Module

for

Mapping Involving Elementary Functions

 

10.3  Mapping Involving Elementary Functions

    In Section 5.1 we saw that the function  [Graphics:Images/MapElementaryFunMod_gr_1.gif]  is a one-to-one mapping of the fundamental period strip  [Graphics:Images/MapElementaryFunMod_gr_2.gif]  in the z-plane onto the w-plane with the point  [Graphics:Images/MapElementaryFunMod_gr_3.gif]  deleted.  Because  [Graphics:Images/MapElementaryFunMod_gr_4.gif],  the mapping  [Graphics:Images/MapElementaryFunMod_gr_5.gif]  is a conformal mapping at each point z in the complex plane.  The family of horizontal lines  [Graphics:Images/MapElementaryFunMod_gr_6.gif],  and segments  [Graphics:Images/MapElementaryFunMod_gr_7.gif]  form an orthogonal grid in the fundamental period strip.  Their images under the mapping  [Graphics:Images/MapElementaryFunMod_gr_8.gif]  are the rays  [Graphics:Images/MapElementaryFunMod_gr_9.gif]  and the circles  [Graphics:Images/MapElementaryFunMod_gr_10.gif],  respectively.  These images form an orthogonal curvilinear grid in the w-plane, as shown in Figure 10.9. If  [Graphics:Images/MapElementaryFunMod_gr_11.gif],  then the rectangle  [Graphics:Images/MapElementaryFunMod_gr_12.gif]  is mapped one-to-one and onto the region  [Graphics:Images/MapElementaryFunMod_gr_13.gif].   The inverse mapping is the principal branch of the logarithm  [Graphics:Images/MapElementaryFunMod_gr_14.gif].  

Figure 10.9  The conformal mapping  [Graphics:Images/MapElementaryFunMod_gr_15.gif].

 

    In this section we show how compositions of conformal transformations are used to construct mappings with specified characteristics.

 

Example 10.8.  Show that the transformation  [Graphics:Images/MapElementaryFunMod_gr_16.gif]  is a one-to-one conformal mapping of the horizontal strip  [Graphics:Images/MapElementaryFunMod_gr_17.gif]  onto the disk  [Graphics:Images/MapElementaryFunMod_gr_18.gif].  Furthermore, the x-axis is mapped onto the lower semicircle bounding the disk, and the line  [Graphics:Images/MapElementaryFunMod_gr_19.gif]  is mapped onto the upper semicircle.

[Graphics:Images/MapElementaryFunMod_gr_20.gif]

Solution.  The function  [Graphics:Images/MapElementaryFunMod_gr_21.gif] is the composition of  [Graphics:Images/MapElementaryFunMod_gr_22.gif]  followed by  [Graphics:Images/MapElementaryFunMod_gr_23.gif].  The transformation [Graphics:Images/MapElementaryFunMod_gr_24.gif] maps the horizontal strip  [Graphics:Images/MapElementaryFunMod_gr_25.gif]  onto the upper half plane  [Graphics:Images/MapElementaryFunMod_gr_26.gif];  the x axis is mapped on to the positive X axis;  and the line  [Graphics:Images/MapElementaryFunMod_gr_27.gif]  is mapped onto the negative X axis.  Then the bilinear transformation  [Graphics:Images/MapElementaryFunMod_gr_28.gif]  maps the upper half plane    [Graphics:Images/MapElementaryFunMod_gr_29.gif]  onto the disk  [Graphics:Images/MapElementaryFunMod_gr_30.gif];  the positive X axis is mapped onto the lower semicircle;  and the negative X axis onto the upper semicircle.  Figure 10.10 illustrates the composite mapping.

Figure 10.10  The composite transformation  [Graphics:Images/MapElementaryFunMod_gr_31.gif].

Explore Solution 10.8.

 

Example10.9.  Show that the transformation  [Graphics:Images/MapElementaryFunMod_gr_47.gif]  is a one-to-one conformal mapping of the unit disk  [Graphics:Images/MapElementaryFunMod_gr_48.gif]  onto the horizontal strip  [Graphics:Images/MapElementaryFunMod_gr_49.gif].  Furthermore, the upper semicircle of the disk is mapped onto the line  [Graphics:Images/MapElementaryFunMod_gr_50.gif]  and the lower semicircle onto  [Graphics:Images/MapElementaryFunMod_gr_51.gif].  

