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.



Steepest descent integration in several dimensions
You can look for **Multivariable Morse Lemma** to get an extension of Steepest Descent into multiple complex variables $z=[z\_1, z\_2, ...,z\_n]$. Higher dimensional asymptotics as $M\rightarrow\infty$ for this multiple integral with $f:\mathbb{C}^n\rightarrow\mathbb{C}$ and $C=C\_1\times C\_2\times ...\times C\_n$ a multiple complex contour domain, $$J(M)=\int\_C e^{M f(z)}dz$$ is obtained taking $z\_0=[z\_{10}, z\_{20}, ...,z\_{n0}]$, the point where $f'(z\_0)=\nabla f(z)|\_{z\_0}=0$ then, under regularity conditions (single non-degenerate saddle point), $$J(M)=\frac{(2\pi/M)^{\frac{n}{2}}e^{Mf(z\_0)}}{\sqrt{\det{[-f''(z\_0)]}}}[1+O(1/M)]$$ here $f''(z)$ is the Hessian matrix having eigenvalues $\lambda=[\lambda\_1,\lambda\_2,...,\lambda\_n]$ with $|\arg(-\lambda\_k)|<\frac{\pi}{2}$ $$\sqrt{\det{[-f''(z\_0)]}}=e^{\frac{i}{2}\phi}\prod\_{k=1}^n|\lambda\_k|^{1/2}\ne0$$ where $\phi=\sum\_{k=1}^n \arg(-\lambda\_k)$. Note that if $z\_0\in\mathbb{R}^n$ and $\Im[f(z)]=0\ \ \forall z\in\mathbb{R}^n\ $ then $\ \phi=0$.
Also if $z\_0\in\mathbb{R}^n,\ \Re[f(z)]=0\ \ \forall z\in\mathbb{R}^n\ $ and $|\arg\sqrt{-\lambda\_k}|\le\frac{\pi}{4}$ then $\ \phi=m\cdot\frac{\pi}{2}\ $ where $m$ is the number of negative eigenvalues minus the number of positive ones (stationary phase method).


The case of multiple saddle points $z\_0^{(\ell)}$ s.t. $f'(z\_0^{(\ell)})=0$ and $\det{[-f''(z\_0^{(\ell)})]}\ne0,\ \ \ell=1,2,...L$ is worked the same way. This gives $$J(M)=\sum\_{\ell=1}^L\frac{(2\pi/M)^{\frac{n}{2}}e^{Mf(z\_0^{(\ell)})}}{\sqrt{\det{[-f''(z\_0^{(\ell)})]}}}[1+O(1/M)]$$


Chapters VIII-IX in


*Wong, R.*, [**Asymptotic approximations of integrals**](http://dx.doi.org/10.1137/1.9780898719260), Classics in Applied Mathematics 34. Philadelphia, PA: SIAM (ISBN 0-89871-497-4/pbk). xvii, 543 p. (2001). [ZBL1078.41001](https://zbmath.org/?q=an:1078.41001).


provide asymptotic methods for multidimensional integrals. Cases of degenerate saddles where Hessian vanishes $\det[-f''(z\_0^{(\ell)})]=0$ for some $\ell$ is more complex. They are found in Ch VIII sect. 5 pg. 435 and Ch IX sect. 4 pg 491.


I hope this helps.


