Cauchy-Riemann Equations

Theorem edit

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


Proof edit

Let  . Then


Let  . Then




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.