Dictionary of Conformal Mapping
Part III

Common Mappings

 

Example 26.  The conformal mapping    [Graphics:Images/ConformalMapDictionary.3.1_gr_1.gif].
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.3.1_gr_2.gif],
also  [Graphics:Images/ConformalMapDictionary.3.1_gr_3.gif].  
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.3.1_gr_4.gif]  is the infinite strip  [Graphics:Images/ConformalMapDictionary.3.1_gr_5.gif]  slit along the horizontal ray  [Graphics:Images/ConformalMapDictionary.3.1_gr_6.gif].  
Remark. This is Exercise 5 in Section 11.9, and is illustrated in Figure 11.79.   
Hint: Set  [Graphics:Images/ConformalMapDictionary.3.1_gr_7.gif]  and  [Graphics:Images/ConformalMapDictionary.3.1_gr_8.gif], and the angles are[Graphics:Images/ConformalMapDictionary.3.1_gr_9.gif] [Graphics:Images/ConformalMapDictionary.3.1_gr_10.gif].

      [Graphics:Images/ConformalMapDictionary.3.1_gr_11.gif]     [Graphics:Images/ConformalMapDictionary.3.1_gr_12.gif]

Details 26.

Example 27.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.3.1_gr_43.gif].
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.3.1_gr_44.gif],  
also  [Graphics:Images/ConformalMapDictionary.3.1_gr_45.gif].  
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.3.1_gr_46.gif]  is the semi-infinite strip  [Graphics:Images/ConformalMapDictionary.3.1_gr_47.gif], and the angles are [Graphics:Images/ConformalMapDictionary.3.1_gr_48.gif].
Remark. Hint: Set  [Graphics:Images/ConformalMapDictionary.3.1_gr_49.gif]  and  [Graphics:Images/ConformalMapDictionary.3.1_gr_50.gif], and the angles are [Graphics:Images/ConformalMapDictionary.3.1_gr_51.gif].

      [Graphics:Images/ConformalMapDictionary.3.1_gr_52.gif]     [Graphics:Images/ConformalMapDictionary.3.1_gr_53.gif]

Details 27.

Example 28.  The conformal mapping   [Graphics:Images/ConformalMapDictionary.3.1_gr_80.gif].
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.3.1_gr_81.gif].  
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.3.1_gr_82.gif]  is the three quadrants IV, I, II slit along the segment  [Graphics:Images/ConformalMapDictionary.3.1_gr_83.gif].
Remark. This is Exercise 11 (Extra Example 4) in Section 11.9 and Exercise 4 in Section 11.10, and is illustrated in Figure 11.85 and Figure 11.91.
Hint: Set  [Graphics:Images/ConformalMapDictionary.3.1_gr_84.gif]  and  [Graphics:Images/ConformalMapDictionary.3.1_gr_85.gif], and the angles are [Graphics:Images/ConformalMapDictionary.3.1_gr_86.gif].

      [Graphics:Images/ConformalMapDictionary.3.1_gr_87.gif]     [Graphics:Images/ConformalMapDictionary.3.1_gr_88.gif]

Details 28.

Example 29.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.3.1_gr_94.gif].
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.3.1_gr_95.gif].
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.3.1_gr_96.gif]  is the upper half-plane slit along the segment from  [Graphics:Images/ConformalMapDictionary.3.1_gr_97.gif].
Remark. This is Exercise 9 in Section 11.9 and Exercise 3 (ExtraExample 3) in  Section 11.10, and is illustrated in Figure 11.83 and  Figure 11.90.
Hint: Set  [Graphics:Images/ConformalMapDictionary.3.1_gr_98.gif].  In the w-plane [Graphics:Images/ConformalMapDictionary.3.1_gr_99.gif], and the exterior angles are  [Graphics:Images/ConformalMapDictionary.3.1_gr_100.gif].

      [Graphics:Images/ConformalMapDictionary.3.1_gr_101.gif]     [Graphics:Images/ConformalMapDictionary.3.1_gr_102.gif]

Details 29.

