Finiteness property for generalized abelian integrals

We study the integrals of real functions which are finite compositions of globally subanalytic maps and real power functions. These functions have finiteness properties very similar to those of subanalytic functions. Our aim is to investigate how such finiteness properties can remain when taking the integrals of such functions. The main result is that for almost all power maps arising in a x λ -function, its integration leads to a non-oscillating function. This can be seen as a generalization of Varchenko and Khovanskii’s finiteness theorems for abelian integrals.