In Apostol's Introduction to Analytic Number Theory, the following equation is derived for real $s>1$:
$$\Gamma(s)\zeta(s)=\int_0^{\infty}\frac{x^{s-1}}{e^x-1}\,\mathrm{d}x$$
Then, the fact that both sides are analytic on the half-plane with $\Re(s)>1$ is used to extend this result to all complex $s$ with $\Re(s)>1$. All it says in the textbook is "by analytic continuation" but it doesn't actually eplain what allows this result to be extended. I know what analytic continuation is, and I know about the principle of analytic continuation, but the real line is not open in the complex plane, so surely this doesn't work; just because two analytic functions are equal for real $s>1$ doesn't mean that they are equal on the half-plane with $\Re(s)>1$ even if they are both defined there. Can somebody please explain what allows this result to be extended?
EDIT: I thought that the identity theorem (what I called the principle of analytic continuation) only works when the two functions are equal on an open subset of the complex plane, but, as a couple of people have helpfully pointed out, it applies when the two are equal on any set containing at least one non-isolated point. Clearly, no point of $(1,\infty)$ is isolated, so it can be applied here.
Best Answer
Since the equality$$\Gamma(s)\zeta(s)=\int_0^\infty\frac{x^{s-1}}{e^x-1}\,\mathrm ds$$holds when $s\in(1,\infty)$, and since $s\mapsto\Gamma(s)\zeta(s)$ and $s\mapsto\int_0^\infty\frac{x^{s-1}}{e^x-1}\,\mathrm ds$ are analytic functions defined on $\{z\in\mathbb C\mid\operatorname{Re}z>1\}$, then we have$$\operatorname{Re}z>1\implies\Gamma(s)\zeta(s)=\int_0^\infty\frac{x^{s-1}}{e^x-1}\,\mathrm ds$$by the identity theorem.