Suppose \(U ( z )\) has continuous second order partial derivatives and \(z = f ( \zeta ) , \zeta = \xi + i \eta\) is a holomorphic function. Show that
\[ \frac { \partial ^ { 2 } U } { \partial \xi ^ { 2 } } + \frac { \partial ^ { 2 } U } { \partial \eta ^ { 2 } } = | f ^ { \prime } ( \zeta ) | ^ { 2 } \left[ \frac { \partial ^ { 2 } U } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } U } { \partial y ^ { 2 } } \right] . \]