Multidimensional Stationary Phase an Alternative Derivation

Abstract

The method of multidimensional stationary phase is derived via a technique which makes strong use of integration by parts. The “diagonalization” of the matrix of second derivatives at the stationary point is carried out here in such a manner as to make all coefficients in the exponent $ \pm 1$. This modification of existing technique allows for the explicit calculation of the nth term of the asymptotic expansion in a closed form which involves the amplitude of the integrand in transformed coordinates. The first correction term in the multidimensional stationary phase formula is readily calculated from this result.