Let
be an open subset. Let the function
be differentiable at a point
. Then all partial derivatives of
and
exist at
and the following Cauchy-Riemann equations hold:
In this case, the derivative of
at
can be represented by the formula
Let
. Then
Let
. Then
Hence:
Equating the real and imaginary parts, we get the Cauchy-Riemann equations. The representation formula follows from the above line and the Cauchy-Riemann equations.