Cauchy-Riemann equations

The following system of partial differentialequations

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

where $u(x,y),v(x,y)$ are real-valued functions defined on someopen subset of ${ℝ}^2$, was introduced by Riemann[1] as adefinition of a holomorphic function. Indeed, if $f(z)$ satisfies thestandard definition of a holomorphic function, i.e. if thecomplex derivative

$$f^{\prime }(z)=\underset{\zeta \to 0}{lim}\frac{f(z+\zeta )-f(z)}{\zeta }$$

exists in the domain of definition, then the real and imaginary partsof $f(z)$satisfy the Cauchy-Riemann equations.Conversely, if $u$ and $v$ satisfy the Cauchy-Riemann equations, and if theirpartial derivatives are continuous, then the complex valued function

$$f(z)=u(x,y)+iv(x,y),z=x+iy,$$

possesses a continuous complex derivative.

References

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