Let $\Phi^{+}(t), \Phi^{-}(t)$ denote the boundary values of functions which are analytic inside and outside a disc of radius $\frac{1}{2}$ centred at the origin. Let $C$ denote the boundary of this disc.

Suppose that $\Phi^{+}, \Phi^{-}$satisfy the jump condition

$$\Phi^{+}(t)=\frac{t}{t^{2}-1} \Phi^{-}(t)+\frac{t^{3}-t^{2}+1}{t^{2}-t}, \quad t \in C .$$

(a) Show that the associated index is 1 .

(b) Find the canonical solution of the homogeneous problem, i.e. the solution satisfying

$$X(z) \sim z^{-1}, \quad z \rightarrow \infty$$

(c) Find the general solution of the Riemann-Hilbert problem satisfying the above jump condition as well as

$$\Phi(z)=O\left(z^{-1}\right), \quad z \rightarrow \infty$$

(d) Use the above result to solve the linear singular integral problem

$$\left(t^{2}+t-1\right) \phi(t)+\frac{t^{2}-t-1}{\pi i} \oint_{C} \frac{\phi(\tau)}{\tau-t} d \tau=\frac{2\left(t^{3}-t^{2}+1\right)(t+1)}{t}, \quad t \in C .$$