Legendre Functions Question

Posted on 2008-03-31 20:35 By dcasimir ( Maple Rating 3

296)

Categories:

Math Advice (Student)

Expand the Dirac delta function in a series of Legendre polynomials using the interval

-1 <= x<= 1.

thanks,

v/r,

Legendre series

Comment on 2008-03-31 21:23 from Robert Israel ( Maple Rating 4

1555)

Hints:

1) P(n,x) are orthogonal on the interval [-1,1] with int(P(n,x)^2, x=-1..1) = 2/(2*n+1)

(int doesn't seem to know this if you use LegendreP or orthopoly[P], but see the help page ?orthopoly[P])

2) LegendreP(n,0) = sqrt(Pi)/(GAMMA(1/2-n/2)*GAMMA(1+n/2))
(which is interpreted as 0 if n is an odd integer).

In a way it knows

Comment on 2008-03-31 21:56 from jakubi ( Maple Rating 4

943)

with(orthopoly):
with(gfun):

l:=[seq(int(P(i,x)^2, x=-1..1), i=1..10)];
    l := [2/3, 2/5, 2/7, 2/9, 2/11, 2/13, 2/15, 2/17, 2/19, 2/21]

rec:=listtorec(l,u(m));

   rec := [{u(0) = 2/3, (-2 m - 3) u(m) + (2 m + 5) u(m + 1)}, ogf]

rsolve(op(1,rec),u(m)):
subs(m=n-1,%);

                                  2
                               -------
                               2 n + 1