Limits of admissibility of normal approximations for the Poisson distribution

Various normal approximations of a Poisson distribution function
$$ \mathbf P\{\mu\le m\mid a\}=\sum_{k=0}^m\frac{a^k}{k!}e^{-a}=\frac1{\Gamma(m+1)}\int_a^\infty x^me^{-x}\,dx $$
are considered in case of large values of the parameter $a$. The least values $A(\varepsilon)$ of a for which the corresponding approximation errors do not exceed given $\varepsilon>0$ are also calculated, the results being compared with the exact values obtained with an electronic computer.