[Graphics:Images/MapElementaryFunMod_gr_52.gif]

Solution.  The function  [Graphics:Images/MapElementaryFunMod_gr_53.gif]  is the composition of the bilinear transformation  [Graphics:Images/MapElementaryFunMod_gr_54.gif]  followed by the logarithmic mapping  [Graphics:Images/MapElementaryFunMod_gr_55.gif].  The image of the disk  [Graphics:Images/MapElementaryFunMod_gr_56.gif]  under the bilinear mapping  [Graphics:Images/MapElementaryFunMod_gr_57.gif]  is the right half-plane  [Graphics:Images/MapElementaryFunMod_gr_58.gif];  the upper semicircle is mapped onto the positive Y axis;  and the lower semicircle is mapped onto the negative Y axis.  The logarithmic function  [Graphics:Images/MapElementaryFunMod_gr_59.gif]  then maps the right half-plane onto the horizontal strip;  the image of the positive Y axis is the line  [Graphics:Images/MapElementaryFunMod_gr_60.gif];  and the image of the negative Y axis is the line  [Graphics:Images/MapElementaryFunMod_gr_61.gif].  Figure 10.11 shows the composite mapping.

Figure 10.11  The composite transformation  [Graphics:Images/MapElementaryFunMod_gr_62.gif].

Explore Solution 10.9.

 

Example 10.10.  Show that the transformation  [Graphics:Images/MapElementaryFunMod_gr_77.gif]  is a one-to-one conformal mapping of the portion of the unit disk  [Graphics:Images/MapElementaryFunMod_gr_78.gif]  that lies in the upper half-plane  [Graphics:Images/MapElementaryFunMod_gr_79.gif]  onto the upper half-plane  [Graphics:Images/MapElementaryFunMod_gr_80.gif].  Furthermore, the upper semicircular portion of the boundary is mapped onto the line negative u-axis, and the segment  [Graphics:Images/MapElementaryFunMod_gr_81.gif]  is mapped onto the positive u-axis.

[Graphics:Images/MapElementaryFunMod_gr_82.gif]

Solution.  The function  [Graphics:Images/MapElementaryFunMod_gr_83.gif]  is the composition of the bilinear transformation  [Graphics:Images/MapElementaryFunMod_gr_84.gif]  followed by the mapping  [Graphics:Images/MapElementaryFunMod_gr_85.gif].  The image of the half-disk under the bilinear mapping  [Graphics:Images/MapElementaryFunMod_gr_86.gif]  is the first quadrant  [Graphics:Images/MapElementaryFunMod_gr_87.gif];  the image of the segment  [Graphics:Images/MapElementaryFunMod_gr_88.gif],  is the positive X axis;  and the image of the semicircle is the positive Y axis.  The mapping  [Graphics:Images/MapElementaryFunMod_gr_89.gif]  then maps the first quadrant in the Z plane onto the upper half-plane  [Graphics:Images/MapElementaryFunMod_gr_90.gif],  as shown in Figure 10.12.

Figure 10.12  The composite transformation  [Graphics:Images/MapElementaryFunMod_gr_91.gif].

Explore Solution 10.10.

 

Example 10.11.  Consider the function  [Graphics:Images/MapElementaryFunMod_gr_107.gif],  which is the composition of the functions  [Graphics:Images/MapElementaryFunMod_gr_108.gif]  and  [Graphics:Images/MapElementaryFunMod_gr_109.gif]  where the branch of the square root is  [Graphics:Images/MapElementaryFunMod_gr_110.gif],  where  [Graphics:Images/MapElementaryFunMod_gr_111.gif] ,  [Graphics:Images/MapElementaryFunMod_gr_112.gif],  and  [Graphics:Images/MapElementaryFunMod_gr_113.gif].  Then the transformation  [Graphics:Images/MapElementaryFunMod_gr_114.gif]  maps the upper half-plane  [Graphics:Images/MapElementaryFunMod_gr_115.gif]  one-to-one and onto the upper-half plane  [Graphics:Images/MapElementaryFunMod_gr_116.gif]  slit along the segment  [Graphics:Images/MapElementaryFunMod_gr_117.gif].  

[Graphics:Images/MapElementaryFunMod_gr_118.gif]

Solution.  The function  [Graphics:Images/MapElementaryFunMod_gr_119.gif]  maps the upper half-plane  [Graphics:Images/MapElementaryFunMod_gr_120.gif]  one-to-one and onto the Z-plane slit along the ray  [Graphics:Images/MapElementaryFunMod_gr_121.gif].  Then the function  [Graphics:Images/MapElementaryFunMod_gr_122.gif]  maps the slit plane onto the slit half-plane, as shown in Figure 10.13.

