Example 3.9.  Show that the function  [Graphics:Images/CauchyRiemannMod_gr_924.gif]  

is differentiable at points that lie on the [Graphics:Images/CauchyRiemannMod_gr_925.gif]-axis, and at points that lie on the [Graphics:Images/CauchyRiemannMod_gr_926.gif]-axis, but  [Graphics:Images/CauchyRiemannMod_gr_927.gif]  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 [Graphics:../Images/CauchyRiemannMod_gr_950.gif] we mean that the function

is differentiable not only at [Graphics:../Images/CauchyRiemannMod_gr_951.gif], but also at every point in some  neighborhood of [Graphics:../Images/CauchyRiemannMod_gr_952.gif].  With this in mind, we proceed

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

                    [Graphics:../Images/CauchyRiemannMod_gr_953.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_954.gif],  

and compute the partial derivatives:

                    [Graphics:../Images/CauchyRiemannMod_gr_955.gif],    [Graphics:../Images/CauchyRiemannMod_gr_956.gif],     and  

                    [Graphics:../Images/CauchyRiemannMod_gr_957.gif],    [Graphics:../Images/CauchyRiemannMod_gr_958.gif].  

Here  [Graphics:../Images/CauchyRiemannMod_gr_959.gif]  are continuous, and  

                    [Graphics:../Images/CauchyRiemannMod_gr_960.gif]  

holds for all points [Graphics:../Images/CauchyRiemannMod_gr_961.gif] in the complex plane.   

But   [Graphics:../Images/CauchyRiemannMod_gr_962.gif]   if and only if   [Graphics:../Images/CauchyRiemannMod_gr_963.gif],   which is equivalent to  

                    [Graphics:../Images/CauchyRiemannMod_gr_964.gif].  

Hence, the Cauchy-Riemann equations hold only at the points where   [Graphics:../Images/CauchyRiemannMod_gr_965.gif].  

According to Theorem 3.4,   [Graphics:../Images/CauchyRiemannMod_gr_966.gif]   is differentiable only when   [Graphics:../Images/CauchyRiemannMod_gr_967.gif],  

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

there contains an [Graphics:../Images/CauchyRiemannMod_gr_968.gif]-neighborhood about it, in which there exist points where  [Graphics:../Images/CauchyRiemannMod_gr_969.gif]  is not differentiable.

Applying Definition 3.1 (from Section 3.1) , we see that the function   [Graphics:../Images/CauchyRiemannMod_gr_970.gif]   

is not analytic on either of the coordinate axes.

Therefore,   [Graphics:../Images/CauchyRiemannMod_gr_971.gif]   is nowhere analytic.

 

We are done.

 

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_972.gif] and [Graphics:../Images/CauchyRiemannMod_gr_973.gif] can assist us in finding the partial derivatives.  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_974.gif]

[Graphics:../Images/CauchyRiemannMod_gr_975.gif]


[Graphics:../Images/CauchyRiemannMod_gr_976.gif]

[Graphics:../Images/CauchyRiemannMod_gr_977.gif]


[Graphics:../Images/CauchyRiemannMod_gr_978.gif]

[Graphics:../Images/CauchyRiemannMod_gr_979.gif]


[Graphics:../Images/CauchyRiemannMod_gr_980.gif]

[Graphics:../Images/CauchyRiemannMod_gr_981.gif]


[Graphics:../Images/CauchyRiemannMod_gr_982.gif]

[Graphics:../Images/CauchyRiemannMod_gr_983.gif]


[Graphics:../Images/CauchyRiemannMod_gr_984.gif]

[Graphics:../Images/CauchyRiemannMod_gr_985.gif]


[Graphics:../Images/CauchyRiemannMod_gr_986.gif]

[Graphics:../Images/CauchyRiemannMod_gr_987.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_988.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_989.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_990.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_991.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_992.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_993.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_994.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_995.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_996.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_997.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_998.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_999.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_1000.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_1001.gif]

Here  [Graphics:../Images/CauchyRiemannMod_gr_1002.gif]  are continuous, and  

                    [Graphics:../Images/CauchyRiemannMod_gr_1003.gif]  

holds for all points [Graphics:../Images/CauchyRiemannMod_gr_1004.gif] in the complex plane.   

But   [Graphics:../Images/CauchyRiemannMod_gr_1005.gif]   if and only if   [Graphics:../Images/CauchyRiemannMod_gr_1006.gif],   which is equivalent to  

                    [Graphics:../Images/CauchyRiemannMod_gr_1007.gif].  

Hence, the Cauchy-Riemann equations hold only at the points where   [Graphics:../Images/CauchyRiemannMod_gr_1008.gif].  

According to Theorem 3.4,   [Graphics:../Images/CauchyRiemannMod_gr_1009.gif]   is differentiable only when   [Graphics:../Images/CauchyRiemannMod_gr_1010.gif],  

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

there contains an [Graphics:../Images/CauchyRiemannMod_gr_1011.gif]-neighborhood about it, in which there exist points where  [Graphics:../Images/CauchyRiemannMod_gr_1012.gif]  is not differentiable.

Applying Definition 3.1 (from Section 3.1) , we see that the function   [Graphics:../Images/CauchyRiemannMod_gr_1013.gif]   

is not analytic on either of the coordinate axes.

Therefore,   [Graphics:../Images/CauchyRiemannMod_gr_1014.gif]   is nowhere analytic.

 

Remark.  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