The Cauchy Riemann equations consist of a system of two partial differential equations which together form a necessary and sufficient condition for a complex function to be complexly differentiable that is holomorphic with a certain continuity and differentiability criteria.
The Cauchy–Riemann equations on a pair of real-valued functions of two real variables u(x,y) and v(x,y) are the two equations. It is given by ∂u/∂x = ∂v/∂y or
∂u/∂y = ∂v/∂x
u and v are taken to be the real and imaginary parts respectively