Revisited Example 3.9.  Given   [Graphics:Images/CauchyRiemannMod_gr_1446.gif]  

is differentiable at points that lie on the  [Graphics:Images/CauchyRiemannMod_gr_1447.gif]  axes but  [Graphics:Images/CauchyRiemannMod_gr_1448.gif]  is nowhere analytic.  

 

Explore Revisited Solution 3.9.

 

Solution.    Recall the identities  [Graphics:../Images/CauchyRiemannMod_gr_1469.gif]  and  [Graphics:../Images/CauchyRiemannMod_gr_1470.gif]  that were used in Section 2.1.

They can be substituted in   [Graphics:../Images/CauchyRiemannMod_gr_1471.gif],   and the result is

                    [Graphics:../Images/CauchyRiemannMod_gr_1472.gif]   

When we view  [Graphics:../Images/CauchyRiemannMod_gr_1473.gif]  as a function of the two variables  [Graphics:../Images/CauchyRiemannMod_gr_1474.gif],  we see that

                    [Graphics:../Images/CauchyRiemannMod_gr_1475.gif].

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

and   [Graphics:../Images/CauchyRiemannMod_gr_1476.gif]   is not analytic.

To determine where  [Graphics:../Images/CauchyRiemannMod_gr_1477.gif]  has a derivative we must solve the equation  [Graphics:../Images/CauchyRiemannMod_gr_1478.gif].

First expand the quantity  [Graphics:../Images/CauchyRiemannMod_gr_1479.gif]  as follows.

                    [Graphics:../Images/CauchyRiemannMod_gr_1480.gif]

Hence, the equivalent equation we need to solve is   [Graphics:../Images/CauchyRiemannMod_gr_1481.gif].  

      So we find that the complex form of the Cauchy-Riemann equations hold only when  [Graphics:../Images/CauchyRiemannMod_gr_1482.gif],  

and according to Theorem 3.4,  [Graphics:../Images/CauchyRiemannMod_gr_1483.gif]  is differentiable only at points that lie on the coordinate axes.  

But this means that  [Graphics:../Images/CauchyRiemannMod_gr_1484.gif]  is nowhere analytic because any [Graphics:../Images/CauchyRiemannMod_gr_1485.gif]-neighborhood about a point on either axis

contains points that are not on those axes.

Therefore   [Graphics:../Images/CauchyRiemannMod_gr_1486.gif]   is only differentiable at points on the [Graphics:../Images/CauchyRiemannMod_gr_1487.gif] and [Graphics:../Images/CauchyRiemannMod_gr_1488.gif] axes.

 

We are done.

 

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_1489.gif] and [Graphics:../Images/CauchyRiemannMod_gr_1490.gif] can assist us with the calculations.  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_1491.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1492.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1493.gif]

 

[Graphics:../Images/CauchyRiemannMod_gr_1494.gif]

 

 

We are really done.

 

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_1495.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_1496.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_1497.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_1498.gif]

 

Both Mathematica and Maple have shown that  

                    [Graphics:../Images/CauchyRiemannMod_gr_1499.gif].

It follows that

                    [Graphics:../Images/CauchyRiemannMod_gr_1500.gif],  

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

Consequently   [Graphics:../Images/CauchyRiemannMod_gr_1501.gif]   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  [Graphics:../Images/CauchyRiemannMod_gr_1502.gif].

 

Examples using Mathematica.  Differentiation with respect to   [Graphics:../Images/CauchyRiemannMod_gr_1503.gif].  

[Graphics:../Images/CauchyRiemannMod_gr_1504.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1505.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1506.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1507.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1508.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1509.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1510.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1511.gif]

Both Mathematica and Maple have shown that  

                    [Graphics:../Images/CauchyRiemannMod_gr_1512.gif]

Notice that  [Graphics:../Images/CauchyRiemannMod_gr_1513.gif]  is a function, and we can differentiate  [Graphics:../Images/CauchyRiemannMod_gr_1514.gif]  with respect to  [Graphics:../Images/CauchyRiemannMod_gr_1515.gif].  

 

[Graphics:../Images/CauchyRiemannMod_gr_1516.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1517.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1518.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1519.gif]

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