Legendre Polynomials
Recurrence Relation
pk 1 ( x) 
p0 ( x)  1
(2k  1)
k
xpk ( x) 
pk 1 ( x),
(k  1)
k 1
the first several Legendre polynomials
p1 ( x)  x
k  1,2,3,.....
Legendre Polynomials
the first several Legendre polynomials
Properties
Legendre’s DE
Legendre’s DE
(1  x 2 ) y' '2 xy'n(n  1) y  0
Legendre’s DE
solution
n0
(1  x 2 ) y' '2 xy'  0
p0 ( x)
n 1
(1  x 2 ) y' '2 xy'2 y  0
p1 ( x)
n2
(1  x 2 ) y' '2 xy'6 y  0
p2 ( x)
n3
(1  x 2 ) y' '2 xy'12 y  0
p3 ( x)


(1  x 2 ) y' '2 xy'n(n  1) y  0
pn (x)
Legendre Polynomials
Legendre’s DE
Legendre polynomials
Norm Square
orthogonal set
Pn ( x)n0
is orthogonal with respect to the
weight function p(x) = 1 on [ -1, l].
The orthogonality relation
expanding a function
Fourier series, Fourier cosine series, and Fourier sine series are
three ways of expanding a function in terms of an orthogonal set
of functions
Fourier-Legendre Series
Fourier-Legendre Series
The Fourier-Legendre series of a function
defined on the interval (-1, 1) is given by

f ( x)   cn Pn ( x)
n 0
cn 
2 n 1 1
2
1

f ( x) Pn ( x)dx
Example
Write out the first few nonzero terms in
the Fourier-Legendre expansion of
Fourier-Legendre Series
Fourier-Legendre Series
The Fourier-Legendre series of a function
defined on the interval (-1, 1) is given by

f ( x)   cn Pn ( x)
n 0
cn 
2 n 1 1
2
1

f ( x) Pn ( x)dx
Example
Write out the first few nonzero terms in
the Fourier-Legendre expansion of
Fourier-Legendre Series
Fourier-Legendre Series
The Fourier-Legendre series of a function
defined on the interval (-1, 1) is given by

f ( x)   cn Pn ( x)
n 0
cn 
2 n 1 1
2
1

f ( x) Pn ( x)dx
Alternative Form of Series
x  cos

f ( x)  f (cos  )   Cn Pn (cos  )
n 0

where
2n  1
Cn 
f (cos  ) Pn (cos  ) sin d

2 0