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.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.