Biharmonic Green function of a ring domain

A biharmonic Green function of a circular ring domain $R=\{z\in \mathsf {C}: 0<r<|z|<1\}$ is found in the form 26741 \widehat{G}_{2}(z,\zeta)=|\zeta-z|^{2}G_{1}(z,\zeta)+\widehat{h}_{2}(z,\zeta), 26741 where $G_{1}(z,\zeta)$ is the harmonic Green function of the ring $R$, and $\widehat{h}_{2}(z,\zeta)$ is a specially constructed biharmonic function.