2007/11/30 by Leonhard Euler, Euler, Leonhard
Arts and Humanities · Mathematics · Social Sciences · #01A50 #33-03 #33E20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Historical Philosophy and Science #Historical and Literary Studies #History and Overview (math.HO) #History and Theory of Mathematics #math.CA #math.HO #msc:01A50 #msc:33-03 #msc:33E20
paper · pdf · doi:10.48550/arxiv.0711.4986
5 pages, E565
arxiv created 2007/11/30 · openalex publication_date 2007/11/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
E565 in the Enestrom index. Translated from the Latin original, "De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet" (1775). Euler does not prove any results in this paper. It seems to me like he is trying to develop some general ideas about special functions. He gives some examples of numbers he claims but does not prove cannot be represented by definite integrals of algebraic functions. Euler has the idea that if we knew more about the function with the power series ∑ xtn where tn is the nth triangular number, this could lead to a proof of Fermat's theorem that every positive integer is the sum of three triangular numbers. This doesn't end of being fruitful for Euler, but in fact later Jacobi proves a lot of results like this with his theta functions. The last paragraph (§9) is not clear to me. My best reading is that there are infinitely many "levels" of transcendental numbers and that this is unexpected or remarkable.