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

 

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.