# When are method of stationary phase and method of steepest descent are equivalent?

Normally, when we deal with "real-valued(or pure imagenary-valued) phase" oscillatory integral, for example,finding the asymptotics of $$\int_{[a,b]}f(s)e^{\rho(s)t}ds$$ when $$t$$ goes to positive infinity, where in general $$f:\mathbb{R}\rightarrow \mathbb{C}$$ and $$\rho:\mathbb{R}\rightarrow \mathbb{R}$$, the Laplace's method, Watson's Lemma and the method of Stationary are often the methods coming first. While when we replace the integral into a contour integral, i.e. complex integral, then there comes the method of steepest descent, which based on the "critical points" including saddle point, endpoints.

Simply, My question is if all the critical points are real, and the contour is also real line, then the method of steepest descent is equal to the method of stationary phase, provided enough regularity for the function $$f$$ in the above example and there are no essential singularities.

More specifically, consider $$I(t)=\int_\mathbb{R}f(s)e^{ip(s)t}ds$$ where $$f\in C^\infty(\mathbb{R})$$ and $$p$$ is a polynomial. Question is about the equivalence between stationary and steepest descent.