# 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.

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, Classics in Applied Mathematics 34. Philadelphia, PA: SIAM (ISBN 0-89871-497-4/pbk). xvii, 543 p. (2001). ZBL1078.41001.

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.

• I just realized that Q was made several years ago. Anyway I won't erase this answer. I expect it will be further useful for someone. Mar 26, 2022 at 4:13
• Thanks, it is very useful even though the question is old. Please don't delete your answer.
– a06e
Mar 26, 2022 at 21:24

You can iterate the the integrals assuming there is a saddle point, using the implicit function theorem and a bit of work, as explained here.

The key is expressing the the variables in terms of each other (using implicit function theorem), and having a non vanishing Hessian determinant.

You can find a referenced C^N saddle point method here:

Pinna, Francesco; Viola, Carlo, The saddle-point method in (\mathbb{C}^N) and the generalized Airy functions, Bull. Soc. Math. Fr. 147, No. 2, 221-257 (2019). ZBL1472.41018.