Module

for

Conformal Mapping

 

Chapter 10  Conformal Mapping

Overview

    The terminology "conformal mapping" should have a familiar sound.  In 1569 the Flemish cartographer Gerardus Mercator (1512--1594) devised a cylindrical map projection that preserves angles.  The Mercator projection is still used today for world maps.  Another map projection known to the ancient Greeks is the stereographic projection. It is also conformal (i.e., angle preserving), and we introduced it in Section 2.5 when we defined the Riemann sphere. In complex analysis a function preserves angles if and only if it is analytic or anti-analytic (i.e., the conjugate of an analytic function).  A significant result, known as Riemann mapping theorem, states that any simply connected domain (other than the entire complex plane) can be mapped conformally onto the unit disk.

 

10.1  Basic Properties of Conformal Mappings

    Let f(z) be an analytic function in the domain D, and let  [Graphics:Images/ConformalMappingMod_gr_1.gif]  be a point in D.  If  [Graphics:Images/ConformalMappingMod_gr_2.gif],  then we can express f(z) in the form  

(10-1)            [Graphics:Images/ConformalMappingMod_gr_3.gif],  

where  [Graphics:Images/ConformalMappingMod_gr_4.gif].  If z is near [Graphics:Images/ConformalMappingMod_gr_5.gif], then the transformation  [Graphics:Images/ConformalMappingMod_gr_6.gif]  has the linear approximation  

            [Graphics:Images/ConformalMappingMod_gr_7.gif],  

where  [Graphics:Images/ConformalMappingMod_gr_8.gif].  Because  [Graphics:Images/ConformalMappingMod_gr_9.gif]  when  [Graphics:Images/ConformalMappingMod_gr_10.gif],  for points near [Graphics:Images/ConformalMappingMod_gr_11.gif] the transformation  [Graphics:Images/ConformalMappingMod_gr_12.gif]  has an effect much like the linear mapping  [Graphics:Images/ConformalMappingMod_gr_13.gif].  The effect of the linear mapping  S  is a rotation of the plane through the angle  [Graphics:Images/ConformalMappingMod_gr_14.gif],  followed by a magnification by the factor  [Graphics:Images/ConformalMappingMod_gr_15.gif],  followed by a rigid translation by the vector  [Graphics:Images/ConformalMappingMod_gr_16.gif].  Consequently, the mapping  [Graphics:Images/ConformalMappingMod_gr_17.gif]  preserves angles at the point  [Graphics:Images/ConformalMappingMod_gr_18.gif].  We now show that the mapping  [Graphics:Images/ConformalMappingMod_gr_19.gif]  also preserves angles at  [Graphics:Images/ConformalMappingMod_gr_20.gif].  


    For a smooth curve that passes through the point  [Graphics:Images/ConformalMappingMod_gr_21.gif],  we use the notation  

            [Graphics:Images/ConformalMappingMod_gr_22.gif],  

for  [Graphics:Images/ConformalMappingMod_gr_23.gif].   
    
    A vector [Graphics:Images/ConformalMappingMod_gr_24.gif] tangent to C at the point [Graphics:Images/ConformalMappingMod_gr_25.gif] is given by  

            [Graphics:Images/ConformalMappingMod_gr_26.gif],  
            
where the complex number  [Graphics:Images/ConformalMappingMod_gr_27.gif]  is expressed as a vector.

    The angle of inclination of  [Graphics:Images/ConformalMappingMod_gr_28.gif]  with respect to the positive x axis is  

            [Graphics:Images/ConformalMappingMod_gr_29.gif].  

The image of C under the mapping [Graphics:Images/ConformalMappingMod_gr_30.gif] is the curve K in the w plane given by the formula

            [Graphics:Images/ConformalMappingMod_gr_31.gif].  

We can use the chain rule to show that a vector  [Graphics:Images/ConformalMappingMod_gr_32.gif]  tangent to K at the point  [Graphics:Images/ConformalMappingMod_gr_33.gif]  is given by  

            [Graphics:Images/ConformalMappingMod_gr_34.gif].  

The angle of inclination of [Graphics:Images/ConformalMappingMod_gr_35.gif] with respect to the positive u axis is

            [Graphics:Images/ConformalMappingMod_gr_36.gif],   

where  [Graphics:Images/ConformalMappingMod_gr_37.gif].  

    Therefore the effect of the transformation [Graphics:Images/ConformalMappingMod_gr_38.gif] is to rotate the angle of inclination of the tangent vector [Graphics:Images/ConformalMappingMod_gr_39.gif] at [Graphics:Images/ConformalMappingMod_gr_40.gif] through the angle [Graphics:Images/ConformalMappingMod_gr_41.gif]  to obtain the angle of inclination of the tangent vector [Graphics:Images/ConformalMappingMod_gr_42.gif] at [Graphics:Images/ConformalMappingMod_gr_43.gif].  This situation is illustrated in Figure 10.1.

