Example 3.5.  Show that   [Graphics:Images/CauchyRiemannMod_gr_337.gif]   is nowhere differentiable.

Solution.  We have   [Graphics:Images/CauchyRiemannMod_gr_338.gif],   where  

                    [Graphics:Images/CauchyRiemannMod_gr_339.gif]     and     [Graphics:Images/CauchyRiemannMod_gr_340.gif].  

Thus, for any point  [Graphics:Images/CauchyRiemannMod_gr_341.gif],  

                    [Graphics:Images/CauchyRiemannMod_gr_342.gif]     and     [Graphics:Images/CauchyRiemannMod_gr_343.gif].  

The Cauchy-Riemann equations (3-16) are not satisfied at any point  [Graphics:Images/CauchyRiemannMod_gr_344.gif],  so we conclude that  

[Graphics:Images/CauchyRiemannMod_gr_345.gif]   is nowhere differentiable.

 

Explore Solution 3.5.

 

Solution.  We have   [Graphics:../Images/CauchyRiemannMod_gr_346.gif],   where  

                    [Graphics:../Images/CauchyRiemannMod_gr_347.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_348.gif].  

Thus, for any point  [Graphics:../Images/CauchyRiemannMod_gr_349.gif],  

                    [Graphics:../Images/CauchyRiemannMod_gr_350.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_351.gif].  

The Cauchy-Riemann equations (3-16) are not satisfied at any point   [Graphics:../Images/CauchyRiemannMod_gr_352.gif],   so we conclude that  

[Graphics:../Images/CauchyRiemannMod_gr_353.gif]   is nowhere differentiable.

 

We are done.

 

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_354.gif] and [Graphics:../Images/CauchyRiemannMod_gr_355.gif] can assist us in finding the partial derivatives.  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_356.gif]

[Graphics:../Images/CauchyRiemannMod_gr_357.gif]


[Graphics:../Images/CauchyRiemannMod_gr_358.gif]

[Graphics:../Images/CauchyRiemannMod_gr_359.gif]


[Graphics:../Images/CauchyRiemannMod_gr_360.gif]

[Graphics:../Images/CauchyRiemannMod_gr_361.gif]


[Graphics:../Images/CauchyRiemannMod_gr_362.gif]

[Graphics:../Images/CauchyRiemannMod_gr_363.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_364.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_365.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_366.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_367.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_368.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_369.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_370.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_371.gif]

Thus,

          [Graphics:../Images/CauchyRiemannMod_gr_372.gif].

The Cauchy-Riemann equations (3-16) are not satisfied at any point   [Graphics:../Images/CauchyRiemannMod_gr_373.gif],   so we conclude that  

[Graphics:../Images/CauchyRiemannMod_gr_374.gif]   is nowhere differentiable.

 

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