**Revisited Example
3.9.** Given

is differentiable at points that lie on the axes
but is
nowhere analytic.

**Explore Revisited Solution
3.9.**

**Solution.** Recall
the identities and that
were used in Section
2.1.

They can be substituted in , and
the result is

When we view as
a function of the two variables , we
see that

.

Therefore, the complex form of the Cauchy-Riemann
equations do not hold

and is
not analytic.

To determine where has
a derivative we must solve the equation .

First expand the quantity as
follows.

Hence, the equivalent equation we need to solve
is .

So we find that the complex
form of the Cauchy-Riemann
equations hold only when ,

and according to Theorem 3.4, is
differentiable
only at points that lie on the coordinate axes.

But this means that is
nowhere analytic because any -neighborhood
about a point on either axis

contains points that are not on those axes.

Therefore is
only differentiable
at points on the
and
axes.

**We are done.**

**Aside.** Both
and
can assist us with the calculations.

**Aside.** The
*Mathematica* solution uses the commands.

**We are really done.**

**Aside.** The Maple
commands are similar.

**>**** **

**>**** **

Both *Mathematica* and Maple have shown that

.

It follows that

,

and the complex form of the Cauchy-Riemann
equations do not hold.

Consequently is
not analytic.

**We are really really done.**

**Remark 1.** In
*Mathematica* it is possible to differentiate with respect to
different formal variables,

for example it is possible to differentiate with respect to a
function .

**Examples using
Mathematica.** Differentiation with respect
to .

Both *Mathematica* and Maple have shown that

Notice that is
a function, and we can differentiate with
respect to .

This is the desired conclusion that we seek.

**Remark 2.** In this
book the use of computers is optional.

Hopefully this text will promote their use and understanding.

This solution is complements of the authors.

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2011 John H. Mathews, Russell W. Howell