Using the Cauchy Riemann Equations to Prove or Disprove Differentiability - Examples
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
Example: Prove thatis differentiable at all
andsoeverywhere andis differentiable everywhere.
Example: Prove thanis differentiable nowhere
The second Cauchy Riemann equations is not satisfied anywhere so the function is not differentiable anywhere.
Some functions are differentiable at a set of points in the complex plane. Suppose for example thatthenandsoandPutting these equal gives
andPuttinggivesso the Cauchy Riemann equations are satisfied at every point on the line y=-x.