  We consider a sub-critical Gaussian multiplicative chaos (GMC) measure
defined on the unit interval [0,1] and prove an exact formula for the
fractional moments of the total mass of this measure. Our formula includes the
case where log-singularities (also called insertion points) are added in 0 and
1, the most general case predicted by the Selberg integral. The idea to perform
this computation is to introduce certain auxiliary functions resembling
holomorphic observables of conformal field theory that will be solutions of
hypergeometric equations. Solving these equations then provides non-trivial
relations that completely determine the moments we wish to compute. We also
include a detailed discussion of the so-called reflection coefficients
appearing in tail expansions of GMC measures and in Liouville theory. Our
theorem provides an exact value for one of these coefficients. Lastly we
mention some additional applications to small deviations for GMC measures, to
the behavior of the maximum of the log-correlated field on the interval and to
random hermitian matrices.
