Cauchy-Riemann equationsCauchyRiemannEquations2002-08-10 06:48:532002-08-10 06:55:56Definition\usepackage{amsmath}
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\newtheorem{definition}[proposition]{Definition}
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\newcommand{\pln}[2]{{\PMlinkname{{#1}}{#2}}}The following system of partial differential
equations
$$
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\quad
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x},
$$
where $u(x,y), v(x,y)$ are real-valued functions defined on some
open subset of $\reals^2$, 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
$$f'(z) = \lim_{\zeta\rightarrow 0} \frac{f(z+\zeta)-f(z)}{\zeta}$$
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
$$f(z) = u(x,y) + i v(x,y),\quad z=x+i y,$$
possesses a continuous complex derivative.
\paragraph{References}
\begin{enumerate}
\item D. Laugwitz, \emph{Bernhard Riemann, 1826-1866:
Turning points in the Conception of
Mathematics}, translated by Abe Shenitzer. Birkhauser, 1999.
\end{enumerate}