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.