The Special Cartesian Limits.

 

        For the Cartesian coordinate form of a complex function

                    [Graphics:Images/CauchyRiemannMod_gr_15.gif],

it is important to determine how the function values change as we move along the horizontal grid line  

                    [Graphics:Images/CauchyRiemannMod_gr_16.gif]    at the point    [Graphics:Images/CauchyRiemannMod_gr_17.gif],  

and how the function values change as we move along the vertical grid line  

                    [Graphics:Images/CauchyRiemannMod_gr_18.gif]    at the point    [Graphics:Images/CauchyRiemannMod_gr_19.gif].  

        We investigate these two approaches: a horizontal approach and a vertical approach to  [Graphics:Images/CauchyRiemannMod_gr_20.gif].  Recall from our graphical

analysis of   [Graphics:Images/CauchyRiemannMod_gr_21.gif]   in Example 2.12, in Section 2.2, that the image of a square is a "curvilinear quadrilateral" and the

images of the horizontal and vertical edges are portions of parabolas in the  [Graphics:Images/CauchyRiemannMod_gr_22.gif]-plane.   For convenience, we let the square have

vertices   [Graphics:Images/CauchyRiemannMod_gr_23.gif],    [Graphics:Images/CauchyRiemannMod_gr_24.gif],    [Graphics:Images/CauchyRiemannMod_gr_25.gif],   and   [Graphics:Images/CauchyRiemannMod_gr_26.gif].    Then the image

points are  [Graphics:Images/CauchyRiemannMod_gr_27.gif],    [Graphics:Images/CauchyRiemannMod_gr_28.gif],    [Graphics:Images/CauchyRiemannMod_gr_29.gif],   and   [Graphics:Images/CauchyRiemannMod_gr_30.gif],  

as shown in Figure 3.1.  

          [Graphics:Images/CauchyRiemannMod_gr_31.gif]          [Graphics:Images/CauchyRiemannMod_gr_32.gif]  

                  Figure 3.1  The image of a small square under the mapping    [Graphics:Images/CauchyRiemannMod_gr_33.gif],  

                                     the vertex  vertex   [Graphics:Images/CauchyRiemannMod_gr_34.gif],   is mapped onto the point   [Graphics:Images/CauchyRiemannMod_gr_35.gif].  

 

Exploration

 

        We know that   [Graphics:../Images/CauchyRiemannMod_gr_60.gif]   is differentiable, so the limit of the difference quotient   [Graphics:../Images/CauchyRiemannMod_gr_61.gif]   exists no matter how

we approach   [Graphics:../Images/CauchyRiemannMod_gr_62.gif].   Let us zoom in on the point   [Graphics:../Images/CauchyRiemannMod_gr_63.gif]   and investigate the two special Cartesian limits.

 

[Graphics:../Images/CauchyRiemannMod_gr_64.gif]     [Graphics:../Images/CauchyRiemannMod_gr_65.gif]  

                    The image of the square   [Graphics:../Images/CauchyRiemannMod_gr_66.gif]   under the mapping   [Graphics:../Images/CauchyRiemannMod_gr_67.gif].  

 

[Graphics:../Images/CauchyRiemannMod_gr_68.gif]     [Graphics:../Images/CauchyRiemannMod_gr_69.gif]  

                    The image of the square   [Graphics:../Images/CauchyRiemannMod_gr_70.gif]   under the mapping   [Graphics:../Images/CauchyRiemannMod_gr_71.gif].  

 

[Graphics:../Images/CauchyRiemannMod_gr_72.gif]     [Graphics:../Images/CauchyRiemannMod_gr_73.gif]  

                    The image of the square   [Graphics:../Images/CauchyRiemannMod_gr_74.gif]   under the mapping   [Graphics:../Images/CauchyRiemannMod_gr_75.gif].  

 

The vertices   [Graphics:../Images/CauchyRiemannMod_gr_76.gif],   [Graphics:../Images/CauchyRiemannMod_gr_77.gif],   [Graphics:../Images/CauchyRiemannMod_gr_78.gif],   and   [Graphics:../Images/CauchyRiemannMod_gr_79.gif]   are mapped onto the

image points   [Graphics:../Images/CauchyRiemannMod_gr_80.gif],   [Graphics:../Images/CauchyRiemannMod_gr_81.gif],   [Graphics:../Images/CauchyRiemannMod_gr_82.gif],   and   [Graphics:../Images/CauchyRiemannMod_gr_83.gif].  

        We know that   [Graphics:../Images/CauchyRiemannMod_gr_84.gif]   is differentiable, so the limit of the difference quotient   [Graphics:../Images/CauchyRiemannMod_gr_85.gif]   exists no

matter how we approach   [Graphics:../Images/CauchyRiemannMod_gr_86.gif].   Let us investigate the two special Cartesian limits.

 

Solution Using Numerical Approximations

 

Method 1.  Investigate numerical approximations of the difference quotients   [Graphics:../Images/CauchyRiemannMod_gr_87.gif]   

along the horizontal line   [Graphics:../Images/CauchyRiemannMod_gr_88.gif].  

First, we can numerically approximate  [Graphics:../Images/CauchyRiemannMod_gr_89.gif]  by using a horizontal increment in [Graphics:../Images/CauchyRiemannMod_gr_90.gif].   

