Prove that for $w(\zeta)=-\frac{(\zeta-1)(\zeta+1)(\zeta+i)}{\pi}\left\{\iint_{\Delta} \frac{\mu(z) d x d y}{(z-1)(z+1)(z+i)(\zeta-z)}+\iint_{\Delta} \frac{i \overline{\mu(z)} d x d y}{(\bar{z}-1)(\bar{z}+1)(\bar{z}-i)(1-\zeta \bar{z})}\right\}$, the Fourier coefficients are $$v_{k}=\frac{i}{\pi} \iint_{\Delta} \overline{\mu(z)} \bar{z}^{k-2} d x d y, \quad \text { for } \quad k \geqq 2$$, where $\Delta$ is unit disc, $\mu$ is a bounded measurable function on $\Delta$, $\zeta=e^{i\theta}$.
My try is to do factorization and then Taylor expansion. But it is difficult to deal with, how can I work it out? Thanks very much!