## 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

&nbsp;&nbsp;&nbsp;

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