Cauchy–Riemann equations

In mathematics, the Cauchy–Riemann equations are two partial differential equations that characterize differentiability of complex functions. The equations are

{\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}

1a

and

{\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}},

1b

where $u (x, y)$ and $v (x, y)$ are real bivariate differentiable functions. Typically, $u$ and $v$ are the real and imaginary parts, respectively, of a complex-valued function $f(z)=f(x+iy)=u(x,y)+iv(x,y)$ of a complex variable $z = x + iy$ .

If $f$ is complex-differentiable at a complex point $z = x + iy$ , then the partial derivatives of $u$ and $v$ exist and satisfy the Cauchy–Riemann equations at that point. Conversely, if the functions $u$ and $v$ are (real) differentiable at $z$ and satisfy the Cauchy-Riemann equations there, then $f$ is complex-differentiable at $z$ .

In this way, the Cauchy-Riemann equations are closely related to differentiability, which is in turn closely related to analyticity. These close relationships are the starting point of complex analysis.