Example 3 (b).  The functions  [Graphics:Images/PainlevePropertyMod_gr_62.gif]   have poles at the point  [Graphics:Images/PainlevePropertyMod_gr_63.gif].  

Solution 3 (b).

[Graphics:../Images/PainlevePropertyMod_gr_83.gif]


[Graphics:../Images/PainlevePropertyMod_gr_84.gif]

[Graphics:../Images/PainlevePropertyMod_gr_85.gif]

 

 

[Graphics:../Images/PainlevePropertyMod_gr_86.gif]


[Graphics:../Images/PainlevePropertyMod_gr_87.gif]

[Graphics:../Images/PainlevePropertyMod_gr_88.gif]

 

 

[Graphics:../Images/PainlevePropertyMod_gr_89.gif]


[Graphics:../Images/PainlevePropertyMod_gr_90.gif]

[Graphics:../Images/PainlevePropertyMod_gr_91.gif]

 

 

Remark 1.  The graphs of   [Graphics:../Images/PainlevePropertyMod_gr_92.gif] look similar (in Red) because they have simple poles at  [Graphics:../Images/PainlevePropertyMod_gr_93.gif].  of order [Graphics:../Images/PainlevePropertyMod_gr_94.gif].  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005