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.

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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   .

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Both Mathematica and Maple have shown that

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

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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.

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