Elementary functions based on elliptic curves (Q1346802)

The author slightly generalizes the notion of elementarity to Weierstrassian elements and proves a new theorem, a particular case of which was Abel's theorem [see \textit{N. H. Abel}, Précis d'une théorie des fonctions elliptiques, J. Math. 1, 185-221 (1826)]. Let \(K\) be an ordinary differential field of characteristic 0 with a single differentiation \(D\) and the field of constants \(C\) of \(K\) is algebraically closed. The main result of the paper is: Theorem. Let \(a\) be an element of \(K\) and suppose that there exists a \(w\)-elementary extension of \(K\) in which an integral of \(a\) is contained. Then \(a\) can be written in the form \[ a = Db + \sum_j \alpha_j {Db_j \over b_j} + \sum_i \beta_i {Du_i \over v_i} + \sum_i \gamma_i u_i {Du_i \over v_i} + \sum_i \sum_{Q \in E_i (C)^*} \delta_i (Q) {v_i + v_i (Q) \over u_i + u_i (Q)} {Du_i \over v_i}, \] where the sums denote finite sums, \(b\), \(b_j \in K,i\) runs through an index set of a family of elliptic curves \(E_i\) defined over \(C\), \((1, u_i, v_i) \in E_i (K)\), and \(\alpha_i, \beta_i, \gamma_i, \delta_i (Q) \in C\).