Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms

Details Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms

2008-11-18

arxiv.org/abs/0811.3004v1

Mathematics

Classical Analysis and ODEs

66 pages, 1 figure

Scientific paper

Let F be a characteristic zero differential field with an algebraically closed field of constants, E be a no-new-constant extension of F by antiderivatives of F and let y1, ..., yn be antiderivatives of E. The antiderivatives y1, ..., yn of E are called J-I-E antiderivatives if the derivatives of yi in E satisfies certain conditions. We will discuss a new proof for the Kolchin-Ostrowski theorem and generalize this theorem for a tower of extensions by J-I-E antiderivatives and use this generalized version of the theorem to classify the finitely differentially generated subfields of this tower. In the process, we will show that the J-I-E antiderivatives are algebraically independent over the ground differential field. An example of a J-I-E tower is extensions by iterated logarithms. We will discuss the normality of extensions by iterated logarithms and produce an algorithm to compute its finitely differentially generated subfields.

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