Cauchy-Riemann equations
The following system of partial differential equations
where are real-valued functions defined on some open subset of , was introduced by Riemann[1] as a definition of a holomorphic function. Indeed, if satisfies the standard definition of a holomorphic function, i.e. if the complex derivative
exists in the domain of definition, then the real and imaginary parts of satisfy the Cauchy-Riemann equations. Conversely, if and satisfy the Cauchy-Riemann equations, and if their partial derivatives are continuous, then the complex valued function
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 |