  In this note, we study the asymptotic of spherical integrals, which are
analytical extension in index of the normalized Schur polynomials for $\beta
=2$ , and of Jack symmetric polynomials otherwise. Such integrals are the
multiplicative counterparts of the Harish-Chandra-Itzykson-Zuber (HCIZ)
integrals, whose asymptotic are given by the so-called $R$-transform when one
of the matrix is of rank one. We argue by a saddle-point analysis that a
similar result holds for all $\beta >0$ in the multiplicative case, where the
asymptotic is governed by the logarithm of the $S$-transform. As a consequence
of this result one can calculate the asymptotic behavior of complete
homogeneous symmetric polynomials.
