The following system of partial differential equations $$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}, $$ where $u(x,y), v(x,y)$ are real-valued functions defined on some open subset of $\reals^2$ , was introduced by Riemann[1 as a definition of a holomorphic function. Indeed, if $f(z)$ satisfies the standard definition of a holomorphic function, i.e. if the complex derivative $$f'(z) = \lim_{\zeta\rightarrow 0} \frac{f(z+\zeta)-f(z)}{\zeta}$$ exists in the domain of definition, then the real and imaginary parts of $f(z)$ satisfy the Cauchy-Riemann equations. Conversely, if $u$ and $v$ satisfy the Cauchy-Riemann equations, and if their partial derivatives are continuous, then the complex valued function $$f(z) = u(x,y) + i v(x,y),\quad z=x+i y,$$ possesses a continuous complex derivative.

References

1. D. Laugwitz, Bernhard Riemann, 1826-1866: Turning points in the Conception of Mathematics, translated by Abe Shenitzer. Birkhauser, 1999.