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.

We are done.

Aside.  Both and can assist us in finding the partial derivatives.

Aside.  The Mathematica solution uses the commands.

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Aside.  The Maple commands are similar.

>

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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.

Remark.  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