I am just doing some research on the saddlepoint approximation and before I delved to deep into actually applying the approximation method, I was just curious as to why it is in fact called a saddlepoint approximation. I know that a saddlepoint in calculus is a point that can be a maximum or a minimum depending on the direction that you are looking at the graph, but I wasn't sure how this had anything to do with the technique. Thanks!

As you probably know, the saddlepoint method (also known as or stationary phase method or method of steepest descent) used for computing integrals of the form

$$I = \int_{\gamma} f\left(z\right) e^{\lambda g\left(z\right)}dz, \qquad z \in \mathbb C, \qquad 0 \ll \lambda \in \mathbb R%, \qquad \gamma \quad \text{ is a contour }.$$

over a contour $\gamma$ in the complex plane, where both $f$ and $g$ are analytic functions. It is essentially the same as Laplace's method but modified for taking integrals over complex plane.

The idea of the method is to deform contour $\gamma$ in a way that it passes through a saddle point $z_0$ of function $g(z)$, so that at this point function $g$ has a maximum if you move along direction of steepest descend. Recall that analytic functions do not have minima or maxima, but only saddle-points in the domain of analyticity, and that the value of a contour integral of an analytical function does not change if you deform contour continuously. Since real constant $\lambda\gg 0$ is supposed to be large, it determines range of magnitude of integrand in the vicinity of saddle-point $z_0$. Thus, deforming $C$ in a way so that it passes through the $z_0$ in the direction of steepest descent we can ensure that the integrand will be decreasing in magnitude nearly as quickly as possible.

To summarize, the reason the it is called "saddle point method" is because finding saddle points is the crucial step in the algorithm:

1. Write integral in the form $\displaystyle\qquad I\left(\lambda\right) = \int_{\gamma}f\left(z\right)e^{\lambda g\left(z\right)}dz$.

2. Since the exponent determines behavior of $\,I(\lambda)$ as $\lambda\to\infty,\,$ investigate $g(z)$ in a following way:

• find saddle point(s) of $g \iff$ find all points where $g'(z)=0$ (since $g$ is analytic)
• construct paths of steepest descent
3. Deform $\gamma$ to match path of steepest descent.

4. Obtain asymptotical approximation of the integral $I(\lambda)$ using Laplace's Method.