The following system of partial differential equations
where are real-valued functions defined on some open subset of , was introduced by Riemann 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.
D. Laugwitz, Bernhard Riemann, 1826-1866: Turning points in the Conception of Mathematics, translated by Abe Shenitzer. Birkhauser, 1999.
|Date of creation||2013-03-22 12:55:36|
|Last modified on||2013-03-22 12:55:36|
|Last modified by||rmilson (146)|