(i) Let $\Phi^{+}(t)$ and $\Phi^{-}(t)$ denote the boundary values of functions which are analytic inside and outside the unit disc centred on the origin, respectively. Let $C$ denote the boundary of this disc. Suppose that $\Phi^{+}(t)$ and $\Phi^{-}(t)$ satisfy the jump condition

$$\Phi^{+}(t)=t^{-2} \Phi^{-}(t)+t^{-1}+\alpha\left(t^{-1}+t-t^{-3}\right), \quad t \in C,$$

where $\alpha$ is a constant.

Find the canonical solution of the associated homogeneous Riemann-Hilbert problem. Write down the orthogonality conditions.

(ii) Consider the linear singular integral equation

$$\left(t+t^{-1}\right) \psi(t)+\frac{t-t^{-1}}{\pi i} \oint_{C} \frac{\psi(\tau)}{\tau-t} d \tau=2+2 \alpha\left(1+t^{2}-t^{-2}\right)$$

where $\oint$ denotes the principal value integral.

Show that the associated Riemann-Hilbert problem has the jump condition defined in Part (i) above. Using this fact, find the value of the constant $\alpha$ that allows equation $(*)$ to have a solution. For this particular value of $\alpha$ find the unique solution $\psi(t)$.