Figure 10.1  The tangents at the points [Graphics:Images/ConformalMappingMod_gr_44.gif], where f(z) is an analytic function and [Graphics:Images/ConformalMappingMod_gr_45.gif].

 

    A mapping   [Graphics:Images/ConformalMappingMod_gr_46.gif]  is said to be angle preserving, or conformal at [Graphics:Images/ConformalMappingMod_gr_47.gif], if it preserves angles between oriented curves in magnitude as well as in orientation.  Theorem 10.1 shows where a mapping by an analytic function is conformal.

 

Theorem 10.1 (Conformal Mapping).  Let f(z) be an analytic function in the domain D, and let [Graphics:Images/ConformalMappingMod_gr_48.gif] be a point in D.  If  [Graphics:Images/ConformalMappingMod_gr_49.gif],  then f(z) is conformal at [Graphics:Images/ConformalMappingMod_gr_50.gif].

Figure 10.2  The analytic mapping [Graphics:Images/ConformalMappingMod_gr_51.gif] is conformal at the point [Graphics:Images/ConformalMappingMod_gr_52.gif], where [Graphics:Images/ConformalMappingMod_gr_53.gif].

Proof.

 

Example 10.1.  Show that the mapping  [Graphics:Images/ConformalMappingMod_gr_54.gif]  is conformal at the points [Graphics:Images/ConformalMappingMod_gr_55.gif], [Graphics:Images/ConformalMappingMod_gr_56.gif], [Graphics:Images/ConformalMappingMod_gr_57.gif] and , [Graphics:Images/ConformalMappingMod_gr_58.gif] and determine the angle of rotation given by [Graphics:Images/ConformalMappingMod_gr_59.gif] at the given points.

          [Graphics:Images/ConformalMappingMod.0_gr_7.gif]          [Graphics:Images/ConformalMappingMod.0_gr_8.gif]

Solution.  Because [Graphics:Images/ConformalMappingMod_gr_61.gif], we conclude that the mapping  [Graphics:Images/ConformalMappingMod_gr_62.gif]  is conformal at all points except [Graphics:Images/ConformalMappingMod_gr_63.gif], where n is an integer.  

Calculation reveals that  

            [Graphics:Images/ConformalMappingMod.0_gr_1.gif]   
 
Therefore the angle of rotation is given by  

            [Graphics:Images/ConformalMappingMod.0_gr_2.gif]     

Explore Solution 10.1.

 

    Let f(z) be a nonconstant analytic function.  If [Graphics:Images/ConformalMappingMod_gr_107.gif], then [Graphics:Images/ConformalMappingMod_gr_108.gif] is called a critical point of f(z), and the mapping [Graphics:Images/ConformalMappingMod_gr_109.gif] is not conformal at [Graphics:Images/ConformalMappingMod_gr_110.gif].  The next result shows what happens at a critical point.

 

Theorem 10.2.  Let f(z) be analytic at the point [Graphics:Images/ConformalMappingMod_gr_111.gif].  If [Graphics:Images/ConformalMappingMod_gr_112.gif] [Graphics:Images/ConformalMappingMod_gr_113.gif]and  [Graphics:Images/ConformalMappingMod_gr_114.gif],  then the mapping [Graphics:Images/ConformalMappingMod_gr_115.gif] magnifies angles at the vertex [Graphics:Images/ConformalMappingMod_gr_116.gif] by the factor k, as shown in Figure 10.3.

Figure 10.3  The analytic mapping [Graphics:Images/ConformalMappingMod_gr_117.gif] at point [Graphics:Images/ConformalMappingMod_gr_118.gif], where [Graphics:Images/ConformalMappingMod_gr_119.gif] [Graphics:Images/ConformalMappingMod_gr_120.gif]and  [Graphics:Images/ConformalMappingMod_gr_121.gif].

Proof.

 

Example 10.2.  Show that the mapping [Graphics:Images/ConformalMappingMod_gr_122.gif] maps the unit square [Graphics:Images/ConformalMappingMod_gr_123.gif] onto the region in the upper half-plane [Graphics:Images/ConformalMappingMod_gr_124.gif], which lies under the parabolas  

            [Graphics:Images/ConformalMappingMod_gr_125.gif]   and   [Graphics:Images/ConformalMappingMod_gr_126.gif]

as shown in Figure 10.4.

          [Graphics:Images/ConformalMappingMod.0_gr_11.gif]          [Graphics:Images/ConformalMappingMod.0_gr_12.gif]

                                        Figure 10.4  The mapping  [Graphics:Images/ConformalMappingMod_gr_128.gif].

