Solution 14.

Solution.   Starting with   [Graphics:../Images/MobiusTranformationModHome_gr_754.gif]   rearrange and obtain  

                    [Graphics:../Images/MobiusTranformationModHome_gr_755.gif]    then get

                    [Graphics:../Images/MobiusTranformationModHome_gr_756.gif]    next we obtain   

                    [Graphics:../Images/MobiusTranformationModHome_gr_757.gif]    and then   

                    [Graphics:../Images/MobiusTranformationModHome_gr_758.gif].   
    
Therefore, equation (10-18) can be written in the form of equation (10-13) :

    [Graphics:../Images/MobiusTranformationModHome_gr_759.gif].  

We are done.   

Aside.  We can let Mathematica double check our work.

Substitute  [Graphics:../Images/MobiusTranformationModHome_gr_760.gif]  into the formula for  [Graphics:../Images/MobiusTranformationModHome_gr_761.gif]  and get:

[Graphics:../Images/MobiusTranformationModHome_gr_762.gif]

[Graphics:../Images/MobiusTranformationModHome_gr_763.gif]

Substitute  [Graphics:../Images/MobiusTranformationModHome_gr_764.gif]  into the formula for  [Graphics:../Images/MobiusTranformationModHome_gr_765.gif]  and get:

[Graphics:../Images/MobiusTranformationModHome_gr_766.gif]

[Graphics:../Images/MobiusTranformationModHome_gr_767.gif]

Substitute  [Graphics:../Images/MobiusTranformationModHome_gr_768.gif]  into the formula for  [Graphics:../Images/MobiusTranformationModHome_gr_769.gif]  and get:

[Graphics:../Images/MobiusTranformationModHome_gr_770.gif]

[Graphics:../Images/MobiusTranformationModHome_gr_771.gif]

We are really done.   

Aside.   The solution could also be expressed  [Graphics:../Images/MobiusTranformationModHome_gr_772.gif],  where  

    [Graphics:../Images/MobiusTranformationModHome_gr_773.gif]  

Since this is a tedious computation, we shall check it out using Mathematica.

[Graphics:../Images/MobiusTranformationModHome_gr_774.gif]
[Graphics:../Images/MobiusTranformationModHome_gr_775.gif]

[Graphics:../Images/MobiusTranformationModHome_gr_776.gif]
[Graphics:../Images/MobiusTranformationModHome_gr_777.gif]



[Graphics:../Images/MobiusTranformationModHome_gr_778.gif]

[Graphics:../Images/MobiusTranformationModHome_gr_779.gif]
[Graphics:../Images/MobiusTranformationModHome_gr_780.gif]



[Graphics:../Images/MobiusTranformationModHome_gr_781.gif]

[Graphics:../Images/MobiusTranformationModHome_gr_782.gif]
[Graphics:../Images/MobiusTranformationModHome_gr_783.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2008 John H. Mathews, Russell W. Howell