= Solution
Take the <Hankel contour> about the negative real axis: it starts at $-\infty$ just below the cut, circles the origin counterclockwise, and returns to $-\infty$ just above the cut. The notation $(0+)$ records this return side of the branch point. Choose the branch $-\pi<\arg t<\pi$ for $t^{z-1}$.
For $\operatorname{Re}z>1$, collapse the contour onto the two banks of the cut. Their phase difference is determined by
$$
(-x)^{z-1}=x^{z-1}e^{\pm i\pi(z-1)},
$$
and the resulting sine factor, together with the <Gamma reflection formula> $\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z)$, recovers the integral representation (‡). The Hankel integral converges exponentially at its unbounded end and, after the standard local subtraction of the singular part at zero, depends holomorphically on $z$. The apparent poles of $\Gamma(1-z)$ are cancelled by zeros of the contour integral except at $z=1$. It therefore gives the <analytic continuation> of the <Riemann zeta function> to $\mathbb C\setminus\{1\}$.
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