I want to calculate the Laplace of this equation: $\ f = a \ln|\sin(px + iqy)| + by + c $.
Laplace $\Delta$ can be written as $\nabla * \nabla =\nabla ^2$, where $\nabla$ is a differential operator $\nabla = (\frac {\partial} {\partial x}, \frac {\partial} {\partial x} \frac {\partial} {\partial x}) $. I am just not sure how to derivate my given function properly because of that imaginary $i$.
I know that $|u + iv| = \sqrt {u^2 + v^2} $, but what about the sinus. Should I use this identity $ \sin(px + iqy) = \sin(px)\cos(iqy) + \cos(px)\sin(iqy)$? Because I am stuck here.