Dictionary of Conformal Mapping
Part IV

Advanced Common Mappings

Example 38.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.4_gr_1.gif].  
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.4_gr_2.gif],
here  [Graphics:Images/ConformalMapDictionary.4_gr_3.gif], [Graphics:Images/ConformalMapDictionary.4_gr_4.gif], [Graphics:Images/ConformalMapDictionary.4_gr_5.gif]  and  [Graphics:Images/ConformalMapDictionary.4_gr_6.gif], [Graphics:Images/ConformalMapDictionary.4_gr_7.gif], [Graphics:Images/ConformalMapDictionary.4_gr_8.gif],
and the angles are  [Graphics:Images/ConformalMapDictionary.4_gr_9.gif][Graphics:Images/ConformalMapDictionary.4_gr_10.gif].

      [Graphics:Images/ConformalMapDictionary.4_gr_11.gif]     [Graphics:Images/ConformalMapDictionary.4_gr_12.gif]

Details 38.

Example 39.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.4_gr_15.gif],  
here  [Graphics:Images/ConformalMapDictionary.4_gr_16.gif], [Graphics:Images/ConformalMapDictionary.4_gr_17.gif], [Graphics:Images/ConformalMapDictionary.4_gr_18.gif], [Graphics:Images/ConformalMapDictionary.4_gr_19.gif]  and  [Graphics:Images/ConformalMapDictionary.4_gr_20.gif], [Graphics:Images/ConformalMapDictionary.4_gr_21.gif], [Graphics:Images/ConformalMapDictionary.4_gr_22.gif], [Graphics:Images/ConformalMapDictionary.4_gr_23.gif].

      [Graphics:Images/ConformalMapDictionary.4_gr_24.gif]          [Graphics:Images/ConformalMapDictionary.4_gr_25.gif]

Details 39.

Example 40.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.4_gr_28.gif].  

      [Graphics:Images/ConformalMapDictionary.4_gr_29.gif]     [Graphics:Images/ConformalMapDictionary.4_gr_30.gif]
Details 40.

Example 41.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.4_gr_31.gif].
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.4_gr_32.gif],
here  [Graphics:Images/ConformalMapDictionary.4_gr_33.gif], [Graphics:Images/ConformalMapDictionary.4_gr_34.gif], [Graphics:Images/ConformalMapDictionary.4_gr_35.gif]  and  [Graphics:Images/ConformalMapDictionary.4_gr_36.gif], [Graphics:Images/ConformalMapDictionary.4_gr_37.gif], [Graphics:Images/ConformalMapDictionary.4_gr_38.gif],
and the angles are  [Graphics:Images/ConformalMapDictionary.4_gr_39.gif].    
Also  [Graphics:Images/ConformalMapDictionary.4_gr_40.gif].  

      [Graphics:Images/ConformalMapDictionary.4_gr_41.gif]     [Graphics:Images/ConformalMapDictionary.4_gr_42.gif]

Details 41.

Example 42.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.4_gr_45.gif].  
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.4_gr_46.gif],
here  [Graphics:Images/ConformalMapDictionary.4_gr_47.gif], [Graphics:Images/ConformalMapDictionary.4_gr_48.gif],  and  [Graphics:Images/ConformalMapDictionary.4_gr_49.gif], [Graphics:Images/ConformalMapDictionary.4_gr_50.gif],  and the angles are  [Graphics:Images/ConformalMapDictionary.4_gr_51.gif],  
(additionally for [Graphics:Images/ConformalMapDictionary.4_gr_52.gif] we would have [Graphics:Images/ConformalMapDictionary.4_gr_53.gif]).  
Also  [Graphics:Images/ConformalMapDictionary.4_gr_54.gif],  also  [Graphics:Images/ConformalMapDictionary.4_gr_55.gif].  

      [Graphics:Images/ConformalMapDictionary.4_gr_56.gif]     [Graphics:Images/ConformalMapDictionary.4_gr_57.gif]

Details 42.

