The Cauchy-Riemann equations are a set of two partial differential equations that are fundamental in complex analysis. They provide a criterion for determining whether a complex function is differentiable at a point. Specifically, if a function is expressed as f(z) = u(x, y) + iv(x, y) , where u and v are real-valued functions of the real variables x and y , the equations state that \frac\partial u\partial x = \frac\partial v\partial y and \frac\partial u\partial y = -\frac\partial v\partial x . When a function satisfies the Cauchy-Riemann equations and is continuous, it is considered holomorphic, meaning it is complex differentiable in a neighborhood around that point. This property is crucial in various fields, including {mathematics