A Liouvillian function is a function lying in a finite tower of differential-field extensions obtained by adjoining algebraic functions,
antiderivatives, and exponentials of antiderivatives, starting from the rational
functions (Rosenlicht 1972). This permits more than elementary closed
forms because integration is allowed. For example, the error
function is Liouvillian, since it is an integral of 
, but is not elementary.
Wang and Lim (2026) proved that even this larger class does not provide general endpoint geodesic formulas for the Stiefel
manifold 
under its canonical metric when 
and 
.
