Cauchy-Riemann Equations

Example 3.9. Show that the function

is differentiable at points that lie on the -axis, and at points that lie on the -axis, but is nowhere analytic.

Explore Solution 3.9.

Solution.  Recall Definition 3.1 (from Section 3.1):  when we say a function is analytic at a point we mean that the function

is differentiable not only at , but also at every point in some  neighborhood of .  With this in mind, we proceed

to determine where the Cauchy-Riemann equations (3-16) are satisfied.  We write

                         and     ,

and compute the partial derivatives:

                    ,    ,     and

                    ,    .

Here    are continuous, and



holds for all points in the complex plane.

But      if and only if   ,   which is equivalent to

                    .

Hence, the Cauchy-Riemann equations hold only at the points where   .

According to Theorem 3.4,      is differentiable only when   ,

which occurs only at points that lie on the coordinate axes.  Furthermore, for any point on the coordinate axes,

there contains an -neighborhood about it, in which there exist points where    is not differentiable.

Applying Definition 3.1 (from Section 3.1) , we see that the function

is not analytic on either of the coordinate axes.

Therefore,      is nowhere analytic.