I would like to compute the integral
$$ \int_{0}^{\infty}\frac{1}{\sqrt{2t}}e^{-\frac{1}{2t}}dt $$
which wolfram alpha says that it does not converge. However by letting $x=1/2t$ I get $dt=\frac{-1}{2x^2}dx$ and that the integral is transformed to $$ \int_{\infty}^{0}x^{0.5}e^{-x}\cdot\frac{-1}{2x^{2}}dx $$ or $$ \frac{1}{2}\int_{0}^{\infty}x^{-1.5}e^{-x}dx=0.5\Gamma(-0.5) $$ which is finite. Something must have gone wrong in the above calculation but I couldn't find out. Any help is appreciated.
The Gamma function is defined for complex $z$ with positive real part as $$ \Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\,dt. $$ This integral does not converge if $\operatorname{Re}{z}\le0$. $\Gamma$ can be extended to the whole complex plane with the non-positive integers removed, but the above integral makes sense only if $\operatorname{Re}{z}>0$.
A similar situation happens for instance with power series. The series $\sum_{n=0}^\infty z^n$ converges on the unit disc to $1/(1-z)$. This function is defined on the whole plane wth the point $1$ removed, but agrees with the series only on the unit disc.
WolphramAlpha is right in telling that your integral does not converge. As $t\to\infty$, the integrand is like $1/\sqrt{2\,t}$.