# Explanation/reference for method of steepest descent (integration method, not gradient descent)

I am wondering if someone could point me to a resource where I could learn the method of steepest descent. Unfortunately, my knowledge of calculus is limited to a college multivariable calculus course. Thus, I am looking for an article that does not assume the knowledge of contour integration.

Unfortunately, the internet search results are cluttered with stuff on gradient descent, which is an optimization method...

The reason I ask this is that an answer by Fabian for my previous question uses the method of steepest descent to justify the following steps:

Let us first rewrite the integral as $$\int_0^\infty\!dy\,y^2e^{-(k+1)y}(2-e^{-y})^k = \int_0^\infty\!dy\, y^2 e^{-y} e^{- k f(y)}, \qquad f(y) =y-\log(2-e^{-y}).$$ The function $f(y)$ assumes its maximum at $y=0$. Thus, the integral is asymptotically dominated for values of $y$ close to $0$. To this end, we expand $f(y) = y^2 +O(y^3)$ and we have $$\int_0^\infty\!dy\, y^2 e^{-y} e^{- k f(y)} \sim \int_0^\infty\!dy\, (y^2 -y^3 +O(y^4)) e^{-k y^2} = \frac{\sqrt\pi}{4 k^{3/2}} - \frac{1}{2k^2} +O(k^{-5/2}).$$

I understand the mechanics of the manipulation (i.e. why $f(y)$ achieves minimum at zero, and the Taylor series exapansion of $f(y)$ and $y^2e^{-y}$) but I don't understand the reason why these manipulations are valid. Thus I would really appreciate any help in both understanding the general method and its specific application above.

I hope that this separately-posted question is appropriate. I think that this is a very useful tool to have to solve some pretty general problems and the specific example from an answer to my question is just an example that is relevant to me.

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You've seen Wong by any chance? – J. M. Aug 19 '12 at 4:46
I wasn't aware of it until you pointed it out. Looks like my university has access through SIAM here. Thanks, I'll look at it! – M.B.M. Aug 19 '12 at 5:58