Method of Steepest descent integral

Question

I am looking to evaluate the following asymptotic integral:

Find the leading term of asymptotics as $\lambda\to\infty$

$$I(\lambda)=\int_0^1\cos(\lambda x^3)dx$$

Using method of steepest descent along a certain contour. I am having trouble approaching this problem as I don't understand it well. Any help would be appreciated.

Answer 1

To start, recognize that

$$ I(\lambda) = \operatorname{Re} \int_0^1 e^{i\lambda x^3}\,dx. $$

Now there are several questions that you can ask to get yourself going:

1. Where is the saddle point?

2. What are the paths of steepest descent away from the saddle point?

3. How can I deform my contour so that it follows this path of steepest descent?

The last one is a bit tricky since the endpoints of the contour are finite. You only need to follow a portion of the path of steepest descent though; you can have the contour return to its start/endpoint afterwards.

Let me know if you get stuck on any of these.