# Steepest descent integration in several dimensions

The method of steepest descent provides an asymptotic approximation for integrals of the form:

$$I = \int_C \exp(M f(z))\mathrm dz$$

for large positive $$M$$, where $$f(z)$$ is analytic in the region of interest, $$C$$ a contour and $$f(z)$$ goes to zero at the endpoints of the contour. The asymptotic approximation is:

$$I \sim \sqrt{\frac{2\pi}{f''(z_0)}}\exp(M f(z_0)+i\theta)$$

where $$z_0$$ is a saddle-point of $$f$$ ($$f'(z_0)=0$$) such that the original contour $$C$$ can be deformed (fixing the endpoints) to pass through $$z_0$$ in a steepest descent direction (which defines the angle $$\theta$$) and assuming that it doesn't go through any other saddle-points for simplicity.

Is there a generalization for higher-dimensional integrals? For example,

$$J = \int_{C_1}\mathrm dz_1\int_{C_2}\mathrm dz_2 \exp(M g(z_1,z_2))$$

I haven't studied much of multi-dimensional complex analysis, so I'm not sure what conditions should be imposed on $$g(z_1,z_2)$$. But as suggested in (https://en.wikipedia.org/wiki/Several_complex_variables), say that $$g(z_1,z_2$$ can be represented as a convergent power series in the region of interest.