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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

and

Example: Prove thatis differentiable at all

and

andso

andsoeverywhere andis differentiable everywhere.

Example: Prove thanis differentiable nowhere

Withsoand

andso

andso

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.

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