11.3 Fourier-Legendre series
Since Legendre’s equation is
self-adjoint, we can show that
forms an orthogonal set of functions. To decompose
functions as series in Legendre polynomials we shall need the
integrals
|
(11.27) |
which can be determined using the relation 5.
twice to obtain a recurrence relation
and the use of a very simple integral to
fix this number for
,
|
(11.29) |
So we can now develop any function on
in a Fourier-Legendre series
Example
11.4:
-
Find the Fourier-Legendre series
for
|
(11.31) |
Solution:
-
We find
All other coefficients for even
are zero, for odd
they can be evaluated explicitly.