Integration in Finite Terms with Special Functions: Polylogarithmic Integrals, Logarithmic Integrals and Error Functions

Abstract

The thesis work concerns the problem of integration in finite terms with spe- cial functions. The main theorem extends the classical theorem of Liouville in the context of elementary functions to various classes of special functions: error functions, logarithmic integrals, dilogarithmic and trilogarithmic inte- grals. The results are important since they provide a necessary and sufficient condition for an element of the base field to have an antiderivative in a field extension generated by transcendental elementary functions and special func- tions. A special case of the theorem simplifies and generalizes Baddoura’s theorem for integration in finite terms with dilogarithmic integrals. The main theorem can be naturally generalized to include polylogarithmic inte- grals and to this end, a conjecture is stated for integration in finite terms with transcendental elementary functions and polylogarithmic integrals.