Figure 10.13  The composite transformation  [Graphics:Images/MapElementaryFunMod_gr_123.gif]  and the intermediate steps  [Graphics:Images/MapElementaryFunMod_gr_124.gif]  and  [Graphics:Images/MapElementaryFunMod_gr_125.gif].

Explore Solution 10.11.

 

Remark 10.1.  The images of the horizontal lines [Graphics:Images/MapElementaryFunMod_gr_144.gif] are curves in the w plane that bend around the segment from  [Graphics:Images/MapElementaryFunMod_gr_145.gif].  The curves represent the streamlines of a fluid flowing across the w plane. We discuss fluid flows in more detail in Section 11.7.  

 

 

10.3.1  The Mapping  [Graphics:Images/MapElementaryFunMod_gr_146.gif]

    The double-valued function  [Graphics:Images/MapElementaryFunMod_gr_147.gif]  has a branch that is continuous for values of z distant from the origin.  This feature is motivated by our desire for the approximation  [Graphics:Images/MapElementaryFunMod_gr_148.gif]  to hold for values of z distant from the origin.  We begin by expressing [Graphics:Images/MapElementaryFunMod_gr_149.gif] as  

(10-22)        [Graphics:Images/MapElementaryFunMod_gr_150.gif],  

where the principal branch of the square root function is used in both factors.  We claim that the mapping  [Graphics:Images/MapElementaryFunMod_gr_151.gif]  is a one-to-one conformal mapping from the domain set [Graphics:Images/MapElementaryFunMod_gr_152.gif], consisting of the z plane slit along the segment  [Graphics:Images/MapElementaryFunMod_gr_153.gif],  onto the range set [Graphics:Images/MapElementaryFunMod_gr_154.gif], consisting of the w plane
slit along the segment  [Graphics:Images/MapElementaryFunMod_gr_155.gif].  

    To verify this claim, we investigate the two formulas on the right side of Equation (10-22) and express them in the form  

            [Graphics:Images/MapElementaryFunMod_gr_156.gif]  

    where  [Graphics:Images/MapElementaryFunMod_gr_157.gif]  and  [Graphics:Images/MapElementaryFunMod_gr_158.gif],  and  
    
                [Graphics:Images/MapElementaryFunMod_gr_159.gif]  

    where  [Graphics:Images/MapElementaryFunMod_gr_160.gif]  and  [Graphics:Images/MapElementaryFunMod_gr_161.gif].

    The discontinuities of  [Graphics:Images/MapElementaryFunMod_gr_162.gif]  are points on the real axis such that  [Graphics:Images/MapElementaryFunMod_gr_163.gif],  respectively.  We now show that [Graphics:Images/MapElementaryFunMod_gr_164.gif] is continuous on the ray  [Graphics:Images/MapElementaryFunMod_gr_165.gif].

    We let  [Graphics:Images/MapElementaryFunMod_gr_166.gif]  denote a point on the ray  [Graphics:Images/MapElementaryFunMod_gr_167.gif]  and then obtain the following limit as z approaches [Graphics:Images/MapElementaryFunMod_gr_168.gif] from the upper half-plane:   

            [Graphics:Images/MapElementaryFunMod_gr_169.gif]    

    We let  [Graphics:Images/MapElementaryFunMod_gr_170.gif]  denote a point on the ray  [Graphics:Images/MapElementaryFunMod_gr_171.gif]  and then obtain the following limits as z approaches [Graphics:Images/MapElementaryFunMod_gr_172.gif] from the lower half-plane:  

            [Graphics:Images/MapElementaryFunMod_gr_173.gif]      

    We can easily find the inverse mapping and express it similarly:  

            [Graphics:Images/MapElementaryFunMod_gr_174.gif],  

    where the branches of the square root function are given by  
    
                [Graphics:Images/MapElementaryFunMod_gr_175.gif]  

    where  [Graphics:Images/MapElementaryFunMod_gr_176.gif],  [Graphics:Images/MapElementaryFunMod_gr_177.gif],  and  [Graphics:Images/MapElementaryFunMod_gr_178.gif],  
    
                [Graphics:Images/MapElementaryFunMod_gr_179.gif]  

    where  [Graphics:Images/MapElementaryFunMod_gr_180.gif],  [Graphics:Images/MapElementaryFunMod_gr_181.gif],  and  [Graphics:Images/MapElementaryFunMod_gr_182.gif].

