Constant of Laplace approximation I'm reading Example 3.16 of Robert & Casella's Monte Carlo Statistical Methods. It uses a Laplace approximation for approximating an integral related with the Gamma distribution namely $$\int_a^b\frac{x^{\alpha-1}}{\Gamma(\alpha)\beta^\alpha}e^{-\frac{x}{\beta}}dx$$
I'm wondering where $\Gamma(\alpha)\beta^\alpha$ does go in the end result. Also, the book didn't mention too much about importance of $n$, can anyone give some explanation on this regard?

 A: 
This example is indeed rather poorly conducted and full of typos, apologies from one author!!!

First, there is no genuine $n$ (or related sample size) in the picture so the Laplace approximation cannot be universally adequate. The only thing that can replace $n$ in the integral 
$$
\int_a^b \dfrac{x^{\alpha-1}}{\Gamma(\alpha)\beta^\alpha}e^{-x/\beta}\text{d}x$$
is $(b-a)^{-1}$, in the sense that the approximation becomes better and better as $(b-a)$ goes to zero. This is under the provision that the mode $\hat{x}_\theta$ belongs to the interval $(a,b)$.
Second, as you correctly wondered, the constant $\Gamma(\alpha)\beta^\alpha$ in the denominator has truly and inexplicably (!) been forgotten in the final formula, which should thus be
\begin{align}\int_a^b &\dfrac{x^{\alpha-1}}{\Gamma(\alpha)\beta^\alpha}e^{-x/\beta}\text{d}x\,\approx \\
&\dfrac{\hat{x}_\theta^{\alpha-1}e^{-\hat{x}_\theta/\beta}}{\Gamma(\alpha)\beta^\alpha}\sqrt{\frac{2\pi \hat{x}_\theta^2}{\alpha-1}}\left\{\Phi\left(\sqrt{\frac{\alpha-1}{2\pi \hat{x}_\theta^2}} (b-\hat{x}_\theta)\right)-\Phi\left(\sqrt{\frac{\alpha-1}{2\pi \hat{x}_\theta^2}} (a-\hat{x}_\theta)\right)\right\}\end{align}
Third, the formula at the top of page 110 is missing two minus ($-$) signs, as it should in truth be
$$h(x)\approx -\frac{\hat{x}_\theta}{\beta}+(\alpha-1)\log \hat{x}_\theta-\frac{\alpha-1}{2\hat{x}_\theta^2}(x-\hat{x}_\theta)^2$$
With these multiple corrections, the table comparing the Laplace version with the exact coverage can be reconstructed, for instance via the following R code:
xop <- function(al,be){(al-1)*be}

lapl <- function(al,be,a,b){
 hatx=xop(al,be)
 hatx^{al-1}*exp(-hatx/be)*sqrt(2*pi)*hatx/sqrt(al-1)*
 (pnorm(sqrt(al-1)*(b-hatx)/hatx)-pnorm(sqrt(al-1)*(a-hatx)/hatx))/
 be^al/factorial(al-1)
 }

xact <- function(al,be,a,b){
  pgamma(b,al,1/be)-pgamma(a,al,1/be)}

> xact(5,2,7,9)
[1] 0.1933414
> lapl(5,2,7,9)
[1] 0.1933507

Once again, my most sincere apologies for this poor attention to details in this example!
