Example 27.  The conformal mapping  .
It can be constructed via the Schwarz-Christoffel integral  ,  
also  .  
The image of the upper half-plane    is the semi-infinite strip  , and the angles are .
Remark. Hint: Set    and  , and the angles are .

Consider the result that we saw in Example 25 and the conformal mapping    
The image of the upper half-plane    is the portion of the upper half-plane    that lies outside the unit circle  .  

Notice that the conformal mapping   is similar and will also map the upper half-plane    onto the portion of the upper half-plane    that lies outside the unit circle  .  

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

Therefore, the desired conformal mapping is the composition mapping  ,
the image of the upper half-plane     is the semi-infinite strip  .

                                                         The conformal mapping  .

Details 27.

Under Construction

Return to the Complex Analysis Project