Use  [Graphics:../Images/CauchyRiemannMod_gr_91.gif]  and  [Graphics:../Images/CauchyRiemannMod_gr_92.gif]   where   [Graphics:../Images/CauchyRiemannMod_gr_93.gif]

to compute the difference quotient.

                    [Graphics:../Images/CauchyRiemannMod_gr_94.gif]   

 

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_95.gif] and [Graphics:../Images/CauchyRiemannMod_gr_96.gif] can assist us with numerical approximations.  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_97.gif]
[Graphics:../Images/CauchyRiemannMod_gr_98.gif]

[Graphics:../Images/CauchyRiemannMod_gr_99.gif]


[Graphics:../Images/CauchyRiemannMod_gr_100.gif]

[Graphics:../Images/CauchyRiemannMod_gr_101.gif]


[Graphics:../Images/CauchyRiemannMod_gr_102.gif]

[Graphics:../Images/CauchyRiemannMod_gr_103.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_106.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_107.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_108.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_109.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_110.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_111.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_112.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_113.gif]

 

Method 2.  Investigate numerical approximations of the difference quotients   [Graphics:../Images/CauchyRiemannMod_gr_114.gif]   

along the vertical line   [Graphics:../Images/CauchyRiemannMod_gr_115.gif].  

Second, we can numerically approximate   [Graphics:../Images/CauchyRiemannMod_gr_116.gif]   by using a vertical increment in [Graphics:../Images/CauchyRiemannMod_gr_117.gif].  

Use   [Graphics:../Images/CauchyRiemannMod_gr_118.gif]   and   [Graphics:../Images/CauchyRiemannMod_gr_119.gif]   where   [Graphics:../Images/CauchyRiemannMod_gr_120.gif]

to compute the difference quotient.

                    [Graphics:../Images/CauchyRiemannMod_gr_121.gif]      

 

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_122.gif] and [Graphics:../Images/CauchyRiemannMod_gr_123.gif] can assist us with numerical approximations.  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_124.gif]
[Graphics:../Images/CauchyRiemannMod_gr_125.gif]

[Graphics:../Images/CauchyRiemannMod_gr_126.gif]


[Graphics:../Images/CauchyRiemannMod_gr_127.gif]

[Graphics:../Images/CauchyRiemannMod_gr_128.gif]


[Graphics:../Images/CauchyRiemannMod_gr_129.gif]

[Graphics:../Images/CauchyRiemannMod_gr_130.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_131.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_132.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_133.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_134.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_135.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_136.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_137.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_138.gif]

 

Comparing these two numerical approximations we see that

                    [Graphics:../Images/CauchyRiemannMod_gr_139.gif],     

                    and  

                    [Graphics:../Images/CauchyRiemannMod_gr_140.gif],  

which leads us to speculate that   [Graphics:../Images/CauchyRiemannMod_gr_141.gif].  

 

We are done.

 

    These numerical approximations lead to the idea of taking limits along the horizontal and vertical directions.  

 

Solution Using Limits

 

Method 1.  Investigate the limit of the difference quotients   [Graphics:../Images/CauchyRiemannMod_gr_142.gif]   

along the horizontal line   [Graphics:../Images/CauchyRiemannMod_gr_143.gif].  

First, we can taking the limit along the horizontal direction.  

                    [Graphics:../Images/CauchyRiemannMod_gr_144.gif]   

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_145.gif] and [Graphics:../Images/CauchyRiemannMod_gr_146.gif] can assist us in calculating limits.  

Aside.  The Mathematica solution uses the commands.  

 

[Graphics:../Images/CauchyRiemannMod_gr_147.gif]
[Graphics:../Images/CauchyRiemannMod_gr_148.gif]

[Graphics:../Images/CauchyRiemannMod_gr_149.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_150.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_151.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_152.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_153.gif]

 

Method 2.  Investigate the limit of the difference quotients   [Graphics:../Images/CauchyRiemannMod_gr_154.gif]   

along the vertical line   [Graphics:../Images/CauchyRiemannMod_gr_155.gif].  

Second, we can taking the limit along the vertical direction.  

                    [Graphics:../Images/CauchyRiemannMod_gr_156.gif]  

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_157.gif] and [Graphics:../Images/CauchyRiemannMod_gr_158.gif] can assist us in calculating limits.  

Aside.  The Mathematica solution uses the commands.  

 

[Graphics:../Images/CauchyRiemannMod_gr_159.gif]
[Graphics:../Images/CauchyRiemannMod_gr_160.gif]

[Graphics:../Images/CauchyRiemannMod_gr_161.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_162.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_163.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_164.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_165.gif]

Comparing these two limits we see that

                    [Graphics:../Images/CauchyRiemannMod_gr_166.gif],     

                    and  

                    [Graphics:../Images/CauchyRiemannMod_gr_167.gif].  

 

Since the above two limits were not taken along all possible approaches to   [Graphics:../Images/CauchyRiemannMod_gr_168.gif],   they alone

are not sufficient to prove that   [Graphics:../Images/CauchyRiemannMod_gr_169.gif],   but they prepare our thinking for Theorem 3.3.

 

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