Solution.  The derivative is  [Graphics:Images/ConformalMappingMod_gr_129.gif],  and we conclude that the mapping  [Graphics:Images/ConformalMappingMod_gr_130.gif]  is conformal for all  [Graphics:Images/ConformalMappingMod_gr_131.gif].   Note that the right angles at the vertices  [Graphics:Images/ConformalMappingMod_gr_132.gif], [Graphics:Images/ConformalMappingMod_gr_133.gif], and [Graphics:Images/ConformalMappingMod_gr_134.gif]  are mapped onto right angles at the vertices  [Graphics:Images/ConformalMappingMod_gr_135.gif], [Graphics:Images/ConformalMappingMod_gr_136.gif], and [Graphics:Images/ConformalMappingMod_gr_137.gif],  respectively.  At the point  [Graphics:Images/ConformalMappingMod_gr_138.gif],  we have  [Graphics:Images/ConformalMappingMod_gr_139.gif]  and  [Graphics:Images/ConformalMappingMod_gr_140.gif].  Hence angles at the vertex  [Graphics:Images/ConformalMappingMod_gr_141.gif]  are magnified by the factor [Graphics:Images/ConformalMappingMod_gr_142.gif].  In particular, the right angle at [Graphics:Images/ConformalMappingMod_gr_143.gif] is mapped onto the straight angle at [Graphics:Images/ConformalMappingMod_gr_144.gif].

          

Explore Solution 10.2.

 

 

    Another property of a conformal mapping   [Graphics:Images/ConformalMappingMod_gr_151.gif]  is obtained by considering the modulus of  [Graphics:Images/ConformalMappingMod_gr_152.gif].  If  [Graphics:Images/ConformalMappingMod_gr_153.gif]  is near  [Graphics:Images/ConformalMappingMod_gr_154.gif],  we can use the equation  

            [Graphics:Images/ConformalMappingMod_gr_155.gif]

and neglect the term  [Graphics:Images/ConformalMappingMod_gr_156.gif].  We then have the approximation

(10-9)            [Graphics:Images/ConformalMappingMod_gr_157.gif].  

    From Equation (10-9), the distance  [Graphics:Images/ConformalMappingMod_gr_158.gif]  between the images of the points  [Graphics:Images/ConformalMappingMod_gr_159.gif]  and  [Graphics:Images/ConformalMappingMod_gr_160.gif]  given approximately by  [Graphics:Images/ConformalMappingMod_gr_161.gif].  Therefore we say that the transformation   [Graphics:Images/ConformalMappingMod_gr_162.gif]  changes small distances near [Graphics:Images/ConformalMappingMod_gr_163.gif] by the scale factor [Graphics:Images/ConformalMappingMod_gr_164.gif].   For example, the scale factor of the transformation  [Graphics:Images/ConformalMappingMod_gr_165.gif]  near the point  [Graphics:Images/ConformalMappingMod_gr_166.gif]  is  [Graphics:Images/ConformalMappingMod_gr_167.gif].

    We also need to say a few things about the inverse transformation   [Graphics:Images/ConformalMappingMod_gr_168.gif]  of a conformal mapping   [Graphics:Images/ConformalMappingMod_gr_169.gif]  near a point [Graphics:Images/ConformalMappingMod_gr_170.gif], where [Graphics:Images/ConformalMappingMod_gr_171.gif].  
A complete justification of the following assertions relies on theorems studied in advanced calculus. (See, for instance, R. Creighton Buck, Advanced Calculus, 3rd ed. (New York, McGraw-Hill), pp. 358-361, 1978.)  

    We express the mapping   [Graphics:Images/ConformalMappingMod_gr_172.gif]  in the coordinate form  

(10-10)        [Graphics:Images/ConformalMappingMod_gr_173.gif].  

    The mapping in Equations (10-10) represents a transformation from the xy plane into the uv plane, and the Jacobian determinant, [Graphics:Images/ConformalMappingMod_gr_174.gif], is defined by  

(10-11)        [Graphics:Images/ConformalMappingMod_gr_175.gif].  

    The transformation in Equations (10-10) has a local inverse, provided  [Graphics:Images/ConformalMappingMod_gr_176.gif].  Expanding Equation (10-11) and using the Cauchy--Riemann equations, we obtain  

            [Graphics:Images/ConformalMappingMod_gr_177.gif]  

    Consequently, Equations (10-11) and (10-11) imply that a local inverse [Graphics:Images/ConformalMappingMod_gr_178.gif] exists in a neighborhood of the point  [Graphics:Images/ConformalMappingMod_gr_179.gif].  The derivative of g(w) at [Graphics:Images/ConformalMappingMod_gr_180.gif] is given by the familiar expression  

            [Graphics:Images/ConformalMappingMod_gr_181.gif]   

 

Exercises for Section 10.1.  Basic Properties of Conformal Mappings

 

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