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 z = x + iy , the equations state that the partial derivatives must satisfy \frac\partial u\partial x = \frac\partial v\partial y and \frac\partial u\partial y = -\frac\partial v\partial x . These equations ensure that the function is not only differentiable but also holomorphic, meaning it is complex differentiable in a neighborhood around that point. The Cauchy-Riemann equations are essential for understanding properties of analytic functions, which are functions that can be represented by power series. They play a crucial role