$$\Gamma(n+1)=n!,\ \ \ \ \ \Gamma(n+2)=(n+1)!$$
for integers, so if $\Delta$ is some real value with
$$n!\ <\ \Gamma(n+1+\Delta)\ <\ (n+1)!,$$
because $\Gamma$ is monotone there and so there is another number $f$ with
How can we make this more precise? Can we find $f(\Delta)$?
Or if we know the value $\Delta$, which will usually be the case, what $f$ will be a good approximation?