Integration in terms of exponential integrals and incomplete gamma functions

Abstract

Indefinite integration means that given f in some set we want to find g from possibly larger set such that f = g '. When f and g are required to be elementary functions due to work of among others Risch, Rothstein, Trager, Bronstein (see [1] for references) integration problem is now solved at least in theory. In his thesis Cherry gave algorithm to integrate transcendental elementary functions in terms of exponential integrals. In [2] he gave algorithm to integrate transcendental elementary functions in so called reduced fields in terms of error functions. Knowles [3] and [4] extended this allowing also liovillian integrands and weakened restrictions on the field containing integrands. We extend previous results allowing incomplete gamma function Γ( a , x ) with rational a . Also, our theory can handle algebraic extensions and is complete jointly (and not only separately for Ei and erf). In purely transcendental case our method should be more efficient and easier to implement than [2]. In fact, it seems that no system currently implements algorithm from [2], while partial implementation of our method in FriCAS works well enough to be turned on by default. With our approach non-reduced case from [2] can be handled easily. We hope that other classes of special functions can be handled in a similar way, in particular irrational case of incomplete gamma function and polylogarithms (however polylogarithms raise tricky theoretical questions).