Extra Example 5.  Use the Schwarz Christoffel formula to show that  [Graphics:Images/SchwarzChristoffelMod_gr_289.gif]  will map the upper-half-plane onto an equilateral triangle.
Hint: For convenience we have chosen  [Graphics:Images/SchwarzChristoffelMod_gr_290.gif].

[Graphics:Images/SchwarzChristoffelMod_gr_291.gif]

Explore Extra Solution 5.

We shall investigate three cases of this formula.

(i)    [Graphics:../Images/SchwarzChristoffelMod_gr_296.gif]

(ii)    [Graphics:../Images/SchwarzChristoffelMod_gr_297.gif]

(iii)    [Graphics:../Images/SchwarzChristoffelMod_gr_298.gif]

[Graphics:../Images/SchwarzChristoffelMod_gr_299.gif]




[Graphics:../Images/SchwarzChristoffelMod_gr_300.gif]

[Graphics:../Images/SchwarzChristoffelMod_gr_301.gif]

 

 

 

[Graphics:../Images/SchwarzChristoffelMod_gr_302.gif]




[Graphics:../Images/SchwarzChristoffelMod_gr_303.gif]

[Graphics:../Images/SchwarzChristoffelMod_gr_304.gif]

 

 

 

For our next construction we will start with the unit disk and map it onto the upper half plane, then onto the triangle.

[Graphics:../Images/SchwarzChristoffelMod_gr_305.gif]




[Graphics:../Images/SchwarzChristoffelMod_gr_306.gif]

[Graphics:../Images/SchwarzChristoffelMod_gr_307.gif]

 

 

 

[Graphics:../Images/SchwarzChristoffelMod_gr_308.gif]




[Graphics:../Images/SchwarzChristoffelMod_gr_309.gif]

[Graphics:../Images/SchwarzChristoffelMod_gr_310.gif]

 

 

 

[Graphics:../Images/SchwarzChristoffelMod_gr_311.gif]




[Graphics:../Images/SchwarzChristoffelMod_gr_312.gif]

[Graphics:../Images/SchwarzChristoffelMod_gr_313.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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