Extra Example 1.  Given  [Graphics:Images/CauchyRiemannMod_gr_704.gif].   

Show that this function is differentiable for all [Graphics:Images/CauchyRiemannMod_gr_705.gif], and find its derivative.

 

Explore Extra Solution 1.

 

Solution.  We compute the partial derivatives and get  

                    [Graphics:../Images/CauchyRiemannMod_gr_715.gif],     and   

                    [Graphics:../Images/CauchyRiemannMod_gr_716.gif],  

so that the Cauchy-Riemann equations (3-16), are satisfied.  Moreover, the partial derivatives

[Graphics:../Images/CauchyRiemannMod_gr_717.gif]  are continuous everywhere.   

By Theorem 3.4,   [Graphics:../Images/CauchyRiemannMod_gr_718.gif],  

is differentiable everywhere, and, from Equation (3-14),  

                    [Graphics:../Images/CauchyRiemannMod_gr_719.gif]  

Alternatively, from Equation (3-15),

                    [Graphics:../Images/CauchyRiemannMod_gr_720.gif]

This result isn't surprising because  [Graphics:../Images/CauchyRiemannMod_gr_721.gif],    

and so the function  [Graphics:../Images/CauchyRiemannMod_gr_722.gif]  is really our old friend   [Graphics:../Images/CauchyRiemannMod_gr_723.gif].  

 

We are done.

 

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_724.gif] and [Graphics:../Images/CauchyRiemannMod_gr_725.gif] can assist us in finding the partial derivatives.  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_726.gif]

[Graphics:../Images/CauchyRiemannMod_gr_727.gif]


[Graphics:../Images/CauchyRiemannMod_gr_728.gif]

[Graphics:../Images/CauchyRiemannMod_gr_729.gif]


[Graphics:../Images/CauchyRiemannMod_gr_730.gif]

[Graphics:../Images/CauchyRiemannMod_gr_731.gif]


[Graphics:../Images/CauchyRiemannMod_gr_732.gif]

[Graphics:../Images/CauchyRiemannMod_gr_733.gif]


[Graphics:../Images/CauchyRiemannMod_gr_734.gif]

[Graphics:../Images/CauchyRiemannMod_gr_735.gif]


[Graphics:../Images/CauchyRiemannMod_gr_736.gif]

[Graphics:../Images/CauchyRiemannMod_gr_737.gif]


[Graphics:../Images/CauchyRiemannMod_gr_738.gif]

[Graphics:../Images/CauchyRiemannMod_gr_739.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_740.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_741.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_742.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_743.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_744.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_745.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_746.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_747.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_748.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_749.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_750.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_751.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_752.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_753.gif]

 

The Cauchy-Riemann equations hold all points  [Graphics:../Images/CauchyRiemannMod_gr_754.gif]  in the complex plane, therefore

[Graphics:../Images/CauchyRiemannMod_gr_755.gif]   is an entire function.  

Verify that the derivative can be calculated with either of the formulas:

(3-14)              [Graphics:../Images/CauchyRiemannMod_gr_756.gif],     or

(3-15)              [Graphics:../Images/CauchyRiemannMod_gr_757.gif].  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_758.gif]

[Graphics:../Images/CauchyRiemannMod_gr_759.gif]


[Graphics:../Images/CauchyRiemannMod_gr_760.gif]

[Graphics:../Images/CauchyRiemannMod_gr_761.gif]


[Graphics:../Images/CauchyRiemannMod_gr_762.gif]

[Graphics:../Images/CauchyRiemannMod_gr_763.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_764.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_765.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_766.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_767.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_768.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_769.gif]

 

We are really done.

 

Therefore, both Mathematica and Maple have shown that if  

                    [Graphics:../Images/CauchyRiemannMod_gr_770.gif],   

then the derivative is

                    [Graphics:../Images/CauchyRiemannMod_gr_771.gif],   

as expected.

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_772.gif]

[Graphics:../Images/CauchyRiemannMod_gr_773.gif]


[Graphics:../Images/CauchyRiemannMod_gr_774.gif]

[Graphics:../Images/CauchyRiemannMod_gr_775.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_776.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_777.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_778.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_779.gif]

 

Remark 1.  You might wonder why we need the Cauchy-Riemann equations to differentiate a well known function.

We are preparing for Section 3.3 where the Cauchy-Riemann equations are used to construct a conjugate harmonic function.

 

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