Legendre関数

2013 Sep 25
Masahiro Mizutani

legendre[x_,n_]:=Table[LegendreP[k,x],{k,0,n}]

Plot[legendre[x,20],{x,-1,1},Frame->True]

正規化されらLegendre関数

LegendrePoly[n_,x_]=Sqrt[n+1/2]LegendreP[n,x]

正規直交性

Integrate[LegendrePoly[8,x]^2,{x,-1,1}]

Integrate[LegendrePoly[8,x]LegendrePoly[9,x],{x,-1,1}]

LegendrePoly[0,x]

LegendrePoly[1,x]

LegendrePoly[2,x]

LegendrePoly[3,x]

[-1, 1] 上の単項式系からGram–Schmidt基底を生成

Orthogonalize[{1,x,x^2,x^3,x^4},Integrate[#1#2,{x,-1,1}]&]