 Solution

A complex power requires a choice of complex logarithm. Taking the principal argument of $-e$ to be $+\pi$ gives $\mathrm{Log}(-e)=1+i\pi$, henceThus $a=0$ and $b=e^{-{\pi }^2/2}$ for this principal value. If the power is interpreted as multivalued, the logarithms are $1+i(2k+1)\pi$ and the possible values are $i\exp [-(2k+1){\pi }^2/2]$, $k\in \mathbb{Z}$. Applying an identity for real powers without specifying this branch would lose the ambiguity.