Example 43.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.4_gr_60.gif].  
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.4_gr_61.gif],
here  [Graphics:Images/ConformalMapDictionary.4_gr_62.gif], [Graphics:Images/ConformalMapDictionary.4_gr_63.gif], [Graphics:Images/ConformalMapDictionary.4_gr_64.gif], [Graphics:Images/ConformalMapDictionary.4_gr_65.gif]  and  [Graphics:Images/ConformalMapDictionary.4_gr_66.gif], [Graphics:Images/ConformalMapDictionary.4_gr_67.gif], [Graphics:Images/ConformalMapDictionary.4_gr_68.gif], [Graphics:Images/ConformalMapDictionary.4_gr_69.gif],
and the angles are  [Graphics:Images/ConformalMapDictionary.4_gr_70.gif].
Also  [Graphics:Images/ConformalMapDictionary.4_gr_71.gif].

      [Graphics:Images/ConformalMapDictionary.4_gr_72.gif]     [Graphics:Images/ConformalMapDictionary.4_gr_73.gif]

Details 43.

Example 44.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.4_gr_76.gif].
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.4_gr_77.gif],
here [Graphics:Images/ConformalMapDictionary.4_gr_78.gif], [Graphics:Images/ConformalMapDictionary.4_gr_79.gif],  and  [Graphics:Images/ConformalMapDictionary.4_gr_80.gif], [Graphics:Images/ConformalMapDictionary.4_gr_81.gif],  and the angles are  [Graphics:Images/ConformalMapDictionary.4_gr_82.gif],  
(additionally for [Graphics:Images/ConformalMapDictionary.4_gr_83.gif] we would have [Graphics:Images/ConformalMapDictionary.4_gr_84.gif]  and  [Graphics:Images/ConformalMapDictionary.4_gr_85.gif], [Graphics:Images/ConformalMapDictionary.4_gr_86.gif]).  
Also  [Graphics:Images/ConformalMapDictionary.4_gr_87.gif].  

      [Graphics:Images/ConformalMapDictionary.4_gr_88.gif]     [Graphics:Images/ConformalMapDictionary.4_gr_89.gif]

Details 44.

Example 45.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.4_gr_92.gif],  
here  [Graphics:Images/ConformalMapDictionary.4_gr_93.gif], [Graphics:Images/ConformalMapDictionary.4_gr_94.gif], [Graphics:Images/ConformalMapDictionary.4_gr_95.gif]  and  [Graphics:Images/ConformalMapDictionary.4_gr_96.gif], [Graphics:Images/ConformalMapDictionary.4_gr_97.gif], [Graphics:Images/ConformalMapDictionary.4_gr_98.gif].  

      [Graphics:Images/ConformalMapDictionary.4_gr_99.gif]     [Graphics:Images/ConformalMapDictionary.4_gr_100.gif]

Details 45.

Example 46.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.4_gr_103.gif],  
here  [Graphics:Images/ConformalMapDictionary.4_gr_104.gif], [Graphics:Images/ConformalMapDictionary.4_gr_105.gif], [Graphics:Images/ConformalMapDictionary.4_gr_106.gif]  and  [Graphics:Images/ConformalMapDictionary.4_gr_107.gif], [Graphics:Images/ConformalMapDictionary.4_gr_108.gif], [Graphics:Images/ConformalMapDictionary.4_gr_109.gif].  

      [Graphics:Images/ConformalMapDictionary.4_gr_110.gif]     [Graphics:Images/ConformalMapDictionary.4_gr_111.gif]

Details 46.

Example 47.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.4_gr_114.gif].  
It can be constructed via the Schwarz-Christoffel integral  [Graphics:Images/ConformalMapDictionary.4_gr_115.gif],
here  [Graphics:Images/ConformalMapDictionary.4_gr_116.gif], [Graphics:Images/ConformalMapDictionary.4_gr_117.gif], [Graphics:Images/ConformalMapDictionary.4_gr_118.gif]  and  [Graphics:Images/ConformalMapDictionary.4_gr_119.gif], [Graphics:Images/ConformalMapDictionary.4_gr_120.gif], [Graphics:Images/ConformalMapDictionary.4_gr_121.gif],  and the angles are  [Graphics:Images/ConformalMapDictionary.4_gr_122.gif].   
Also  [Graphics:Images/ConformalMapDictionary.4_gr_123.gif].  

      [Graphics:Images/ConformalMapDictionary.4_gr_124.gif]     [Graphics:Images/ConformalMapDictionary.4_gr_125.gif]

Details 47.

Remark.  When we compare Examples 31, 41, 45, 46, 47 we see that there are many conformal mappings whose range is a right angle channel. Many of the other examples can also be done in several alternate ways.

 

  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