Example 3.4.  We know that      is differentiable and that   .   

Furthermore, the Cartesian coordinate form for    is

                    .  

Use the Cartesian coordinate form of the Cauchy-Riemann equations and find  .  

Solution.  It is easy to verify that Cauchy-Riemann equations (3-16) are indeed satisfied:

                         and     .  

Using Equations (3-14) and (3-15), respectively, to compute    gives

                    ,     

                    and   

                    ,
                    
as expected.

Explore Solution 3.4.

Solution.  It is easy to verify that Cauchy-Riemann equations (3-16) are indeed satisfied:

                         and     .  

Using Equations (3-14) and (3-15), respectively, to compute    gives

                    ,     

                    and   

                    ,
                    
as expected.

We are done.

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

Aside.  The Mathematica solution uses the commands.  

Aside.  The Maple commands are similar.  

          >  

                              

          >  

                              

          >  

                              

          >    

                              

          >    

                              

          >    

                              

          >    

                              

The Cauchy-Riemann equations hold  all points      in the complex plane,

therefore      is an entire function.  

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

(3-14)              ,     

                        or  

(3-15)              .  

Aside.  The Mathematica solution uses the commands.  

Aside.  The Maple commands are similar.  

          >    

                              

          >    

                              

          >    

                              

Therefore, both Mathematica and Maple have shown that if  

                    ,   

then the derivative is

                    ,   

as expected.

We are really done.

Aside.  Figure E.3.4 a, shows the graphs of      and   .  

The partial derivatives of    are  

                         and     ,   

and the partial derivatives of    are

                         and     .  
        
At the point   ,   we have      and   ,   and these partial derivatives appear along

the edges of the surfaces for    at the points      and   ,   respectively.

Similarly,  at the point   ,   we have      and      and these partial derivatives appear

along the edges of the surfaces for at the points and  ,  respectively.

                    .                                                                          .
                                                                                  Figure E.3.4 a

                    ,                                                                         ,  

                    at we have .                                                at we have  .  
                                                                                  Figure E.3.4 b

                    ,                                                                         ,  

                    at we have .                                                at we have  .  
                                                                                  Figure E.3.4 c

                                        For the function      we see that

                                             and     .  

                                                                                  Figure E.3.4

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