Example 30.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.3.1_gr_104.gif].
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.3.1_gr_105.gif],  
also  [Graphics:Images/ConformalMapDictionary.3.1_gr_106.gif].  
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.3.1_gr_107.gif]  is the exterior of the semi-infinite strip  [Graphics:Images/ConformalMapDictionary.3.1_gr_108.gif].
Remark. This is Exercise 6 in Section 11.9 and Exercise 2 (Extra Example 2) in  Section 11.10, and is illustrated in Figure 11.80 and Figure 11.89.
Hint: Set  [Graphics:Images/ConformalMapDictionary.3.1_gr_109.gif].  In the w-plane [Graphics:Images/ConformalMapDictionary.3.1_gr_110.gif], and the exterior angles are  [Graphics:Images/ConformalMapDictionary.3.1_gr_111.gif].

                 [Graphics:Images/ConformalMapDictionary.3.1_gr_112.gif]          [Graphics:Images/ConformalMapDictionary.3.1_gr_113.gif]

Details 30.

Example 31.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.3.1_gr_119.gif].
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.3.1_gr_120.gif],
also  [Graphics:Images/ConformalMapDictionary.3.1_gr_121.gif].
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.3.1_gr_122.gif]  is the portion of the first quadrant  [Graphics:Images/ConformalMapDictionary.3.1_gr_123.gif]  that lies below the region where  [Graphics:Images/ConformalMapDictionary.3.1_gr_124.gif],
the right angle channel in the first quadrant, which is bounded by the coordinate axes and the rays  [Graphics:Images/ConformalMapDictionary.3.1_gr_125.gif].
Remark. This is Example 11.28 in Section 11.9 and Exercise 9 (Extra Example 3) in Section 11.1, and is illustrated in Figure 11.74 and Figure 11.107.  
Hint: Set [Graphics:Images/ConformalMapDictionary.3.1_gr_126.gif],  [Graphics:Images/ConformalMapDictionary.3.1_gr_127.gif],  [Graphics:Images/ConformalMapDictionary.3.1_gr_128.gif],  [Graphics:Images/ConformalMapDictionary.3.1_gr_129.gif],  and  [Graphics:Images/ConformalMapDictionary.3.1_gr_130.gif],  [Graphics:Images/ConformalMapDictionary.3.1_gr_131.gif],  [Graphics:Images/ConformalMapDictionary.3.1_gr_132.gif],  [Graphics:Images/ConformalMapDictionary.3.1_gr_133.gif],  
and the angles are[Graphics:Images/ConformalMapDictionary.3.1_gr_134.gif] [Graphics:Images/ConformalMapDictionary.3.1_gr_135.gif].

            [Graphics:Images/ConformalMapDictionary.3.1_gr_136.gif]          [Graphics:Images/ConformalMapDictionary.3.1_gr_137.gif]

Details 31.  

Example 32.  The conformal mapping   [Graphics:Images/ConformalMapDictionary.3.2_gr_1.gif].
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.3.2_gr_2.gif],  
also  [Graphics:Images/ConformalMapDictionary.3.2_gr_3.gif].  
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.3.2_gr_4.gif]  is the region in the [Graphics:Images/ConformalMapDictionary.3.2_gr_5.gif]-plane bounded by the negative  [Graphics:Images/ConformalMapDictionary.3.2_gr_6.gif]-axis,  the segment [Graphics:Images/ConformalMapDictionary.3.2_gr_7.gif],  and the ray  [Graphics:Images/ConformalMapDictionary.3.2_gr_8.gif].  
Remark. This is Exercise 3 (Extra Example 1) in Section 11.9 and Example 11.29 in  Section 11.10, and is illustrated in Figure 11.77 and Figure 11.87.
Hint: Set  [Graphics:Images/ConformalMapDictionary.3.2_gr_9.gif]  and   [Graphics:Images/ConformalMapDictionary.3.2_gr_10.gif],  and the angles are [Graphics:Images/ConformalMapDictionary.3.2_gr_11.gif].

