I've been struggling to solve this the whole day today . Basically, I followed the same process as demonstrated in this post, but when I got to this equation

I cannot find anyway to solve it.$\displaystyle \Gamma(-1/2)= \int_{0 }^{ \infty } {u }^{-2 }{e }^{ {-u }^{2 } }du $

Any help would be really appreciated. Thanks a lot.