extend the rational functions by the natural logarithm and
exponential function (see algebraic expression domains )
Theorem (Liouville) Let be a differential
field and be from . Then, an
elementary extension of the field , which has the
same field of constants as and contains an element
such that , exists if and only
if there exist constants from and functions from such that
i.e.,
Risch 1968-1969 - first decision procedure for the integration
of elementary transcendental and algebraic functions; the
procedure determines if the integral exists within a given class
of functions and if so, then calculates the value of the integral