(a) Legendre's equation for $w(z)$ is

$$\left(z^{2}-1\right) w^{\prime \prime}+2 z w^{\prime}-\ell(\ell+1) w=0, \quad \text { where } \quad \ell=0,1,2, \ldots$$

Let $\mathcal{C}$ be a closed contour. Show by direct substitution that for $z$ within $\mathcal{C}$

$$\int_{\mathcal{C}} d t \frac{\left(t^{2}-1\right)^{\ell}}{(t-z)^{\ell+1}}$$

is a non-trivial solution of Legendre's equation.

(b) Now consider

$$Q_{\nu}(z)=\frac{1}{4 i \sin \nu \pi} \int_{\mathcal{C}^{\prime}} d t \frac{\left(t^{2}-1\right)^{\nu}}{(t-z)^{\nu+1}}$$

for real $\nu>-1$ and $\nu \neq 0,1,2, \ldots$. The closed contour $\mathcal{C}^{\prime}$ is defined to start at the origin, wind around $t=1$ in a counter-clockwise direction, then wind around $t=-1$ in a clockwise direction, then return to the origin, without encircling the point $z$. Assuming that $z$ does not lie on the real interval $-1 \leqslant x \leqslant 1$, show by deforming $\mathcal{C}^{\prime}$ onto this interval that functions $Q_{\ell}(z)$ may be defined as limits of $Q_{\nu}(z)$ with $\nu \rightarrow \ell=0,1,2, \ldots$.

Find an explicit expression for $Q_{0}(z)$ and verify that it satisfies Legendre's equation with $\ell=0$.