Niels Henrik Abel Et lidet Bidrag til Læren om adskillige transcendente Functioner

Abstract

Abel’s paper, titled «Et lidet Bidrag til Læren om adskillige transcendente Functioner» (which in Lie’s and Sylow’s edition from 1881 of Abel’s Oeuvres Complètes got the title «Petite contribution à la théorie de quelques fonctions transcendantes»), was submitted to DKNV’s publication series March 22, 1826. It was communicated by Christopher Hansteen, one of Abel’s mentors in Oslo (at that time named Christiania), and it was published in volume 2 (1824-1827), p.177-207. The paper was written by Abel just before he embarked on his journey abroad, which lasted from September 1825until May 1827, where he visited Berlin and Paris, but not Göttingen and Gauss as was originally planned.

The main result of his paper is a vast generalization of a result proved by Legendre for elliptic integrals of the third kind, which later in German mathematical literature was named «Die Vertauschung von Parameter und Argument» (the exchange of parameter and argument). In Legendre’s two-volume monograph «Exercises de Calcul intégral», which Abel studied assiduously in the autumn of 1823, elliptic integrals were reduced to three normal types, the first, second and third kind, and many beautiful identifies between these were shown. Abel soon went way beyond Legendre’s results. He inverted the elliptic integral of the first kind and showed that it was a meromorphic function with two periods, what subsequently was called an elliptic function. Then he generalized Legendre’s «Vertauschungsatz» to hyperelliptic integrals, (..., see the pdf).

Abel goes far beyond the hyperelliptic case, though. He realized that what made Legendre’s proof work was that the square root under the integral satisfies a homogeneous linear differential equation of degree one with polynomials in the variable as coefficients. He therefore proved a much more general, but analogous identity as (*, see the pdf), and this is contained in the publication under consideration.

Later he generalized this further by considering solutions of homogeneous linear differential equations of any degree (with polynomials in the variable as coefficients). He wrote a paper on this which was posthumously published. This was later taken up Jacobi, Fuchs and Frobenius, and they published papers which largely reformulated Abel’s results in terms of new settings and structures that had been introduced in the meantime.