I have an upper bound for a guassian type integral in terms of the integral in this post and I feel like I have seen something like this as an application of a change of variables and using the definition of the Gamma function.
Let $s \in (0,1)$ and $z$ be any large positive real number, that is $ z \gg 1$
Does the integral $\int_z^\infty t^{s} \exp ( -\frac{1}{2t^{2s}}) \exp(-\frac{t^2}{2})\;dt$ have a known simplification in terms of elementary functions or gamma functions or does it appear in a table of integrals?
If it helps any the upper bound we are trying to derive (and I don't know if there is an estimate of this form applicable to the inegral above) is a bound in terms of $\mathrm{polynomial} (z) \cdot e^{-z^{2-\frac{2s}{1+s}}}$.