On the integration of the differential \(\frac{\rho dx}{\sqrt R}\), where \(R\) and \(\rho\) are polynomials. (Ueber die Integration der Differential-Formel \(\frac{\rho dx}{\sqrt R}\), wenn \(R\) und \(\rho\) ganze Functionen sind.) (German) ERAM 001.0021cj

Abel, N. H.

J. Reine Angew. Math. 1, 185-221 (1826).

Abel classifies all differentials of the form \(\frac{\rho\, dx}{\sqrt{R}}\), where \(\rho\) and \(R\) are polynomials of \(x\), whose integral can be expressed by a function of the form \(\log \frac{p+q\sqrt{R}}{p-q\sqrt{R}}\). In his investigations he is led to “Pell equations” for polynomials of the form \(Np^2 - Rq^2 = 1\), which he solves using continued fractions. Abel’s results were taken up again by P. Tchebicheff (Chebyshev), J. Math. Pures Appl. 2, 168–192 (1857); ibid. 9, 225–246 (1864)], and have found applications in the theory of hyperelliptic curves (see B. H. Gross [Acta Math. Vietnam. 37, No. 4, 579–588 (2012; Zbl 1294.11107)].

Reviewer: Franz Lemmermeyer (Jagstzell) (2014)

MSC:

34A05

Explicit solutions, first integrals of ordinary differential equations

Keywords:

polynomials; rational functions; continued fractions; polynomial Pell equation; hyperelliptic curves

Citations:

Zbl 1294.11107