Integrals whose saddle-point expansions terminate

Abstract

The saddle-point expansion for integrals with integrand exp(−kf(x)) is a series in powers of 1/k. Usually this series diverges, but there is a family of exponent functions f(x), defining a family of canonical integrals, for which the series terminates and the saddle-point expansion is exact. For this family, the transformation x → X such that f(x) = X 2 possesses a Jacobian that is a polynomial in X, whose coefficients parameterise the canonical integrals.