Definition of the gamma function I know that the Gamma function with argument $(-\frac{1}{ 2})$ -- in other words $\Gamma(-\frac{1}{2})$ is equal to $-2\pi^{1/2}$. However, the definition of $\Gamma(k)=\int_0^\infty t^{k-1}e^{-t}dt$ but how can $\Gamma(-\frac{1}{2})$ be obtained from the definition? WA says it does not converge... 

The functional equation $\Gamma(z+1)=z\Gamma(z)$ allows you to define $\Gamma(z)$ for all $z$ with real part greater than $-1$, other than $z=0$: just set $\Gamma(z) = \Gamma(z+1)/z$, and the integral definition of $\Gamma(z+1)$ does converge. The value $\Gamma(\frac12)=\sqrt\pi$ is well known and can be derived from Euler's reflection formula.
(Repeated use of the functional equation shows that $\Gamma$ can be defined for all complex numbers other than nonpositive integers.)
