The Cauchy Riemann equations enable us to determine if a function is or is not differentiable at a point. The equations state that if
where
and if
is differentiable at a point
then
all exist at
and
and
Example: Prove that
is differentiable at all
and
and
so
and
so
everywhere andis differentiable everywhere.
Example: Prove than
is differentiable nowhere
With
so
and
and
so
and
so
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 that
then
and
so
and
Putting these equal gives
and
Puttinggives
so the Cauchy Riemann equations are satisfied at every point on the line y=-x.