I have two terminology questions regarding the method of steepest descent:
- This section of the Wikipedia page defines
A non-degenerate saddle point, $z^0 ∈ \mathbb{C}^n$, of a holomorphic function $S(z)$ is a point where the function reaches an extremum (i.e., $∇S(z^0) = 0$) and has a non-vanishing determinant of the Hessian (i.e., $\det S_{zz}″(z^0) ≠ 0$).
I understand the "non-degenerate" part regarding the invertibility of the Hessian matrix. But why is a point satisfying $∇S(z^0) = 0$ called a "saddle point" instead of a "critical point", as is standard in other contexts? This terminology conflicts with the usual definition of a "saddle point", which is basically a special case of a critical point which is not a local extremum (although the notion of an "extremum" is somewhat ambiguous for complex-valued functions).
- Why is it called the method of "steepest descent"? What is descending steeply? It seems to me that the whole point of the method is to deform the contour to a stationary point, where the integrand is not descending (or ascending) at all.
Best Answer
$∇S(z^0) = 0$ automatically implies a saddle point for holomorphic functions because of the maximum modulus principle if $z^0$ is not a zero of $S$. Consider any open set $U$ containing $z^0$, then $S(z^0)$ cannot be a minimum or a maximum of $Z$ in $U$ according to that theorem. If $z^0$ is indeed a zero of $S$, then thanks to the fact the hessian has a non vanishing determinant at this point implies that $z^0$ is still a saddle point.
That point is more complex and would require a long answer. You can find some information with this post .