Cauchy-Riemann 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 real-valued functions defined on some open subset of ${ℝ}^2$, 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 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

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

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.

Title	Cauchy-Riemann equations
Canonical name	CauchyRiemannEquations
Date of creation	2013-03-22 12:55:36
Last modified on	2013-03-22 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