Legendre Differential Equation

Author

Eric W. Weisstein
October 19, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/LegendreDifferentialEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LegendreDifferentialEquation.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Solution

Usual

DSolve[(1-x^2)y''[x]-2x y'[x]+l(l+1)y[x]==0,y[x],x]

Usual Angular

x=Cos[θ];

dydx=-1/Sin[θ]y'[θ]

d2ydx2=FullSimplify[-1/Sin[θ]D[dydx,θ]]

FullSimplify[(1-x^2) d2ydx2-2x dydx+(l(l+1)-(m/(1-x^2)))y[θ]]

DSolve[y''[θ]+Cot[θ]y'[θ]+l(l+1)y[θ]==0,y[θ],θ]

Associated

DSolve[(1-x^2)y''[x]-2x y'[x]+(l(l+1)-((m^2)/(1-x^2)))y[x]==0,y[x],x]

Associated Angular

DSolve[y''[θ]+Cot[θ]y'[θ]+(l(l+1)-((m^2)/((Sin[θ])^2)))y[θ]==0,y[θ],θ]

Moon and Spencer

DSolve[(1-(x^2))y''[x]-2x y'[x]-((k^2)(a^2)((x^2)-1)-p(p+1)-((q^2)/((x^2)-1)))y[x]==0,y[x],x]

Series

Even l

Product[l-2k,{k,0,(n-1)}]//FullSimplify

Product[l+2k-1,{k,1,n}]//FullSimplify

FullSimplify[((((-1))^n)/((2n)!))(((((-2))^n) Gamma[-(l/2)+n])/(Gamma[-(l/2)]))(2^n) Pochhammer[((1+l)/2),n],n∈Integers&&n>0]

1+Sum[(((4^n) Gamma[-(l/2)+n] Pochhammer[((1+l)/2),n])/((2 n)! Gamma[-(l/2)]))(x^(2n)),{n,∞}]

FunctionExpand[%]

l1=Table[Hypergeometric2F1[-(l/2),((1+l)/2),(1/2),(x^2)],{l,0,15,2}]//FunctionExpand

l2=Table[LegendreP[l,x],{l,0,15,2}]

l1/l2//Factor

Hypergeometric2F1[-(l/2),((1+l)/2),(1/2),(x^2)]/.x->1

Table[Hypergeometric2F1[-(l/2),((1+l)/2),(1/2),1],{l,-10,10}]

Table[Hypergeometric2F1[-(l/2),((1+l)/2),(1/2),1]/.l->RandomComplexDisk[],{1000}]//Union

Odd l

Product[l-2k+1,{k,n}]

Product[l+2k,{k,n}]

FullSimplify[((((-1))^n)/((2n+1)!))(((-2))^n) Pochhammer[(1/2)-(l/2),n](2^n) Pochhammer[1+(l/2),n],n∈Integers&&n>0]

x+Sum[(((4^n) Pochhammer[(1/2)-(l/2),n] Pochhammer[1+(l/2),n])/(Gamma[2+2 n]))(x^(2n+1)),{n,∞}]//FullSimplify

l1=Table[x Hypergeometric2F1[1+(l/2),(1/2)-(l/2),(3/2),(x^2)],{l,1,10,2}]

l2=Table[LegendreP[l,x],{l,1,10,2}]