Proof of the Cauchy Riemann Equations

The Cauchy Riemann equations enable us to determine if a function is or is not differentiable at a point. The equations state that ifwhereand ifis differentiable at a pointthenall exist atand

and

Proof

Suppose that the complex derivative
exists for someThis means that for allthere exists asuch that for all complexwithwe have

   

Now setwith

Ifis real, then the above limit reduces to a partial derivative ini.e.

Similarly forpurely imaginary we have

Setting these two expressions equal, since the differential of a function at a point is independent of the path taken to the point, we have

Now match real and imaginary parts to get the Cauchy Riemann equations

and