    A similar argument shows that  [Graphics:Images/MapElementaryFunMod_gr_183.gif]  is continuous for all w except those points that lie on the segment  [Graphics:Images/MapElementaryFunMod_gr_184.gif].  Verification that  

            [Graphics:Images/MapElementaryFunMod_gr_185.gif]   and   [Graphics:Images/MapElementaryFunMod_gr_186.gif]

hold for z in [Graphics:Images/MapElementaryFunMod_gr_187.gif] and w in [Graphics:Images/MapElementaryFunMod_gr_188.gif], respectively, is straightforward.  Therefore we conclude that  [Graphics:Images/MapElementaryFunMod_gr_189.gif]  is a one-to-one mapping from [Graphics:Images/MapElementaryFunMod_gr_190.gif] onto [Graphics:Images/MapElementaryFunMod_gr_191.gif].  Verifying that [Graphics:Images/MapElementaryFunMod_gr_192.gif] is also analytic on the ray  [Graphics:Images/MapElementaryFunMod_gr_193.gif],  is tedious.  We leave it as a challenging exercise.  

 

10.3.2  The Riemann Surface for  [Graphics:Images/MapElementaryFunMod_gr_194.gif]

    Using the other branch of the square root, we find that [Graphics:Images/MapElementaryFunMod_gr_195.gif], is a one-to-one conformal mapping from the domain set [Graphics:Images/MapElementaryFunMod_gr_196.gif] consisting of the z-plane slit along the segment [Graphics:Images/MapElementaryFunMod_gr_197.gif], onto the range set [Graphics:Images/MapElementaryFunMod_gr_198.gif] consisting of the w-plane slit along the segment [Graphics:Images/MapElementaryFunMod_gr_199.gif].  The sets [Graphics:Images/MapElementaryFunMod_gr_200.gif] and [Graphics:Images/MapElementaryFunMod_gr_201.gif] for [Graphics:Images/MapElementaryFunMod_gr_202.gif] and [Graphics:Images/MapElementaryFunMod_gr_203.gif] and [Graphics:Images/MapElementaryFunMod_gr_204.gif] form the Riemann surface for the mapping, as shown in Figure 10.14.  

Figure 10.14  The mappings  [Graphics:Images/MapElementaryFunMod_gr_205.gif]  and  [Graphics:Images/MapElementaryFunMod_gr_206.gif].  

 

    We obtain the Riemann surface for  [Graphics:Images/MapElementaryFunMod_gr_207.gif]  by gluing the edges of [Graphics:Images/MapElementaryFunMod_gr_208.gif][Graphics:Images/MapElementaryFunMod_gr_209.gif] together and the edges of [Graphics:Images/MapElementaryFunMod_gr_210.gif] together.  In the domain set, we glue edges  [Graphics:Images/MapElementaryFunMod_gr_211.gif],  [Graphics:Images/MapElementaryFunMod_gr_212.gif],  [Graphics:Images/MapElementaryFunMod_gr_213.gif],  and  [Graphics:Images/MapElementaryFunMod_gr_214.gif].  In the image set, we glue edges  [Graphics:Images/MapElementaryFunMod_gr_215.gif],  [Graphics:Images/MapElementaryFunMod_gr_216.gif],  [Graphics:Images/MapElementaryFunMod_gr_217.gif],  and   [Graphics:Images/MapElementaryFunMod_gr_218.gif].  The result is a Riemann domain surface and Riemann image surface for the mapping, as illustrated in Figures 10.15(a) and 10.15(b), respectively.

 

[Graphics:Images/MapElementaryFunMod_gr_219.gif] [Graphics:Images/MapElementaryFunMod_gr_220.gif]

Figure 10.15  The Riemann surfaces for the mapping  [Graphics:Images/MapElementaryFunMod_gr_221.gif].

 

Exercises for Section 10.3.  Mappings Involving Elementary Functions

 

Library Research Experience for Undergraduates

Conformal Mapping

Exponential Function

Complex Logarithms

Mobius - Bilinear Transformation

Smith Chart

Quasiconformal Mapping

Riemann Surfaces

 

 

 

 

The Next Module is

Mapping Involving Trigonometric Functions

 

 

Return to the Complex Analysis Modules  

 

 

Return to the Complex Analysis Project

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2012 John H. Mathews, Russell W. Howell