An integral which occurs in statistics

In this note we give a direct evaluation of the integral whose value has been inferred from the theory of statistics. Here A = A p = (α μν ) and C = C p = ( C μν ) are real symmetrical matrices, of which A is positive definite; there are ½ p ( p + 1) independent variables of integration t μν (1 ≤ μ ≤ ν ≤ p ), and t μν is written also as t νμ for symmetry of notation; in the summation ∑ the variables μ, ν run independently from 1 to p; k is a real number. A word of explanation is necessary with regard to the determination of the power | A − iT | − k . Since A is positive definite and T real and symmetric, the roots of the equation