  We give a Fourier-type formula for computing the orthogonal Weingarten
formula. The Weingarten calculus was introduced as a systematic method to
compute integrals of polynomials with respect to Haar measure over classical
groups. Although a Fourier-type formula was known in the unitary case, the
orthogonal counterpart was not known. It relies on the Jack polynomial
generalization of both Schur and zonal polynomials. This formula substantially
reduces the complexity involved in the computation of Weingarten formulas. We
also describe a few more new properties of the Weingarten formula, state a
conjecture and give a table of values.
