Saddle-Point Method for a Complex Function of Several Arguments

Abstract

Let x represent M real variables x ₁,..., x _m, and let F(x) and G(x) be complex-valued functions of x with Re F ≤ 0 in the volume V in $\mathcal{R}^M$ over which we are going to integrate. Suppose that V contains a single nondegenerate stationary point x ₀ with ∂ _xi F(x ₀) = 0, i = 1,..., M. Let us denote by Q _m×m the matrix of the negative second derivatives, i.e. (Q _m×m) _ik = - ∂ _xi ∂ _xk F(x) taken at x = x ₀. The condition of nondegeneracy means detQ _m × m ≠ 0. We then have, for J → ∞ [190],

$$ \int_V {d^M xe^{JF(x)} G(x)} = G(x_0 )\sqrt {\frac{{(2\pi )^M }} {{J^M |\det Q_{M \times M} |}}} e^{JF(x_0 ) - (i/2)IndQ_{M \times M} } [1 + \mathcal{O}(1/J)]. $$

((A.1))

Here IndQ _{m × m} is the index of the complex quadratic form χ of M real variables,

$$ \chi = \sum\limits_{i,j = 1}^M {(Q_{M \times M} )_{ij} x_i x_j .} $$

((A.2))