Example 3 (a).  The functions  [Graphics:Images/PainlevePropertyMod_gr_60.gif]   have poles at the point  [Graphics:Images/PainlevePropertyMod_gr_61.gif].  

Solution 3 (a).

[Graphics:../Images/PainlevePropertyMod_gr_64.gif]


[Graphics:../Images/PainlevePropertyMod_gr_65.gif]

[Graphics:../Images/PainlevePropertyMod_gr_66.gif]

 

 

[Graphics:../Images/PainlevePropertyMod_gr_67.gif]


[Graphics:../Images/PainlevePropertyMod_gr_68.gif]

[Graphics:../Images/PainlevePropertyMod_gr_69.gif]

 

 

[Graphics:../Images/PainlevePropertyMod_gr_70.gif]


[Graphics:../Images/PainlevePropertyMod_gr_71.gif]

[Graphics:../Images/PainlevePropertyMod_gr_72.gif]

 

 

[Graphics:../Images/PainlevePropertyMod_gr_73.gif]


[Graphics:../Images/PainlevePropertyMod_gr_74.gif]

[Graphics:../Images/PainlevePropertyMod_gr_75.gif]

 

 

[Graphics:../Images/PainlevePropertyMod_gr_76.gif]


[Graphics:../Images/PainlevePropertyMod_gr_77.gif]

[Graphics:../Images/PainlevePropertyMod_gr_78.gif]

 

 

Remark 1.  The graphs of   [Graphics:../Images/PainlevePropertyMod_gr_79.gif] look similar (in Red) because they have simple poles of order [Graphics:../Images/PainlevePropertyMod_gr_80.gif].  

Remark 2.  The graphs of   [Graphics:../Images/PainlevePropertyMod_gr_81.gif] look similar (in Blue) because they have poles of order [Graphics:../Images/PainlevePropertyMod_gr_82.gif].  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005