Asymptotic behavior of $\int_{- \pi}^{\pi} \frac{\mathrm{d}\hat{m}}{2\pi} [e^{i \hat{m} m} \cos (\hat{m})]^N$ Given a real value $-1 \leq m \leq 1$, I need to determine the asymptotic $\left(~\mbox{large}\ N~\right)$ behavior of the following integral:
$$
\int_{- \pi}^{\pi} \frac{\mathrm{d}\hat{m}}{2\pi}\,
\left[\,{\mathrm{e}^{\mathrm{i}\,\hat{m}\,m}\,
\cos\left(\hat{m}\right)}\,\right]^{N}
$$
It should satisfy some large deviation principle. How can I derive it $?$.
 A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\on}[1]{\operatorname{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
$\ds{\bbox[5px,#ffd]{}}$

\begin{align}
&\bbox[5px,#ffd]{%
\left.\int_{- \pi}^{\pi}
{\dd\theta \over 2\pi}\,
\bracks{\expo{\ic m\theta}\,
\cos\pars{\theta}}^{N}
\,\right\vert_{\ m\ \in\ \bracks{-1,1}}}
\\[5mm] = &\
{1 \over \pi}
\int_{0}^{\pi}\cos\pars{Nm\theta}
\cos^{N}\pars{\theta}\,\dd \theta 
\\[5mm] = &\
{\pars{-1}^{N} \over \pi}
\int_{-\pi/2}^{\pi/2}\cos\pars{Nm\theta +
Nm\,{\pi \over 2}}
\sin^{N}\pars{\theta}\,\dd \theta
\\[5mm] = &\
{\pars{-1}^{N} \over \pi}
\\[2mm] &\
\int_{0}^{\pi/2}\left[%
\cos\pars{Nm\theta + Nm\,{\pi \over 2}}
\sin^{N}\pars{\theta}\right.
\\[2mm] &\ \left.
\cos\pars{-Nm\theta + Nm\,{\pi \over 2}}
\sin^{N}\pars{-\theta}
\right]\dd \theta
\\[5mm] = &\
{\pars{-1}^{N} \over \pi}
\int_{0}^{\pi/2}\left[%
\cos\pars{Nm\pi - Nm\theta}
\cos^{N}\pars{\theta}\right.
\\[2mm] &\ \phantom{{\pars{-1}^{N} \over \pi}
\int_{0}^{\pi/2}\left[\,\,\right.}
\left.
\cos\pars{Nm\theta}\pars{-1}^{N}
\cos^{N}\pars{\theta}
\right]\dd \theta
\\[5mm] = &\
{\pars{-1}^{N} \over \pi}
\\[2mm] &\ \int_{0}^{\pi/2}
\bracks{\cos\pars{Nm\theta - Nm\pi} +
\pars{-1}^{N}\cos\pars{Nm\theta}}
\\[2mm] &\
\expo{N\ln\pars{\cos\pars{\theta}}}
\,\,\,\dd\theta
\\[5mm] &\
\stackrel{\color{red}{\mrm{as}\ N\ \to\ \infty}}{\sim}\,\,\,
{\pars{-1}^{N} \over \pi}
\bracks{\cos\pars{Nm\pi} + \pars{-1}^{N}}
\\[2mm] &\
\int_{0}^{\infty}
\exp\pars{-N\theta^{2} \over 2}\dd\theta
\\[5mm] = &\
{1 \over \root{2\pi}}
{1 + \pars{-1}^{N}\cos\pars{Nm\pi} \over N^{1/2}}
\end{align}
