Dictionary of Conformal Mapping
Part II

Common Mappings

 

Example 16.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.2_gr_1.gif].  
The image of the horizontal strip  [Graphics:Images/ConformalMapDictionary.2_gr_2.gif]  is the punctured plane  [Graphics:Images/ConformalMapDictionary.2_gr_3.gif].  

           [Graphics:Images/ConformalMapDictionary.2_gr_4.gif]          [Graphics:Images/ConformalMapDictionary.2_gr_5.gif]

Details 16.

Example 17.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.2_gr_19.gif].
The image of the  plane  [Graphics:Images/ConformalMapDictionary.2_gr_20.gif]  is the horizontal strip  [Graphics:Images/ConformalMapDictionary.2_gr_21.gif].  

           [Graphics:Images/ConformalMapDictionary.2_gr_22.gif]          [Graphics:Images/ConformalMapDictionary.2_gr_23.gif]

Details 17.

Example 18.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.2_gr_38.gif].
The image of the vertical strip  [Graphics:Images/ConformalMapDictionary.2_gr_39.gif]  is the w-plane slit along the two rays [Graphics:Images/ConformalMapDictionary.2_gr_40.gif] and [Graphics:Images/ConformalMapDictionary.2_gr_41.gif].

           [Graphics:Images/ConformalMapDictionary.2_gr_42.gif]          [Graphics:Images/ConformalMapDictionary.2_gr_43.gif]

Details 18.

Example 19.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.2_gr_55.gif].  
The image of the vertical strip  [Graphics:Images/ConformalMapDictionary.2_gr_56.gif]  is the unit disk  [Graphics:Images/ConformalMapDictionary.2_gr_57.gif].

           [Graphics:Images/ConformalMapDictionary.2_gr_60.gif]          [Graphics:Images/ConformalMapDictionary.2_gr_61.gif]

Details 19.

Example 20.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.2_gr_73.gif].
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.2_gr_74.gif]  is the semi-infinite strip  [Graphics:Images/ConformalMapDictionary.2_gr_75.gif].

           [Graphics:Images/ConformalMapDictionary.2_gr_76.gif]          [Graphics:Images/ConformalMapDictionary.2_gr_77.gif]

Details 20.

Example 21.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.2_gr_82.gif].  
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.2_gr_83.gif]  is the semi-infinite strip  [Graphics:Images/ConformalMapDictionary.2_gr_84.gif].

      [Graphics:Images/ConformalMapDictionary.2_gr_85.gif]     [Graphics:Images/ConformalMapDictionary.2_gr_86.gif]

Details 21.

Example 22.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.2_gr_91.gif].  
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.2_gr_92.gif]  is the horizontal strip  [Graphics:Images/ConformalMapDictionary.2_gr_93.gif],  
and  [Graphics:Images/ConformalMapDictionary.2_gr_94.gif], [Graphics:Images/ConformalMapDictionary.2_gr_95.gif], and [Graphics:Images/ConformalMapDictionary.2_gr_96.gif] and  [Graphics:Images/ConformalMapDictionary.2_gr_97.gif], [Graphics:Images/ConformalMapDictionary.2_gr_98.gif], and [Graphics:Images/ConformalMapDictionary.2_gr_99.gif].  

      [Graphics:Images/ConformalMapDictionary.2_gr_100.gif]     [Graphics:Images/ConformalMapDictionary.2_gr_101.gif]

 Details 22.

 Example 23.  The conformal mapping    [Graphics:Images/ConformalMapDictionary.2.2_gr_1.gif].  
The image of the portion of the unit disk  [Graphics:Images/ConformalMapDictionary.2.2_gr_2.gif]  that lies in the upper half-plane  [Graphics:Images/ConformalMapDictionary.2.2_gr_3.gif] is the upper half-plane  [Graphics:Images/ConformalMapDictionary.2.2_gr_4.gif].  Furthermore, the upper semicircular portion of the boundary is mapped onto the line negative u-axis, and the segment  [Graphics:Images/ConformalMapDictionary.2.2_gr_5.gif]  is mapped onto the positive u-axis.

[Graphics:Images/ConformalMapDictionary.2.2_gr_6.gif]     [Graphics:Images/ConformalMapDictionary.2.2_gr_7.gif]

Details 23.

Example 24.  The conformal mapping    [Graphics:Images/ConformalMapDictionary.2.2_gr_15.gif].   
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.2.2_gr_16.gif]  is the upper-half plane  [Graphics:Images/ConformalMapDictionary.2.2_gr_17.gif]  slit along the vertical segment  [Graphics:Images/ConformalMapDictionary.2.2_gr_18.gif].  

[Graphics:Images/ConformalMapDictionary.2.2_gr_19.gif]     [Graphics:Images/ConformalMapDictionary.2.2_gr_20.gif]

Details 24.

Example 25.  The conformal mapping  [Graphics:Images/ConformalMapDictionary.2.2_gr_34.gif].   
The image of the upper half-plane  [Graphics:Images/ConformalMapDictionary.2.2_gr_35.gif]  is the portion of the upper half-plane  [Graphics:Images/ConformalMapDictionary.2.2_gr_36.gif]  that lies outside the unit circle  [Graphics:Images/ConformalMapDictionary.2.2_gr_37.gif].  

[Graphics:Images/ConformalMapDictionary.2.2_gr_38.gif]     [Graphics:Images/ConformalMapDictionary.2.2_gr_39.gif]

Details 25.

 

  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