Saddle point method in complex plane

Summary of the saddle point method [1].

The saddle point method is also known as the method of steepest descent or stationary phase. It estimates the asymptotic behavior of an integral of the following form

Eq. (1)

as $𝜆\to +\infty .$ Here, $𝐶$ is a contour in the complex $𝑧$ plane and $𝑆$ is a holomorphic function.

In order to evaluate this integral, we deform $𝐶$ such that it passes through the saddle points of $𝑆$ (where ${𝑆}^{\prime }\left(𝑧\right)=0)$ in the steepest descent directions. The idea of the saddle point method is that, as $𝜆\to +\infty ,$ the main contribution to the integral comes from the neighborhood of the saddle points. For simplicity, let’s assume that $𝑆$ has only one saddle point $𝑧={𝑧}_0;$ if $𝑆$ has multiple saddle points, we need only to sum the contributions from each. Taylor expanding [2] $𝑆$ about ${𝑧}_0$ and noting that the Gaussian profile has width $\left|𝑧-{𝑧}_0\right|=\mathrm{𝒪︀}\left({𝜆}^{-1/2}\right),$ we have

Eq. (2)

Now we need to determine the direction of the steepest descent. Let $𝑧-{𝑧}_0=\left(𝜌/\sqrt{𝜆}\right){\mathrm{e}}^{\mathrm{i}𝜃}$ and ${𝑆}^{″}\left({𝑧}_0\right)=\left|{𝑆}^{″}\left({𝑧}_0\right)\right|{\mathrm{e}}^{\mathrm{i}𝛼},$ then the real part of the exponent

Eq. (3)

decreases most rapidly when $cos\left(2𝜃+𝛼\right)=-1,$ i.e., when $𝜃=\pm \mathrm{\pi }/2-𝛼/2\equiv {𝜃}_0$ [3]. Choosing $𝐶$ to pass through ${𝑧}_0$ along the direction determined by the proper $𝜃={𝜃}_0,$ we find

Eq. (4)

In physical problems, one often encounter integrals of the following form

Eq. (5)

with $𝑓$ being holomorphic. In this case, we may repeat the above derivation to obtain

Eq. (6)

For more physical applications of the saddle point method, see [1] and [4].