  We consider a function $U=e^{-f_0}\prod_j^N f_j^{\alpha_j}$ on a real affine
space, here $f_0,..,f_N$ are linear functions, $\alpha_1, ...,\alpha_N$ complex
numbers. The zeros of the functions $f_1, ..., f_N$ form an arrangement of
hyperplanes in the affine space. We study the period matrix of the
hypergeometric integrals associated with the arrangement and the function $U$
and compute its determinant as an alternating product of gamma functions and
critical points of the functions $f_0,..., f_N$ with respect to the
arrangement. In the simplest example, $N=1, f_0=f_1=t$, the determinant formula
takes the form $\int_0^\infty e^{-t} t^{\alpha -1} dt=\Gamma (\alpha).$
  We also give a determinant formula for Selberg type exponential integrals. In
this case the arangements of hyperplanes is special and admits a symmetry
group, the period matrix is decomposed into blocks corresponding to different
representations of the symmetry group on the space of the hypergeometric
integrals associated with the arrangement. We compute the determinant of the
block corresponding to the trivial representation.
