CauchyRiemann equations
The following system of partial differential equations^{}
$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x},$$ 
where $u(x,y),v(x,y)$ are realvalued functions defined on some open subset of ${\mathbb{R}}^{2}$, 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^{}
$${f}^{\prime}(z)=\underset{\zeta \to 0}{lim}\frac{f(z+\zeta )f(z)}{\zeta}$$ 
exists in the domain of definition, then the real and imaginary parts^{} of $f(z)$ satisfy the CauchyRiemann equations^{}. Conversely, if $u$ and $v$ satisfy the CauchyRiemann equations, and if their partial 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, 18261866: Turning points in the Conception of Mathematics, translated by Abe Shenitzer. Birkhauser, 1999.
Title  CauchyRiemann equations 

Canonical name  CauchyRiemannEquations 
Date of creation  20130322 12:55:36 
Last modified on  20130322 12:55:36 
Owner  rmilson (146) 
Last modified by  rmilson (146) 
Numerical id  5 
Author  rmilson (146) 
Entry type  Definition 
Classification  msc 30E99 
Related topic  Holomorphic 