Question on cauchy Riemann equation

Question

Let u(x, y) = x² + y² and v(x, y) = xy be two functions. Then; (a) Check the existence, continuity and CR-equations of partial derivatives of u and v.

(b) Check the differentiability and analyticity of the function f = u + iv

Answer 1

The Cauchy-Rieman equations are simply to establish: ∂u/∂x = ∂v/∂y and ∂u/∂y = - ∂v/∂x.

Function f is analytical if the CR equations hold true and ∂²u/∂x² + ∂²u/∂y² =0.