Question :
If $\phi$ and $\psi$ ae function of $ x $ and $y$ satisfying laplace's equation . Show that $f(z) = s + it$ is analytic , where $$ s = \frac{\partial \phi}{\partial y} - \frac{\partial \psi}{\partial x} \text{and} \text{ t} = \frac{\partial \phi}{\partial x} - \frac{\partial \psi}{\partial y} $$
To prove $f(z) $ is analytic, we have to show that $f$ satisfies C-R equation and $ \frac{\partial s}{\partial x}, \frac{\partial s}{\partial y}, \frac{\partial t}{\partial x}, \frac{\partial t}{\partial y}$ are all continous
we can easily prove that $f$ satisfies C-R equation .
Also we know that $\phi$ and $\psi$ satisfies laplace equation . Also $\phi$ and $\psi$ are harmonic function . So they possesses continous partial derivative upto second order.
Thus $\frac{\partial s}{\partial x} = \frac{ \partial^2 \phi}{\partial y \partial x} - \frac{ \partial^2 \phi}{ \partial^2 x} $ is continous. Similarly $\frac{\partial s}{\partial y}, \frac{\partial t}{\partial x}, \frac{\partial t}{\partial y}$ are continous.