      [Graphics:Images/ConformalMapDictionary.3.2_gr_12.gif]     [Graphics:Images/ConformalMapDictionary.3.2_gr_13.gif]

Details 32.

Example 33.  The conformal mapping   [Graphics:Images/ConformalMapDictionary.3.2_gr_20.gif].
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.3.2_gr_21.gif],  
also  [Graphics:Images/ConformalMapDictionary.3.2_gr_22.gif].  
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.3.2_gr_23.gif]  is the upper half-plane together with the semi-infinite strip -1<u<1, v<0.
Remark. This is Exercise 4 (Extra Example 2) in Section 11.9 and Example 11.34 in Section 11.11, and is illustrated in Figure 11.78. and Figure 11.98.
Hint: Set  [Graphics:Images/ConformalMapDictionary.3.2_gr_24.gif]  and  [Graphics:Images/ConformalMapDictionary.3.2_gr_25.gif], where  [Graphics:Images/ConformalMapDictionary.3.2_gr_26.gif],  [Graphics:Images/ConformalMapDictionary.3.2_gr_27.gif],  [Graphics:Images/ConformalMapDictionary.3.2_gr_28.gif]  and   [Graphics:Images/ConformalMapDictionary.3.2_gr_29.gif],  [Graphics:Images/ConformalMapDictionary.3.2_gr_30.gif],  [Graphics:Images/ConformalMapDictionary.3.2_gr_31.gif],  and the angles are  [Graphics:Images/ConformalMapDictionary.3.2_gr_32.gif].

      [Graphics:Images/ConformalMapDictionary.3.2_gr_33.gif]     [Graphics:Images/ConformalMapDictionary.3.2_gr_34.gif]

Details 33.

Example 34.  The conformal mapping   [Graphics:Images/ConformalMapDictionary.3.2_gr_44.gif].  
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.3.2_gr_45.gif],  
also  [Graphics:Images/ConformalMapDictionary.3.2_gr_46.gif].  
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.3.2_gr_47.gif]  is the portion of the upper half plane that lies outside the quadrant [Graphics:Images/ConformalMapDictionary.3.2_gr_48.gif].
Remark. This is Exercise 8 in Section 11.9 and Exercise 7 (Extra Example 1) in Section 11.11, and is illustrated in Figure 11.82 and Figure Figure 11.105.
Hint: Set  [Graphics:Images/ConformalMapDictionary.3.2_gr_49.gif] and [Graphics:Images/ConformalMapDictionary.3.2_gr_50.gif], and the angles are  [Graphics:Images/ConformalMapDictionary.3.2_gr_51.gif].

      [Graphics:Images/ConformalMapDictionary.3.2_gr_52.gif]     [Graphics:Images/ConformalMapDictionary.3.2_gr_53.gif]

Details 34.

Example 35.  The conformal mapping   [Graphics:Images/ConformalMapDictionary.3.2_gr_63.gif].
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.3.2_gr_64.gif],  
also  [Graphics:Images/ConformalMapDictionary.3.2_gr_65.gif].
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.3.2_gr_66.gif]  is the portion of the upper half plane that lies outside the region bounded by the rays [Graphics:Images/ConformalMapDictionary.3.2_gr_67.gif] and   [Graphics:Images/ConformalMapDictionary.3.2_gr_68.gif].
Remark. This is Exercise 10 (Extra Example 3) in Section 11.9 and Exercise 8 (Extra Example 2) in Section 11.11, and is illustrated in Figure 11.84 and Figure 11.106.
Hint: Set  [Graphics:Images/ConformalMapDictionary.3.2_gr_69.gif] and [Graphics:Images/ConformalMapDictionary.3.2_gr_70.gif],  and the angles are  [Graphics:Images/ConformalMapDictionary.3.2_gr_71.gif].
Use the change of variable  [Graphics:Images/ConformalMapDictionary.3.2_gr_72.gif]  in the resulting integral.

      [Graphics:Images/ConformalMapDictionary.3.2_gr_73.gif]     [Graphics:Images/ConformalMapDictionary.3.2_gr_74.gif]

Details 35.

