Example 5 (c).  Find the inner product of   [Graphics:Images/LegendrePolyMod_gr_208.gif]  and  [Graphics:Images/LegendrePolyMod_gr_209.gif]  over [Graphics:Images/LegendrePolyMod_gr_210.gif].   

Solution 5 (c).

[Graphics:../Images/LegendrePolyMod_gr_219.gif]



[Graphics:../Images/LegendrePolyMod_gr_220.gif]

 

 

The details of this computation.

[Graphics:../Images/LegendrePolyMod_gr_221.gif]



[Graphics:../Images/LegendrePolyMod_gr_222.gif]

 

Remark.  The endpoints of the interval are important,  if we change  [Graphics:../Images/LegendrePolyMod_gr_223.gif]  to  [Graphics:../Images/LegendrePolyMod_gr_224.gif]  we will get a different answer which will be discussed in the next example.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005