Question1. Let the function $$f(z)=u(x, y)+i v(x, y)$$ and it satisfies the Cauchy-Riemann conditions: $$\begin{array}{l} \frac{\partial u(x, y)}{\partial x}=\frac{\partial v(x, y)}{\partial y} \\ \frac{\partial u(x, y)}{\partial y}=-\frac{\partial v(x, y)}{\partial x} \end{array}$$ then $f(z)$ is said to be analytical and $v(x, y)$ is said to be harmonic conjugate of $u(x, y)$. It is said to be harmonic if $$\begin{array}{l} \frac{\partial^{2} u(x, y)}{\partial x^{2}}+\frac{\partial^{2} u(x, y)}{\partial y^{2}}=0 \\ \frac{\partial^{2} v(x, y)}{\partial x^{2}}+\frac{\partial^{2} v(x, y)}{\partial y^{2}}=0 \end{array}$$
Show that the following $u(x, y)$ are harmonic and find its harmonic conjugate (a) $u(x, y)=2 x(1-y)$ (b) $u(x, y)=\sinh (x) \sin (y)$