Two representations of the zeta function are

$$\zeta(z)=\frac{\Gamma(1-z)}{2 \pi i} \int_{-\infty}^{\left(0^{+}\right)} \frac{t^{z-1}}{e^{-t}-1} d t \quad \text { and } \quad \zeta(z)=\sum_{1}^{\infty} n^{-z}$$

where, in the integral representation, the path is the Hankel contour and the principal branch of $t^{z-1}$, for which $|\arg z|<\pi$, is to be used. State the range of $z$ for which each representation is valid.

Evaluate the integral

$$\int_{\gamma} \frac{t^{z-1}}{e^{-t}-1} d t$$

where $\gamma$ is a closed path consisting of the straight line $z=\pi i+x$, with $|x|<2 N \pi$, and the semicircle $|z-\pi i|=2 N \pi$, with $\operatorname{Im} z>\pi$, where $N$ is a positive integer.

Making use of this result and assuming, when necessary, that the integral along the curved part of $\gamma$ is negligible when $N$ is large, derive the functional equation

$$\zeta(z)=2^{z} \pi^{z-1} \sin (\pi z / 2) \Gamma(1-z) \zeta(1-z)$$

for $z \neq 1$.