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I am trying to understand an example introducing Laplace's Method -- Found in 'Analytic Combinatorics' pg 755. There are three parts for the entire exercise that I'm very confused on.

The goal is to estimate: $I_n = \int_{-\pi/2}^{\pi/2} (\cos{x})^n$

First, a substitution is made $x=\frac{\omega}{\sqrt{n}}$, which I don't completely understand why this was chosen. This choice was made after observing that the plots show the unimodal nature of the function, and that the bulk of the contribution to the integral will be moving towards 0.

Second, the integrand is rewritten as $exp(n\log{\cos{x}}) = exp(-\frac{\omega}{2} + O(n^{-1}\omega^{4}))$ I understand how we got here; the author just expanded about x=0, then made the substitution for x as above. My issue here is I don't quite understand what O means when it has two variables $n$ and $\omega$.

Finally, there is a place where we have $\frac{1}{\sqrt{n}}\int_{-n^{1/10}}^{n^{1/10}} e^{-\omega^2/2}(1+O(n^{-1}\omega^{4}))dw = \frac{1}{\sqrt{n}}\int_{-n^{1/10}}^{n^{1/10}} e^{-\omega^2/2}dw+O(n^{-\frac{3}{5}}) $ The only way I see how to get the $O(n^{-\frac{3}{5}})$ is by directly plugging in $w=n^{\frac{1}{10}}$, and using the leading $\frac{1}{\sqrt{n}}$ -- but how this is made rigorous, I'm stuck.

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    $O(n^{-1}\omega^4)$ stands for a function $f_n(\omega)$ for which there is a uniform constant $C$ such that $|f_n(\omega)| \leq Cn^{-1}\omega^4$ as long as $n$ and $\omega$ remain in a certain range. Note that this range may be very large; it depends on the context. – 2017-01-17
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    In this case it would probably have been clearer to just write $n^{-1} O(\omega^4)$. It's the same thing here. – 2017-01-17

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