Example 6.  Show that the Legendre polynomials  [Graphics:Images/LegendrePolyMod_gr_225.gif],  [Graphics:Images/LegendrePolyMod_gr_226.gif],  [Graphics:Images/LegendrePolyMod_gr_227.gif],  [Graphics:Images/LegendrePolyMod_gr_228.gif],  [Graphics:Images/LegendrePolyMod_gr_229.gif]  and  [Graphics:Images/LegendrePolyMod_gr_230.gif]  are orthogonal on [Graphics:Images/LegendrePolyMod_gr_231.gif].  

Solution 6.

[Graphics:../Images/LegendrePolyMod_gr_232.gif]

[Graphics:../Images/LegendrePolyMod_gr_233.gif]

 

 

 

For  [Graphics:../Images/LegendrePolyMod_gr_234.gif] we have.  

[Graphics:../Images/LegendrePolyMod_gr_235.gif]

[Graphics:../Images/LegendrePolyMod_gr_236.gif]

 

 

The details for showing that [Graphics:../Images/LegendrePolyMod_gr_237.gif] are orthogonal are given below.  

[Graphics:../Images/LegendrePolyMod_gr_238.gif]

[Graphics:../Images/LegendrePolyMod_gr_239.gif]

 

 

 

For  [Graphics:../Images/LegendrePolyMod_gr_240.gif] we have.  

[Graphics:../Images/LegendrePolyMod_gr_241.gif]

[Graphics:../Images/LegendrePolyMod_gr_242.gif]

[Graphics:../Images/LegendrePolyMod_gr_243.gif]

Therefore, the Legendre polynomials  [Graphics:../Images/LegendrePolyMod_gr_244.gif]  are orthogonal on [Graphics:../Images/LegendrePolyMod_gr_245.gif].   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005