Extra Example 2.  Show that  [Graphics:Images/SchwarzChristoffelMod_gr_210.gif]  maps the upper half-plane onto the domain indicated in Figure 11.78.   
Hint: Set  [Graphics:Images/SchwarzChristoffelMod_gr_211.gif]  and  [Graphics:Images/SchwarzChristoffelMod_gr_212.gif]

[Graphics:Images/SchwarzChristoffelMod_gr_213.gif]

Figure 11.78  The region with [Graphics:Images/SchwarzChristoffelMod_gr_214.gif] and [Graphics:Images/SchwarzChristoffelMod_gr_215.gif].

Explore Extra Solution 2.

Enter the formula [Graphics:../Images/SchwarzChristoffelMod_gr_223.gif] and integrate it to construct  f(z).

[Graphics:../Images/SchwarzChristoffelMod_gr_224.gif]




[Graphics:../Images/SchwarzChristoffelMod_gr_225.gif]

 

 

 

We will use this form of the integral to continue the computations.

[Graphics:../Images/SchwarzChristoffelMod_gr_226.gif]



[Graphics:../Images/SchwarzChristoffelMod_gr_227.gif]

 

 

Now solve for the coefficients  A  and  B.

[Graphics:../Images/SchwarzChristoffelMod_gr_228.gif]




[Graphics:../Images/SchwarzChristoffelMod_gr_229.gif]

 

 

 

Use Mathematica to graph conformal mapping  w = f(z).

[Graphics:../Images/SchwarzChristoffelMod_gr_230.gif]




[Graphics:../Images/SchwarzChristoffelMod_gr_231.gif]

[Graphics:../Images/SchwarzChristoffelMod_gr_232.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell