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.
Best Answer
As you have said, $\displaystyle \int_0^{\infty}t^{z-1}e^{-t}dt$ does not even make sense on the left half plane.
However, if we define $\Gamma(z) = \dfrac{\Gamma(z+1)}{z}$ on the left half plane, (and this is not equal to the integral above on the left half plane), this is consistent with the fact that $\Gamma(z) = \displaystyle \int_0^{\infty}t^{z-1}e^{-t}dt$ on the right half plane and also provides us a possible extension on the left half plane except at negative integers.
Further, this has to be the only analytic continuation since $\Gamma(z) = \dfrac{\Gamma(z+1)}{z}$ is analytic everywhere except at negative integers and matches with the integral on the right half plane and hence by uniqueness of analytic continuation this is the only possible extension.
EDIT
To see that the extension, is in fact analytic except at negative integers, you proceed strip by strip. On the right half plane, you have $\Gamma(z)$ defined by the integral, $\displaystyle \int_0^{\infty} t^{z-1} \exp(-t) dt$. You can verify that the integral satisfies the functional equation and is in fact analytic.
We will now extend the function to the strip corresponding to $(-1,0)$ i.e. the strip $\text{Re}(z) \in (-1,0)$ based on the functional equation.
Define $\Gamma(z) = \dfrac{\Gamma(z+1)}{z}$ on the strip $\text{Re}(z) \in (-1,0)$.
First note that $\Gamma(z+1)$ is analytic on the strip $\text{Re}(z) \in (-1,0)$ since $z+1 \in$ right half plane. Also, $\dfrac1z$ is analytic on the strip $\text{Re}(z) \in (-1,0)$. Hence, $\Gamma(z) = \dfrac{\Gamma(z+1)}{z}$ is analytic on the strip $\text{Re}(z) \in (-1,0)$. Hence, now we have extended the definition of the $\Gamma$ function to the region $\text{Re}(z) > -1$.
Now repeat the same argument for the strip $\text{Re}(z) \in (-2,-1)$, since we now have that the $\Gamma$-function is analytic on the region $\text{Re}(z) > -1$. And so on...