Example 36.  The conformal mapping   [Graphics:Images/ConformalMapDictionary.3.2_gr_84.gif].  
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.3.2_gr_85.gif],
also  [Graphics:Images/ConformalMapDictionary.3.2_gr_86.gif].  
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.3.2_gr_87.gif]  is first quadrant slit along the ray  [Graphics:Images/ConformalMapDictionary.3.2_gr_88.gif].
Remark. This is Exercise 15 in Section 11.9 and Exercise 10 (Extra Example 4) in Section 11.11, and is illustrated in Figure 11.86 and Figure 11.108.
Hint:  Hint: Set  [Graphics:Images/ConformalMapDictionary.3.2_gr_89.gif]  and  [Graphics:Images/ConformalMapDictionary.3.2_gr_90.gif]. , and the angles are  [Graphics:Images/ConformalMapDictionary.3.2_gr_91.gif].

      [Graphics:Images/ConformalMapDictionary.3.2_gr_92.gif]     [Graphics:Images/ConformalMapDictionary.3.2_gr_93.gif]

Details 36.

Example 37.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.3.2_gr_103.gif].  
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.3.2_gr_104.gif],  
also,  w = [Graphics:Images/ConformalMapDictionary.3.2_gr_105.gif].
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.3.2_gr_106.gif]  is the portion of the upper half plane  [Graphics:Images/ConformalMapDictionary.3.2_gr_107.gif]   that lies above the segment  [Graphics:Images/ConformalMapDictionary.3.2_gr_108.gif]  and the ray  [Graphics:Images/ConformalMapDictionary.3.2_gr_109.gif].  
Remark. This is Exercise 5 (Extra Example 5) in  Section 11.10, and is illustrated in Figure 11.92.
Hint:  Set  [Graphics:Images/ConformalMapDictionary.3.2_gr_110.gif].  In the w-plane [Graphics:Images/ConformalMapDictionary.3.2_gr_111.gif], and the exterior angles are  [Graphics:Images/ConformalMapDictionary.3.2_gr_112.gif].

      [Graphics:Images/ConformalMapDictionary.3.2_gr_113.gif]     [Graphics:Images/ConformalMapDictionary.3.2_gr_114.gif]

Details 37.

 

  1. Conformal Mapping Dictionary - Part I
  2. Conformal Mapping Dictionary - Part II
  3. Conformal Mapping Dictionary - Part III
  4. Conformal Mapping Dictionary - Part IV
  5. Conformal Mapping Dictionary - Part V

 

Chapter 2. Complex Functions
  1. Complex Functions and Linear Mappings
  2. The Mappings [Graphics:Images/ComplexFunPowerRoot_gr_1.gif] and [Graphics:Images/ComplexFunPowerRoot_gr_2.gif]
  3. Complex Limits and Continuity
  4. Branches of Complex Functions
  5. The Reciprocal Transformation [Graphics:Images/ComplexFunReciprocalMod_gr_1.gif]

    Chapter 5. Elementary Functions

  6. The Complex Exponential Function
  7. The Complex Logarithm Function
  8. Complex Exponents and Powers
  9. Trigonometric and Hyperbolic Functions
  10. Inverse Trigonometric and Hyperbolic Functions

    Chapter 10. Conformal Mapping

  11. Basic Properties of Conformal Mappings
  12. Mobius Transformations - Bilinear Transformations
  13. Mappings Involving Elementary Functions
  14. Mappings by Trigonometric Functions

    Chapter 11. Applications of Harmonic Functions

  15. Preliminaries
  16. Invariance of Laplace's Equation and the Dirichlet Problem
  17. Poisson's Integral Formula for the Upper Half Plane
  18. Two-Dimensional Mathematical Models
  19. Steady State Temperatures
  20. Two-Dimensional Electrostatics
  21. Two-Dimensional Fluid Flow
  22. The Joukowski Airfoil
  23. The Schwarz-Christoffel Transformation
  24. Image of a Fluid Flow
  25. Sources and Sinks

 

 

Return to the Complex Analysis Project

 

(c) 2008 John H. Mathews, Russell W. Howell