As I warned everybody in class, the method of steepesct descents can be kind of tricky. It's one of the hallmark ideas of this class because it generalizes both Laplace's method and stationary phase.
Here is an example posted online that might help.
The book by Bender & Orszag has many other steepest decents examples and I strongly encourage that you study them. (In fact, I strongly encourage that you purchase this book because you'll be referring to it many times in your mathematical career.)
If there is time, I'll try to present another example of steepest descents in lecture before the end of the term. As all of you know, we are very much restricted by the limited time we have.