As part of an assignment, we had to investigate the asymptotic behaviour as $n$ tends to $+\infty$ of the ratio $$\frac{\Gamma(n,n|z|^2)}{\Gamma(n)},$$ where $\Gamma(s,x)$ is defined as $$\Gamma(s,x)=\int_x^{+\infty}t^{s-1}e^{-t}\mathrm{d} t.$$ In the cases $|z|>1$ and $|z|<1,$ I was able to use a steepest descent analysis on the integral to prove $$\lim_{n\rightarrow+\infty}\left(\frac{\Gamma(n,n|z|^2)}{\Gamma(n)}\right)=\begin{cases} 1 &\text{ if } |z|<1,\\ 0 &\text{ if } |z|>1. \end{cases}$$
The case $|z|=1$ is not part of the assignment, but I am curious about the limit in this case. Numerical reseach using WolframAlpha suggests $1/2$, but I don't see how to prove this value. Is the numerical value correct and how can one proof this?