Dictionary of Conformal Mapping
Part I

Common Mappings

Example 1.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_1.gif]  in polar coordinates.

            [Graphics:Images/ConformalMapDictionary.1_gr_2.gif]          [Graphics:Images/ConformalMapDictionary.1_gr_3.gif]

Details 1.

Example 2.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_8.gif]  in rectangular coordinates.

           [Graphics:Images/ConformalMapDictionary.1_gr_9.gif]          [Graphics:Images/ConformalMapDictionary.1_gr_10.gif]

Details 2.

Example 3.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_19.gif]  in polar coordinates.

 [Graphics:Images/ConformalMapDictionary.1_gr_20.gif]     [Graphics:Images/ConformalMapDictionary.1_gr_21.gif]

Details 3.

Example 4.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_30.gif]  in rectangular coordinates.

 [Graphics:Images/ConformalMapDictionary.1_gr_31.gif]     [Graphics:Images/ConformalMapDictionary.1_gr_32.gif]

Details 4.

Example 5.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_41.gif].  The image of the half-plane  [Graphics:Images/ConformalMapDictionary.1_gr_42.gif]  is the unit disk  [Graphics:Images/ConformalMapDictionary.1_gr_43.gif].

           [Graphics:Images/ConformalMapDictionary.1_gr_44.gif]          [Graphics:Images/ConformalMapDictionary.1_gr_45.gif]

Details 5.

Example 6.   The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_55.gif].  The image of the right half-plane  [Graphics:Images/ConformalMapDictionary.1_gr_56.gif]  is the unit disk  [Graphics:Images/ConformalMapDictionary.1_gr_57.gif]

           [Graphics:Images/ConformalMapDictionary.1_gr_58.gif]          [Graphics:Images/ConformalMapDictionary.1_gr_59.gif]

Details 6.

Example 7.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_60.gif].  The image of  the unit disk  [Graphics:Images/ConformalMapDictionary.1_gr_61.gif]  is the left half-plane  [Graphics:Images/ConformalMapDictionary.1_gr_62.gif].

           [Graphics:Images/ConformalMapDictionary.1_gr_63.gif]          [Graphics:Images/ConformalMapDictionary.1_gr_64.gif]

Details 7.

Example 8.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_65.gif].  The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.1_gr_66.gif]  is the upper half-plane  [Graphics:Images/ConformalMapDictionary.1_gr_67.gif].

           [Graphics:Images/ConformalMapDictionary.1_gr_68.gif]          [Graphics:Images/ConformalMapDictionary.1_gr_69.gif]

Details 8.

Example 9.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_72.gif].  The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.1_gr_73.gif]  is the upper half-plane  [Graphics:Images/ConformalMapDictionary.1_gr_74.gif].

           [Graphics:Images/ConformalMapDictionary.1_gr_75.gif]          [Graphics:Images/ConformalMapDictionary.1_gr_76.gif]

Details 9.

Example 10.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_81.gif].  The image of the disk  [Graphics:Images/ConformalMapDictionary.1_gr_82.gif]  is the right half plane  [Graphics:Images/ConformalMapDictionary.1_gr_83.gif].

           [Graphics:Images/ConformalMapDictionary.1_gr_84.gif]          [Graphics:Images/ConformalMapDictionary.1_gr_85.gif]

Details 10.

Example 11.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_88.gif].  The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.1_gr_89.gif]  is the exterior of the circle  [Graphics:Images/ConformalMapDictionary.1_gr_90.gif].

[Graphics:Images/ConformalMapDictionary.1_gr_91.gif]     [Graphics:Images/ConformalMapDictionary.1_gr_92.gif]

Details 11.

Example 12.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_95.gif].  

The image of the region inside the unit disk  [Graphics:Images/ConformalMapDictionary.1_gr_96.gif]  and outside the circle [Graphics:Images/ConformalMapDictionary.1_gr_97.gif]  is the annulus  [Graphics:Images/ConformalMapDictionary.1_gr_98.gif].  

For this mapping  [Graphics:Images/ConformalMapDictionary.1_gr_99.gif]  and  [Graphics:Images/ConformalMapDictionary.1_gr_100.gif]  and  [Graphics:Images/ConformalMapDictionary.1_gr_101.gif].

           [Graphics:Images/ConformalMapDictionary.1_gr_102.gif]          [Graphics:Images/ConformalMapDictionary.1_gr_103.gif]

Details 12.

Example 13.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_104.gif].  

The image of the region that lies outside the the unit circle  [Graphics:Images/ConformalMapDictionary.1_gr_105.gif]  and outside the circle [Graphics:Images/ConformalMapDictionary.1_gr_106.gif]  is the annulus  [Graphics:Images/ConformalMapDictionary.1_gr_107.gif].  

For this mapping  [Graphics:Images/ConformalMapDictionary.1_gr_108.gif]  and  [Graphics:Images/ConformalMapDictionary.1_gr_109.gif]  and  [Graphics:Images/ConformalMapDictionary.1_gr_110.gif].

           [Graphics:Images/ConformalMapDictionary.1_gr_111.gif]          [Graphics:Images/ConformalMapDictionary.1_gr_112.gif]

Details 13.

Example 14.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_113.gif].  

The image of the region that lies in the left half plane  [Graphics:Images/ConformalMapDictionary.1_gr_114.gif]  and outside the the unit circle [Graphics:Images/ConformalMapDictionary.1_gr_115.gif]  is the annulus  [Graphics:Images/ConformalMapDictionary.1_gr_116.gif].  

For this mapping  [Graphics:Images/ConformalMapDictionary.1_gr_117.gif]  and  [Graphics:Images/ConformalMapDictionary.1_gr_118.gif]  and  [Graphics:Images/ConformalMapDictionary.1_gr_119.gif].
           [Graphics:Images/ConformalMapDictionary.1_gr_120.gif]          [Graphics:Images/ConformalMapDictionary.1_gr_121.gif]

Details 14.

Example 15.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.1_gr_122.gif],  where  [Graphics:Images/ConformalMapDictionary.1_gr_123.gif], [Graphics:Images/ConformalMapDictionary.1_gr_124.gif], [Graphics:Images/ConformalMapDictionary.1_gr_125.gif]  and  [Graphics:Images/ConformalMapDictionary.1_gr_126.gif], [Graphics:Images/ConformalMapDictionary.1_gr_127.gif], [Graphics:Images/ConformalMapDictionary.1_gr_128.gif]

           [Graphics:Images/ConformalMapDictionary.1_gr_129.gif]          [Graphics:Images/ConformalMapDictionary.1_gr_130.gif]

Details 15.

  

  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

  6. Complex Functions and Linear Mappings
  7. The Mappings [Graphics:Images/ComplexFunPowerRoot_gr_1.gif] and [Graphics:Images/ComplexFunPowerRoot_gr_2.gif]
  8. Complex Limits and Continuity
  9. Branches of Complex Functions
  10. The Reciprocal Transformation [Graphics:Images/ComplexFunReciprocalMod_gr_1.gif]

    Chapter 5. Elementary Functions

  11. The Complex Exponential Function
  12. The Complex Logarithm Function
  13. Complex Exponents and Powers
  14. Trigonometric and Hyperbolic Functions
  15. Inverse Trigonometric and Hyperbolic Functions

    Chapter 10. Conformal Mapping

  16. Basic Properties of Conformal Mappings
  17. Mobius Transformations - Bilinear Transformations
  18. Mappings Involving Elementary Functions
  19. Mappings by Trigonometric Functions

    Chapter 11. Applications of Harmonic Functions

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

 

 

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