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]

 

[Graphics:../Images/ConformalMapDictionary.3.1_gr_103.gif]

 

Details 29.

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