(i) The function $f$ is defined by

$$f(z)=\int_{C} t^{z-1} d t$$

where $C$ is the circle $|t|=r$, described anti-clockwise starting on the positive real axis and where the value of $t^{z-1}$ at each point on $C$ is determined by analytic continuation along $C$ with $\arg t=0$ at the starting point. Verify by direct integration that $f$ is an entire function, the values of which depend on $r$.

(ii) The function $J$ is defined by

$$J(z)=\int_{\gamma} e^{t}\left(t^{2}-1\right)^{z} d t$$

where $\gamma$ is a figure of eight, starting at $t=0$, looping anti-clockwise round $t=1$ and returning to $t=0$, then looping clockwise round $t=-1$ and returning again to $t=0$. The value of $\left(t^{2}-1\right)^{z}$ is determined by analytic continuation along $\gamma$ with $\arg \left(t^{2}-1\right)=-\pi$ at the start. Show that, for $\operatorname{Re} z>-1$,

$$J(z)=-2 i \sin \pi z I(z),$$

where

$$I(z)=\int_{-1}^{1} e^{t}\left(t^{2}-1\right)^{z} d t$$

Explain how this provides the analytic continuation of $I(z)$. Classify the singular points of the analytically continued function, commenting on the points $z=0,1, \ldots$.

Explain briefly why the analytic continuation could not be obtained by this method if $\gamma$ were replaced by the circle $|t|=2$.