2023/08/01 by Kumbhakar, Partha, Srinivasan, Varadharaj R. · 1 citation
#12Fxx #12H05 #14H52 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2308.00659
Let k be a differential field of characteristic zero and the field of constants C of k be an algebraically closed field. Let E be a differential field extension of k having C as its field of constants and that E=Em⊇ Em-1⊇⋯⊇ E1⊇ E0=k, where Ei is either an elementary extension of Ei-1 or Ei=Ei-1(ti, t'i) and ti is weierstrassian (in the sense of Kolchin ([Page 803, Kolchin1953]) over Ei-1 or Ei is a Picard-Vessiot extension of Ei-1 having a differential Galois group isomorphic to either the special linear group SL2(C) or the infinite dihedral subgroup D_∞ of SL2(C). In this article, we prove that Liouville's theorem on integration in finite terms ([Theorem, Rosenlicht1968]) holds for E. That is, if η∈ E and η'∈ k then there is a positive integer n and for i=1,2,…,n, there are elements ci∈ C, ui∈ k∖ \0\ and v∈ k such that η'=∑ni=1ci(u'i)/(ui)+v'.