Distributions of the form
$$F\left(x,\lambda \right)=\frac{1}{\Gamma \left(\frac{\lambda +1}{2}\right)}\int \phantom{\rule{0.3em}{0ex}}f\left(x,u\right){}^{\lambda}g\left(x,u\right)\phantom{\rule{0.3em}{0ex}}du$$  (1) 
are considered, where
$x$
and
$u$ belong
to
${R}^{p}$ and
${R}^{n}$ respectively. The
parameter
$\lambda $ is
complex, and
$F\left(x,\lambda \right)$ is
evaluated for
$Re\left(\lambda \right)<0$ by
analytic continuation. Such integrals arise in solution formulas for partial differential equations.
In case
$n=1$
or
$n=2$,
$F$
is expressed in terms of homogeneous distributions of degree
$>\lambda +\alpha $,
where
$\alpha $
is nonnegative and depends upon the geometry of the roots of
$f$. The case of general
$n$ is also treated, in
case the Hessian of
$f$
with respect to
$u$
is different from zero. The results lead to asymptotic expansions of analogous
multiple integrals.
