From analytic to algebraic methods, Liouville's approach to integration in finite terms (Q1267404)

Let \(y(x)\) be an algebraic function. Decide if its indefinite integral can be expressed in terms of algebraic functions, logarithms and exponentials, and if the answer is positive, find the integral. Liouville, and before him Abel, proved that the integral must be algebraic or of the first order, no exponentials are needed, and the logarithms enter in a simple linear way. According to Abel the logarithmic terms have rational functions of \(x\) and \(y\) as their arguments. Liouville also dealt with the more general problem of integration of differential equations by quadrature. In his early and previously unpublished notes he prefered an analytical approach, considering polynomials as functions and relying on arguments using limits, asymptotic behavior and series expansions. These are absent in his later publications where algebraic techniques on polynomials prevail. The subjects are treated in greater detail in the author's book `Joseph Liouville, 1809-1882: master of pure and applied mathematics. Springer Verlag (1990; Zbl 0701.01015)'.