Example 26.  The conformal mapping    .
It can be constructed via the Schwarz-Christoffel integral  ,
also  .  
The image of the upper half-plane    is the infinite strip    slit along the horizontal ray  .  
Remark. This is Exercise 5 in Section 11.9, and is illustrated in Figure 11.79.   
Hint: Set    and  , and the angles are .

Use the result that we saw in Example 24 and the conformal mapping    .   
The image of the upper half-plane    is the upper-half plane    slit along the vertical segment  .  

                            The mapping  .

Use familiar properties of the logarithm  , which maps the upper half-plane    onto the infinite strip  .

                            The mapping  .

Therefore, the desired conformal mapping is the composition mapping  ,
the image of the upper half-plane    is the infinite strip    slit along the horizontal ray  .  

                                                        The mapping  .

Details 26.

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