Example 3.8.  Show that the function   [Graphics:Images/CauchyRiemannMod_gr_780.gif]  

is differentiable for all  [Graphics:Images/CauchyRiemannMod_gr_781.gif]  and find its derivative.

 

Explore Solution 3.8.

 

Solution.  We first observe that   

                    [Graphics:../Images/CauchyRiemannMod_gr_791.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_792.gif].   

Then compute the partial derivatives and get  

                    [Graphics:../Images/CauchyRiemannMod_gr_793.gif],     and

                    [Graphics:../Images/CauchyRiemannMod_gr_794.gif].  

Moreover, the partial derivatives  [Graphics:../Images/CauchyRiemannMod_gr_795.gif]  are continuous everywhere.   

By Theorem 3.4,   [Graphics:../Images/CauchyRiemannMod_gr_796.gif],  is differentiable everywhere.  

Therefore, using Equation (3-14) and (3-15), we have    

                    [Graphics:../Images/CauchyRiemannMod_gr_797.gif],     and  

                    [Graphics:../Images/CauchyRiemannMod_gr_798.gif].

 

We are done.

 

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_799.gif] and [Graphics:../Images/CauchyRiemannMod_gr_800.gif] can assist us in finding the partial derivatives.  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_801.gif]

[Graphics:../Images/CauchyRiemannMod_gr_802.gif]


[Graphics:../Images/CauchyRiemannMod_gr_803.gif]

[Graphics:../Images/CauchyRiemannMod_gr_804.gif]


[Graphics:../Images/CauchyRiemannMod_gr_805.gif]

[Graphics:../Images/CauchyRiemannMod_gr_806.gif]


[Graphics:../Images/CauchyRiemannMod_gr_807.gif]

[Graphics:../Images/CauchyRiemannMod_gr_808.gif]


[Graphics:../Images/CauchyRiemannMod_gr_809.gif]

[Graphics:../Images/CauchyRiemannMod_gr_810.gif]


[Graphics:../Images/CauchyRiemannMod_gr_811.gif]

[Graphics:../Images/CauchyRiemannMod_gr_812.gif]


[Graphics:../Images/CauchyRiemannMod_gr_813.gif]

[Graphics:../Images/CauchyRiemannMod_gr_814.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_815.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_816.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_817.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_818.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_819.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_820.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_821.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_822.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_823.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_824.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_825.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_826.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_827.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_828.gif]

The Cauchy-Riemann equations hold all points  [Graphics:../Images/CauchyRiemannMod_gr_829.gif]  in the complex plane,

therefore    [Graphics:../Images/CauchyRiemannMod_gr_830.gif]   is an entire function.  

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

(3-14)              [Graphics:../Images/CauchyRiemannMod_gr_831.gif],     or

(3-15)              [Graphics:../Images/CauchyRiemannMod_gr_832.gif].  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_833.gif]

[Graphics:../Images/CauchyRiemannMod_gr_834.gif]


[Graphics:../Images/CauchyRiemannMod_gr_835.gif]

[Graphics:../Images/CauchyRiemannMod_gr_836.gif]


[Graphics:../Images/CauchyRiemannMod_gr_837.gif]

[Graphics:../Images/CauchyRiemannMod_gr_838.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_839.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_840.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_841.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_842.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_843.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_844.gif]

 

Remark 1.  In Section 5.1, we will develop formula (5-1) for the complex exponential function, which is  

                    [Graphics:../Images/CauchyRiemannMod_gr_845.gif].  

Then we can replace  [Graphics:../Images/CauchyRiemannMod_gr_846.gif]  in the above formula and see that our function [Graphics:../Images/CauchyRiemannMod_gr_847.gif] really is

                    [Graphics:../Images/CauchyRiemannMod_gr_848.gif].

The rules for differentiation of the real exponential function will be and extended for  [Graphics:../Images/CauchyRiemannMod_gr_849.gif]  and we will have  

                    [Graphics:../Images/CauchyRiemannMod_gr_850.gif].  

Combining this with the previous relation will permit us to write

                    [Graphics:../Images/CauchyRiemannMod_gr_851.gif]  

Which is exactly the result we obtained by using formulas (3-14) and (3-15).  

We still have important material to learn before we get to Section 5.1 and the complex exponential function.

Do not worry about these formulas.  There will be plenty of time to get them.

 

Remark 2.  This example shows how the Cauchy-Riemann equations are useful for differentiating some functions

that are given in the Cartesian coordinate form.  In Section 3.3 we show that the Cauchy-Riemann equations are useful

for constructing a conjugate harmonic function.

 

Remark 3.  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