Understanding the Analytic Continuation of the Gamma Function

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...

The recurrence is used to define Gamma in places where the integral doesn't work. What's needed is to prove that a function defined that way is analytic; then it follows that it's the analytic continuation of Gamma.

An excellent introduction to this topic can be found in the book The Gamma Function by James Bonnar. An entire chapter is devoted to analytic continuation of the factorials, as well as why the Gamma function is defined as it is -- Hölder's theorem and the Bohr-Mullerup theorem are discussed.