[This page is deprecated in favor of User:Dnessett/Legendre/Associated Legendre Functions Orthonormality for fixed m, since the theorem not only shows the orthogonality of the Associated Legendre Functions, but also provides the normalization constant. All future modifications will be made to the referenced page.]
This article provides a proof that the associated Legendre functions are orthogonal for fixed m.
[Note: This article uses the more common
notation, rather than
]
Where:
The Associated Legendre Functions are regular solutions to the
general Legendre equation:
, where
This equation is an example of a more general class of equations
known as the Sturm-Liouville equations. Using Sturm-Liouville
theory, one can show that
vanishes when
However, one can find
directly from the above definition, whether or not
Since k and ℓ occur symmetrically, one can without loss of generality assume
that
Integrate by parts
times, where the curly brackets in the integral indicate the
factors, the first being
and the second
For each of the first
integrations by parts,
in the
term contains the factor
;
so the term vanishes. For each of the remaining ℓ integrations,
in that term contains the factor
;
so the term also vanishes. This means:
Expand the second factor using Leibnitz' rule:
The leftmost derivative in the sum is non-zero only when
(remembering that
). The other derivative is non-zero only when
,
that is, when
Because
these two conditions imply that the only non-zero term in the
sum occurs when
and
So:
To evaluate the differentiated factors, expand
using the binomial theorem:
The only thing that survives differentiation
times is the
term, which (after differentiation) equals:
. Therefore:
................................................. (1)
Evaluate
by a change of variable:
![{\displaystyle x=\cos \theta \Rightarrow dx=-\sin \theta \,d\theta {\text{ and }}1-x^{2}=\sin \theta .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8d4fc7af67b2e91c82c8db9822cb128dde5a21)
Thus,
[To eliminate the negative sign on the second integral, the limits are switched
from
to
, recalling that
and
].
A table of standard trigonometric integrals shows:
Since
for
Applying this result to
and changing the variable back to
yields:
for
Using this recursively:
Applying this result to (1):
QED.
- Kenneth Franklin Riley, Michael Paul Hobson, Stephen John Bence, "Mathematical methods for physics and engineering", (2006) 3
Edition, Cambridge University Press, ISBN 0-521-67971-0.