Explain the method of stationary phase for determining the behaviour of the integral

$$I(x)=\int_{a}^{b} d u e^{i x f(u)}$$

for large $x$. Here, the function $f(u)$ is real and differentiable, and $a, b$ and $x$ are all real.

Apply this method to show that the first term in the asymptotic behaviour of the function

$$\Gamma(m+1)=\int_{0}^{\infty} d u u^{m} e^{-u}$$

where $m=i n$ with $n>0$ and real, is

$$\Gamma(i n+1) \sim \sqrt{2 \pi} e^{-i n} \exp \left[\left(i n+\frac{1}{2}\right)\left(\frac{i \pi}{2}+\log n\right)\right]$$

as $n \rightarrow \infty$