Let

$$I(z)=i \oint_{C} \frac{u^{z-1}}{u^{2}-4 u+1} d u$$

where $C$ is a closed anti-clockwise contour which consists of the unit circle joined to a loop around a branch cut along the negative axis between $-1$ and 0 . Show that

$$I(z)=F(z)+G(z)$$

where

$$F(z)=2 \sin (\pi z) \int_{0}^{1} \frac{x^{z-1}}{x^{2}+4 x+1} d x, \quad \operatorname{Re} z>0$$

and

$$G(z)=\frac{1}{2} \int_{-\pi}^{\pi} \frac{e^{i(z-1) \theta}}{1+2 \sin ^{2} \frac{\theta}{2}} d \theta, \quad z \in \mathbb{C}$$

Evaluate $I(z)$ using Cauchy's theorem. Explain how this may be used to obtain an analytic continuation of $F(z)$ valid for all $z \in \mathbb{C}$.