The Weyl integration formula is a result in the field of functional analysis and operator theory that relates the eigenvalues and eigenvectors of self-adjoint operators to the integration of certain functions over the spectrum of the operator. Specifically, it provides a way to express the integral of a function of an operator in terms of its eigensystem.
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Let be a connected compact Lie group, a maximal torus, and its Weyl group. With normalized Haar measures and , every continuous class function satisfies . For a general continuous function, replace by its average . Conjugation from covers the regular elements times, and its real Jacobian determinant is .