The following integral appears in certain physical problem:
$$ \lambda_m = \int_{0}^{2\pi} \frac{d\phi}{2\pi} e^{K\cos \phi + im\phi} $$ where $m$ is an integer and $K$ is a large real number. I'm asked to evaluate it using the saddle point approximation (which in the math literature is called Laplace's method as far as I'm aware). The real part of the argument of the exponential is the largest when $\phi = 0$, so I approximated it by
$$ e^{K \cos \phi + im\phi} \approx e^{K-\frac{1}{2}\phi^{2}K} $$
which means that
$$ \lambda_{m}\approx \frac{e^{K}}{2\pi}\int\limits _{0}^{2\pi}d\phi\,e^{-\frac{1}{2}\phi^{2}K} $$
But apparently this derivation is wrong. The biggest problem is that $m$ doesn't appear in the approximation. Presumably, the correct approximation should be:
$$ \lambda_{m}\approx\frac{e^{K}}{2\pi}\int_{-\infty}^{\infty}d\theta\,e^{-K\theta^{2}/2+im\theta}=\frac{e^{K}}{\sqrt{2\pi}K}e^{-\frac{m^{2}}{2K}} $$
However I don't understand what saddle point was used here and why the limits of the integral were changed.
Without Laplace method
If $m$ is an integer
$$\lambda_m = \frac{1}{2\pi}\int_{0}^{2\pi} e^{K\cos (\phi) + im\phi}\,d\phi=I_m(K)$$ where appears the modified Bessel function of the first kind.
For large values of $K$, use the asymptotic expansions developed by Hankel $$I_m(K) =\frac{e^K}{\sqrt{2 \pi K}}\Bigg[1+(4m^2-1)\sum_{p=0}^\infty 4^p\,\frac{\left(\frac{3}{2}-m\right)_p \left(\frac{3}{2}+m\right)_p}{p!\,(8K)^p}\Bigg]$$
that is to say $$I_m(K) =\frac{e^K}{\sqrt{2 \pi K}}\Bigg[1-\frac{(4m^2-1)}{1!\,(8K)}+\frac{(4m^2-1)(4m^2-9)}{2!\,(8K)^2} +\cdots\Bigg] $$