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