Example 11.28.  Show that  [Graphics:Images/SchwarzChristoffelMod_gr_150.gif]  maps the upper half plane  [Graphics:Images/SchwarzChristoffelMod_gr_151.gif]  onto the right angle channel in the first quadrant, which is bounded by the coordinate axes and the rays  [Graphics:Images/SchwarzChristoffelMod_gr_152.gif], as depicted in Figure 11.74(b).

[Graphics:Images/SchwarzChristoffelMod_gr_161.gif]

Figure 11.74  The region with [Graphics:Images/SchwarzChristoffelMod_gr_162.gif] and [Graphics:Images/SchwarzChristoffelMod_gr_163.gif].

Explore Solution 11.28.

Enter the formula [Graphics:../Images/SchwarzChristoffelMod_gr_174.gif] and integrate it to construct  f(z).

[Graphics:../Images/SchwarzChristoffelMod_gr_175.gif]




[Graphics:../Images/SchwarzChristoffelMod_gr_176.gif]

 

 

 

This is one, formula for the integral.   However, we will use the following form of the integral to continue the computations.

[Graphics:../Images/SchwarzChristoffelMod_gr_177.gif]



[Graphics:../Images/SchwarzChristoffelMod_gr_178.gif]

 

 

Now solve for the coefficients  A  and  B.

[Graphics:../Images/SchwarzChristoffelMod_gr_179.gif]




[Graphics:../Images/SchwarzChristoffelMod_gr_180.gif]

 

 

 

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

[Graphics:../Images/SchwarzChristoffelMod_gr_181.gif]




[Graphics:../Images/SchwarzChristoffelMod_gr_182.gif]

[Graphics:../Images/SchwarzChristoffelMod_gr_183.gif]

We see that  [Graphics:../Images/SchwarzChristoffelMod_gr_184.gif]  maps the upper half plane  [Graphics:../Images/SchwarzChristoffelMod_gr_185.gif]  onto the channel in the right half plane bounded by the coordinate axes and rays  [Graphics:../Images/SchwarzChristoffelMod_gr_186.gif].  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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