So my book proves the convergence of $\Gamma(z) = \int_0^{\infty}t^{z-1}e^{-t}dt$ in the right half plane $Re(z) > 0$, and then goes on to prove the initial recurrence relation $\Gamma(z+1)=z\Gamma(z)$ by applying integration by parts to $\Gamma(z+1)$:
$$\int_0^{\infty}t^{z}e^{-t}dt = -t^ze^{-t}|_0^{\infty} + z\int_0^{\infty}t^{z-1}e^{-t}dt$$
The book explicitly states this equality to be true only in the right half plane, since otherwise $-t^ze^{-t}|_0^{\infty} = \infty$, instead of equaling zero. With this initial recurrence relation we are 'supposably' able to analytically continue the Gamma function to $Re(z) > -1$ (not including the origin) by writing the relation in the form:
$$\Gamma(z) = \frac{\Gamma(z+1)}{z}$$
What I don't understand is this relation is still only true in the right half plane, since otherwise $-t^ze^{-t}|_0^{\infty}\neq 0$. I don't see what reason we have to believe that, for instance, $\Gamma(-\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{-\frac{1}{2}}$.
Furthermore $\int_0^{\infty}t^{z-1}e^{-t}dt$ is clearly not convergent in the left half plane, so I can't even imagine why it would be plausible to think that a recurrence relation directly based on it could possibly lead to a genuine analytic continuation of its domain.