Determining well-definedness for functions

How does one establish well - definedness in logical extension for $\Gamma(s)\zeta(s)$ function?

To start with:

$$\Gamma(s)\zeta(s) = \int_0^\infty dt \frac{t^{s-1}}{e^t - 1},\quad Re(s) > 1$$

Using estimate for tiny $t$, I can expand the formula to:

$\displaystyle \Gamma(s)\zeta(s) = \int_0^\infty dt t^{s-1}\left(\frac{1}{e^t - 1} - \frac{1}{t} + \frac{1}{2} - \frac{t}{12}\right) + \frac{1}{s - 1} - \frac{1}{2s} + \frac{1}{12(s + 1)} + \int_1^\infty dt \frac{t^{s-1}}{e^t - 1}$

Now, I require to show that the appropriate - hand side is well - specified for $Re(s) > -2$. I can see that there are straightforward posts at $-1$, $0$ and also $1$, yet just how do I establish well - definedness for terms having integrals?

EDIT: Let is consider $\displaystyle \int_1^\infty dt \frac{t^{s-1}}{e^t - 1}$. Given that it does not have any kind of posts, we just need to examine the actions at 0 and also $\infty$. Consequently:

$\displaystyle \lim_{t\rightarrow 1}\frac{t^{s-1}}{e^t - 1} = \frac{1}{e - 1}$ for any kind of $s$ and also $\displaystyle \lim_{t\rightarrow \infty}\frac{t^{s-1}}{e^t - 1} = ???$

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2022-06-07 14:35:50
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