “non esiste” la primitiva di $e^{x^2}$

De Lellis, Camillo

La Matematica nella società e nella cultura, Série 1, Tome 7 (2014) no. 1, pp. 55-97

Résumé

It is well known that the primitive of the real function $x \mapsto e^{x^2}$ cannot be expressed in terms of “elementary functions”. This theorem was first proved by Liouville in the nineteenth century. Nonetheless it seems that very few mathematicians know how it is proved. In this manuscript I explain the proof, following a modern version of Liouville’s ideas and based on the works of Rosenlicht and Ostrowski. The most important points are explained in a quite elementary way, but on the other hand the interested reader will find also a complete and quite detailed account of all the aspects, including the most technical ones. This note is indeed the written acount of a lecture given in the Liceo Cantonale di Bellinzona, as part of the conference “L’eredità di Evariste Galois, matematico e rivoluzionario. Convegno sulla teoria di Galois e le sue applicazioni”, organized by the Commissione di Matematica della Svizzera Italiana. It has already appeared in “Il Volterriano”, a journai published by the mathematics teachers from the Liceo di Mendrisio, who have kindly allowed me to publish it also here in a slightly modified form.