Theorem 3.3 (Cauchy-Riemann Equations).  Suppose that  

(3-14)              [Graphics:Images/CauchyRiemannMod_gr_170.gif],  

is differentiable at the point  [Graphics:Images/CauchyRiemannMod_gr_171.gif].   Then the partial derivatives of  [Graphics:Images/CauchyRiemannMod_gr_172.gif]  exist at the point  [Graphics:Images/CauchyRiemannMod_gr_173.gif],  

and can be used to calculate the derivative at [Graphics:Images/CauchyRiemannMod_gr_174.gif].  That is,    

(3-14)              [Graphics:Images/CauchyRiemannMod_gr_175.gif],     

                        and also  

(3-15)              [Graphics:Images/CauchyRiemannMod_gr_176.gif].  

Equating the real and imaginary parts of Equations (3-14) and (3-15) gives the so-called Cauchy-Riemann Equations:

(3-16)              [Graphics:Images/CauchyRiemannMod_gr_177.gif]     and     [Graphics:Images/CauchyRiemannMod_gr_178.gif].  

 

Exploration for the Cauchy-Riemann Equations.

 

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_179.gif] and [Graphics:../Images/CauchyRiemannMod_gr_180.gif] can assist us in calculating limits.  

 

Aside.  The Mathematica solution uses the command.  

[Graphics:../Images/CauchyRiemannMod_gr_181.gif]

[Graphics:../Images/CauchyRiemannMod_gr_182.gif]


[Graphics:../Images/CauchyRiemannMod_gr_183.gif]
[Graphics:../Images/CauchyRiemannMod_gr_184.gif]

[Graphics:../Images/CauchyRiemannMod_gr_185.gif]



[Graphics:../Images/CauchyRiemannMod_gr_186.gif]

[Graphics:../Images/CauchyRiemannMod_gr_187.gif]

Looking at the above limits, and equating the real and imaginary parts we have the following equations.

                    [Graphics:../Images/CauchyRiemannMod_gr_188.gif],    

                    and  

                    [Graphics:../Images/CauchyRiemannMod_gr_189.gif] .  

 

In Mathematica the syntax for partial derivatives can be explained as follows.

In the expression  [Graphics:../Images/CauchyRiemannMod_gr_190.gif],  the superscript  [Graphics:../Images/CauchyRiemannMod_gr_191.gif]  means, take one derivative with respect to the first variable [Graphics:../Images/CauchyRiemannMod_gr_192.gif].

In the expression  [Graphics:../Images/CauchyRiemannMod_gr_193.gif],  the superscript  [Graphics:../Images/CauchyRiemannMod_gr_194.gif]  means, take one derivative with respect to the second variable [Graphics:../Images/CauchyRiemannMod_gr_195.gif].

Similarly, the expressions  [Graphics:../Images/CauchyRiemannMod_gr_196.gif]  and  [Graphics:../Images/CauchyRiemannMod_gr_197.gif]  are the  [Graphics:../Images/CauchyRiemannMod_gr_198.gif]  and  [Graphics:../Images/CauchyRiemannMod_gr_199.gif]  partial derivatives, respectively.

        Therefore, we see that Mathematica can establish the Cauchy-Riemann equations  

                    [Graphics:../Images/CauchyRiemannMod_gr_200.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_201.gif].   

 

We are done.

 

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_202.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_203.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_204.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_205.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_206.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_207.gif]

Looking at the above limits, and equating the real and imaginary parts we have the following equations.

                    [Graphics:../Images/CauchyRiemannMod_gr_208.gif],    and  

                    [Graphics:../Images/CauchyRiemannMod_gr_209.gif] .  

 

In Maple the syntax for partial derivatives can be explained as follows.

In the expression  [Graphics:../Images/CauchyRiemannMod_gr_210.gif],  the subscript  [Graphics:../Images/CauchyRiemannMod_gr_211.gif]  means, take one derivative with respect to the first variable [Graphics:../Images/CauchyRiemannMod_gr_212.gif].

In the expression  [Graphics:../Images/CauchyRiemannMod_gr_213.gif],  the subscript  [Graphics:../Images/CauchyRiemannMod_gr_214.gif]  means, take one derivative with respect to the second variable [Graphics:../Images/CauchyRiemannMod_gr_215.gif].

Similarly, the expressions  [Graphics:../Images/CauchyRiemannMod_gr_216.gif]  and  [Graphics:../Images/CauchyRiemannMod_gr_217.gif]  are the  [Graphics:../Images/CauchyRiemannMod_gr_218.gif]  and  [Graphics:../Images/CauchyRiemannMod_gr_219.gif]  partial derivatives, respectively.

        Therefore, we see that Maple can establish the Cauchy-Riemann equations  

                    [Graphics:../Images/CauchyRiemannMod_gr_220.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_221.gif].   

 

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