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

The proof can be decomposed into 3 main steps:
- calculate the partial derivative for the real part,
- calculate the partial derivative for the imaginary part,
- due to property of
being complex differentiable both derivatives yield the same complex value. This leads to Cauchy-Riemann equations.
Let
. Then
Let
. Then
Both partial derivatives must provide a same complex value due to the fact that
is complex differentiable:
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.
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